diff --git a/egison.cabal b/egison.cabal
--- a/egison.cabal
+++ b/egison.cabal
@@ -1,5 +1,5 @@
 Name:                egison
-Version:             3.7.7
+Version:             3.7.8
 Synopsis:            Programming language with non-linear pattern-matching against non-free data
 Description:
   An interpreter for Egison, a **pattern-matching-oriented**, purely functional programming language.
@@ -60,7 +60,7 @@
 Extra-Source-Files:  benchmark/Benchmark.hs
 
 Data-files:          lib/core/*.egi lib/math/*.egi lib/math/common/*.egi lib/math/algebra/*.egi lib/math/analysis/*.egi lib/math/geometry/*.egi
-                     sample/*.egi sample/io/*.egi
+                     sample/*.egi sample/io/*.egi sample/math/algebra/*.egi sample/math/analysis/*.egi sample/math/geometry/*.egi sample/math/number/*.egi sample/math/others/*.egi
                      elisp/egison-mode.el
 
 source-repository head
diff --git a/hs-src/Language/Egison/Types.hs b/hs-src/Language/Egison/Types.hs
--- a/hs-src/Language/Egison/Types.hs
+++ b/hs-src/Language/Egison/Types.hs
@@ -1822,3 +1822,9 @@
 isHash' (Intermediate (IIntHash _)) = return $ Value $ Bool True
 isHash' (Intermediate (IStrHash _)) = return $ Value $ Bool True
 isHash' _ = return $ Value $ Bool False
+
+readUTF8File :: FilePath -> (IO String)
+readUTF8File name = do
+  h <- openFile name ReadMode
+  hSetEncoding h utf8
+  hGetContents h
diff --git a/sample/math/algebra/cubic-equation.egi b/sample/math/algebra/cubic-equation.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/algebra/cubic-equation.egi
@@ -0,0 +1,45 @@
+(define $cubic-formula c-f)
+
+(define $c-f
+  (lambda [$f $x]
+    (match (coefficients f x) (list math-expr)
+      {[<cons $a_0 <cons $a_1 <cons $a_2 <cons $a_3 <nil>>>>>
+        (c-f' a_3 a_2 a_1 a_0)]})))
+
+(define $c-f'
+  (lambda [$a $b $c $d]
+    (match [a b c d] [math-expr math-expr math-expr math-expr]
+      {[[,1 ,0 $p $q]
+        (let {[[$s1 $s2] (2#[(rt 3 %1) (rt 3 %2)] (q-f' 1 (* 27 q) (* -27 p^3)))]}
+          [(/ (+ s1 s2) 3)               ; r1
+           (/ (+ (* w^2 s1) (* w s2)) 3) ; r2
+           (/ (+ (* w s1) (* w^2 s2)) 3) ; r3
+           ])]
+       [[,1 _ _ _]
+        (3#[(- %1 (/ b 3)) (- %2 (/ b 3)) (- %3 (/ b 3))]
+           (with-symbols {x y}
+             (c-f (substitute {[x (- y (/ b 3))]} (+ x^3 (* b x^2) (* c x) d)) y)))]
+       [[_ _ _ _] (c-f' 1 (/ b a) (/ c a) (/ d a))]})))
+
+(define $w (/ (+ -1 (* i (sqrt 3))) 2))
+
+(* (- x 1) (- x 2) (- x 3))
+;=>(+ x^3 (* -6 x^2) (* 11 x) -6)
+
+(c-f (+ x^3 (* -6 x^2) (* 11 x) -6) x)
+;=>[2 1 3]
+
+(3#%1 (c-f (+ x^3 (* p x) q) x))
+;=>
+;(/ (+ (rt 3 (+ (* -108 q)
+;               (* 12 (sqrt (+ (* 81 q^2) (* 12 p^3))))))
+;      (rt 3 (+ (* -108 q)
+;               (* -12 (sqrt (+ (* 81 q^2) (* 12 p^3)))))))
+;   6)
+
+(3#%1 (c-f (+ (* a x^3) (* b x^2) (* c x) d) x))
+;=>
+;(/ (+ (* (rt 3 (/ (+ (* -108 d a^3) (* 36 c b a^2) (* -8 b^3 a) (* 12 (sqrt (+ (* 81 a^6 d^2) (* -54 a^5 d c b) (* 12 a^4 d b^3) (* -3 a^4 c^2 b^2) (* 12 a^5 c^3))))) a^4)) a)
+;      (* (rt 3 (/ (+ (* -108 d a^3) (* 36 c b a^2) (* -8 b^3 a) (* -12 (sqrt (+ (* 81 a^6 d^2) (* -54 a^5 d c b) (* 12 a^4 d b^3) (* -3 a^4 c^2 b^2) (* 12 a^5 c^3))))) a^4)) a)
+;      (* -2 b))
+;    (* 6 a))
diff --git a/sample/math/algebra/quadratic-equation.egi b/sample/math/algebra/quadratic-equation.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/algebra/quadratic-equation.egi
@@ -0,0 +1,36 @@
+(define $quadratic-formula q-f)
+
+(define $q-f
+  (lambda [$f $x]
+    (match (coefficients f x) (list math-expr)
+      {[<cons $a_0 <cons $a_1 <cons $a_2 <nil>>>>
+        (q-f' a_2 a_1 a_0)]})))
+
+(define $q-f'
+  (lambda [$a $b $c]
+    (match [a b c] [math-expr math-expr math-expr]
+      {[[,1 ,0 _]
+        [(sqrt (* -1 c)) (* -1 (sqrt (* -1 c)))]]
+       [[,1 _ _]
+        (2#[(+ (* -1 (/ b 2)) %1) (+ (* -1 (/ b 2)) %2)]
+           (with-symbols {x y}
+             (q-f (substitute {[x (- y (/ b 2))]} (+ x^2 (* b x) c)) y)))]
+       [[_ _ _] (q-f' 1 (/ b a) (/ c a))]})))
+
+
+(q-f (+ x^2 x 1) x)
+;=>
+;[(/ (+ -1 (* i (sqrt 3))) 2)
+; (/ (+ -1 (* -1 i (sqrt 3))) 2)]
+
+(q-f (+ (* a x^2) (* b x) c) x)
+;=>
+;[(/ (+ (* -1 b) (sqrt (+ b^2 (* -4 c a))))
+;    (* 2 a))
+; (/ (+ (* -1 b) (* -1 (sqrt (+ b^2 (* -4 c a)))))
+;    (* 2 a))]
+
+(q-f (+ (* a x^2) (* 2 b x) c) x)
+;=>
+;[(/ (+ (* -1 b) (sqrt (+ b^2 (* -1 c a)))) a)
+; (/ (+ (* -1 b) (* -1 (sqrt (+ b^2 (* -1 c a))))) a)]
diff --git a/sample/math/algebra/quartic-equation.egi b/sample/math/algebra/quartic-equation.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/algebra/quartic-equation.egi
@@ -0,0 +1,34 @@
+(define $quartic-formula qt-f)
+
+(define $qt-f
+  (lambda [$f $x]
+    (match (coefficients f x) (list math-expr)
+      {[<cons $a_0 <cons $a_1 <cons $a_2 <cons $a_3 <cons $a_4 <nil>>>>>>
+        (qt-f' a_4 a_3 a_2 a_1 a_0)]})))
+
+(define $qt-f'
+  (lambda [$a $b $c $d $e]
+    (match [a b c d e] [math-expr math-expr math-expr math-expr math-expr]
+      {[[,1 ,0 $p ,0 $q]
+        (let* {[[$s1 $s2] (q-f' 1 p q)]
+               [[$r1 $r2] (q-f' 1 0 (* -1 s1))]
+               [[$r3 $r4] (q-f' 1 0 (* -1 s2))]}
+          [r1 r2 r3 r4])]
+       [[,1 ,0 $p $q $r]
+        (let* {[$u (3#%1 (with-symbols {u} (c-f (+ (* u (+ p u)^2) (* -4 r u) (* -1 q^2)) u)))]
+               [[$r1 $r2] (q-f (+ y^2 (/ (+ p u) 2) (* (sqrt u) (- y (/ q (* 2 u))))) y)]
+               [[$r3 $r4] (q-f (+ y^2 (/ (+ p u) 2) (* -1 (sqrt u) (- y (/ q (* 2 u))))) y)]}
+          [r1 r2 r3 r4])]
+       [[,1 _ _ _ _]
+        (4#[(- %1 (/ b 4)) (- %2 (/ b 4)) (- %3 (/ b 4)) (- %4 (/ b 4))]
+           (with-symbols {x y}
+             (qt-f (substitute {[x (- y (/ b 4))]} (+ x^4 (* b x^3) (* c x^2) (* d x) e)) y)))]
+       [[_ _ _ _ _] (qt-f' 1 (/ b a) (/ c a) (/ d a) (/ e a))]})))
+
+(define $w (/ (+ -1 (* i (sqrt 3))) 2))
+
+(* (- x 1) (- x 2) (- x 3) (- x 4))
+;=>(+ x^4 (* -10 x^3) (* 35 x^2) (* -50 x) 24)
+
+(qt-f (+ x^4 (* -10 x^3) (* 35 x^2) (* -50 x) 24) x)
+;=>[4 1 3 2]
diff --git a/sample/math/algebra/quartic-formula.egi b/sample/math/algebra/quartic-formula.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/algebra/quartic-formula.egi
@@ -0,0 +1,28 @@
+(define $quartic-formula qt-f)
+
+(define $qt-f
+  (lambda [$f $x]
+    (match (coefficients f x) (list math-expr)
+      {[<cons $a_0 <cons $a_1 <cons $a_2 <cons $a_3 <cons $a_4 <nil>>>>>>
+        (qt-f' a_4 a_3 a_2 a_1 a_0)]})))
+
+(define $qt-f'
+  (lambda [$a $b $c $d $e]
+    (match [a b c d e] [math-expr math-expr math-expr math-expr math-expr]
+      {[[,1 ,0 $p ,0 $q]
+        (let* {[[$s1 $s2] (q-f' 1 p q)]
+               [[$r1 $r2] (q-f' 1 0 (* -1 s1))]
+               [[$r3 $r4] (q-f' 1 0 (* -1 s2))]}
+          [r1 r2 r3 r4])]
+       [[,1 ,0 $p $q $r]
+        (let* {[$u '(3#%1 (with-symbols {u} (c-f (+ (* u (+ p u)^2) (* -4 r u) (* -1 q^2)) u)))]
+               [[$r1 $r2] (q-f (+ y^2 (/ (+ p u) 2) (* (sqrt u) (- y (/ q (* 2 u))))) y)]
+               [[$r3 $r4] (q-f (+ y^2 (/ (+ p u) 2) (* -1 (sqrt u) (- y (/ q (* 2 u))))) y)]}
+          [r1 r2 r3 r4])]
+       [[,1 _ _ _ _]
+        (4#[(- %1 (/ b 4)) (- %2 (/ b 4)) (- %3 (/ b 4)) (- %4 (/ b 4))]
+           (with-symbols {x y}
+             (qt-f (substitute {[x (- y (/ b 4))]} (+ x^4 (* b x^3) (* c x^2) (* d x) e)) y)))]
+       [[_ _ _ _ _] (qt-f' 1 (/ b a) (/ c a) (/ d a) (/ e a))]})))
+
+;(define $w (/ (+ -1 (* i (sqrt 3))) 2))
diff --git a/sample/math/analysis/complex-analysis.egi b/sample/math/analysis/complex-analysis.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/analysis/complex-analysis.egi
@@ -0,0 +1,47 @@
+;;;;;
+;;;;; Complex Integration
+;;;;;
+
+(define $C1 (dSd x 0 a x))
+(define $C2 (dSd y 0 b (* i (- a (* i y)))))
+
+(define $C2' (dSd y 0 b (* i (* -1 (* i y)))))
+(define $C1' (dSd x 0 a (- x (* i b))))
+
+C1 ;=>(/ a^2 2)
+C2 ;=>(/ (+ (* 2 i a b) b^2) 2)
+
+C2';=>(/ b^2 2)
+C1';=>(/ (+ a^2 (* -2 i b a)) 2)
+
+(+ C1 C2);=>(/ (+ a^2 (* 2 i a b) b^2) 2)
+(+ C2' C1');=>(/ (+ b^2 a^2 (* -2 i b a)) 2)
+
+(- (+ C1 C2)
+   (+ C2' C1'))
+;=>(* 2 i a b)
+
+
+(define $D1 (dSd x 0 a x))
+(define $D2 (dSd y 0 b (* i (+ a (* i y)))))
+
+(define $D2' (dSd y 0 b (* i (* i y))))
+(define $D1' (dSd x 0 a (+ x (* i b))))
+
+D1 ;=>(/ a^2 2)
+D2 ;=>(/ (+ (* 2 i a b) (* -1 b^2)) 2)
+
+D2';=>(/ (* -1 b^2) 2)
+D1';=>(/ (+ a^2 (* 2 i b a)) 2)
+
+(- (+ D1 D2)
+   (+ D2' D1'))
+;=>0
+
+(define $E (dSd t 0 (* 2 pi) (* r (** e (* -1 i t)) i r (** e (* i t)))))
+
+E;=>(* 2 i r^2 pi)
+
+(define $F (dSd t 0 (* 2 pi) (exp (* i t))))
+
+F;=>0
diff --git a/sample/math/analysis/eulers-formula.egi b/sample/math/analysis/eulers-formula.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/analysis/eulers-formula.egi
@@ -0,0 +1,11 @@
+(take 8 (taylor-expansion (** e (* i x)) x 0))
+;{1 (* i x) (/ (* -1 x^2) 2) (/ (* -1 i x^3) 6) (/ x^4 24) (/ (* i x^5) 120) (/ (* -1 x^6) 720) (/ (* -1 i x^7) 5040)}
+
+(take 8 (taylor-expansion (cos x) x 0))
+;{1 0 (/ (* -1 x^2) 2) 0 (/ x^4 24) 0 (/ (* -1 x^6) 720) 0}
+
+(take 8 (taylor-expansion (* i (sin x)) x 0))
+;{0 (* i x) 0 (/ (* -1 i x^3) 6) 0 (/ (* i x^5) 120) 0 (/ (* -1 i x^7) 5040)}
+
+(take 8 (map2 + (taylor-expansion (cos x) x 0) (taylor-expansion (* i (sin x)) x 0)))
+;{1 (* i x) (/ (* -1 x^2) 2) (/ (* -1 i x^3) 6) (/ x^4 24) (/ (* i x^5) 120) (/ (* -1 x^6) 720) (/ (* -1 i x^7) 5040)}
diff --git a/sample/math/analysis/laplacian-hessian-jacobian.egi b/sample/math/analysis/laplacian-hessian-jacobian.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/analysis/laplacian-hessian-jacobian.egi
@@ -0,0 +1,33 @@
+(define $parameters [| x y z |]) 
+
+(define $∂ (∂/∂ $ parameters))
+
+(∂_i [| x^2 y^2 z^2 |]_i)
+;[| (* 2 x) (* 2 y) (* 2 z) |]_i
+
+(∂_i [| x^2 y^2 z^2 |]_j)
+;[| [| (* 2 x) 0 0 |] [| 0 (* 2 y) 0 |] [| 0 0 (* 2 z) |] |]_i_j
+
+(define $Δ
+  (lambda [%f]
+    (with-symbols {i}
+      (contract + (∂~i (∂_i f))))))
+
+(define $Hessian
+  (lambda [%f]
+    (with-symbols {i j}
+      (∂_i (∂_j f)))))
+
+(define $Jacobian
+  (lambda [%v]
+    (with-symbols {i j}
+      (M.det (∂_i v_j)))))
+
+(Δ (+ x^2 y^2 z^2))
+;6
+
+(Hessian (+ x^2 y^2 z^2))
+;[| [| 2 0 0 |] [| 0 2 0 |] [| 0 0 2 |] |]
+
+(Jacobian [| x^2 y^2 z^2 |])
+;(* 8 x y z)
diff --git a/sample/math/analysis/leibniz-formula.egi b/sample/math/analysis/leibniz-formula.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/analysis/leibniz-formula.egi
@@ -0,0 +1,41 @@
+(define $f (lambda [$x] x))
+
+(define $multSd
+  (lambda [$x $f $G]
+    (let {[$F (Sd x f)]}
+      (- (* F G)
+         (Sd x (* f (d/d G x)))))))
+
+(multSd x (cos x) (f x));(+ (* (sin x) x) (* -1 (sin x)))
+(multSd x (cos (* 2 x)) (f x));(/ (+ (* 2 (sin (* 2 x)) x) (* -2 (sin (* 2 x)))) 4)
+(multSd x (cos (* n x)) (f x));(/ (+ (* (sin (* n x)) x n) (* -1 (sin (* n x)) n)) n^2)
+
+(multSd x (sin x) (f x));(+ (* -1 (cos x) x) (cos x))
+(multSd x (sin (* 2 x)) (f x));(/ (+ (* -1 (cos (* 2 x)) x) (cos (* 2 x))) 2)
+(multSd x (sin (* n x)) (f x));(/ (+ (* -1 (cos (* n x)) x) (cos (* n x))) n)
+
+
+(define $as (map (lambda [$n] (let {[$F (multSd x (cos (* n x)) (f x))]}
+                                (/ (- (substitute {[x π]} F) (substitute {[x (* -1 π)]} F))
+                                   π)))
+                 nats))
+(take 10 as)
+;{0 0 0 0 0 0 0 0 0 0}
+
+(define $bs (map (lambda [$n] (let {[$F (multSd x (sin (* n x)) (f x))]}
+                                (/ (- (substitute {[x π]} F) (substitute {[x (* -1 π)]} F))
+                                   π)))
+                 (take 10 nats)))
+(take 10 bs)
+;{2 -1 (/ 2 3) (/ -1 2) (/ 2 5) (/ -1 3) (/ 2 7) (/ -1 4) (/ 2 9) (/ -1 5)}
+
+(define $f' (map (lambda [$k $b] (* b (sin (* k x)))) (zip nats bs)))
+
+(take 10 f')
+;{(* 2 (sin x)) (* -1 (sin (* 2 x))) (/ (* 2 (sin (* 3 x))) 3) (/ (* -1 (sin (* 4 x))) 2) (/ (* 2 (sin (* 5 x))) 5) (/ (* -1 (sin (* 6 x))) 3) (/ (* 2 (sin (* 7 x))) 7) (/ (* -1 (sin (* 8 x))) 4) (/ (* 2 (sin (* 9 x))) 9) (/ (* -1 (sin (* 10 x))) 5)}
+
+(take 10 (map (substitute {[x (/ π 2)]} $) f'))
+;{2 0 (/ -2 3) 0 (/ 2 5) 0 (/ -2 7) 0 (/ 2 9) 0} ; = (/ pi 2)
+
+(map (/ $ 2) (take 10 (map (substitute {[x (/ π 2)]} $) f')))
+;{1 0 (/ -1 3) 0 (/ 1 5) 0 (/ -1 7) 0 (/ 1 9) 0} ; = (/ pi 4)
diff --git a/sample/math/analysis/order-of-partial-derivative.egi b/sample/math/analysis/order-of-partial-derivative.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/analysis/order-of-partial-derivative.egi
@@ -0,0 +1,13 @@
+(define $f
+  (lambda [$x $y $z]
+    (/ (* x^5 y^3) z)))
+
+(f x y z);(/ (* x^5 y^3) z)
+
+(∂/∂x (f x y z));(/ (* 5 x^4 y^3) z)
+(∂/∂y (∂/∂x (f x y z)));(/ (* 15 x^4 y^2) z)
+(∂/∂z (∂/∂y (∂/∂x (f x y z))));(/ (* 15 x^4 y^2) z)
+
+(∂/∂x (∂/∂y (∂/∂z (f x y z))));(/ (* 15 x^4 y^2) z)
+(∂/∂y (∂/∂z (∂/∂x (f x y z))));(/ (* 15 x^4 y^2) z)
+(∂/∂z (∂/∂y (∂/∂x (f x y z))));(/ (* 15 x^4 y^2) z)
diff --git a/sample/math/analysis/vector-analysis.egi b/sample/math/analysis/vector-analysis.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/analysis/vector-analysis.egi
@@ -0,0 +1,112 @@
+;;
+;; Tensor Arithmetics
+;;
+(+ 1 [| 1 2 3 |])
+;=>[|2 3 4|]
+
+(+ [| 1 2 3 |] 1)
+;=>[|2 3 4|]
+
+(+ [| 1 2 3 |]_i [| 1 2 3 |]_i)
+;=>[|2 4 6|]_i
+
+(+ [| 10 20 30 |] [| 1 2 3 |])
+;=>[| [| 11 12 13 |] [| 21 22 23 |] [| 31 32 33 |] |]
+
+(+ [| 100 200 300 |]_i
+   [|[| 1 2 3 |]
+     [| 10 20 30 |]|]_j_i)
+;=>[| [| 101 110 |] [| 202 220 |] [| 303 330 |] |]_i_j
+
+(+ [|[| 11 12 |]
+     [| 21 22 |]
+     [| 31 32 |]|]_i_j
+   [| 100 200 300 |]_i)
+;=>[| [| 111 112 |] [| 221 222 |] [| 331 332 |] |]_i_j
+
+(+ [| 100 200 300 |]_i
+   [|[| 11 12 |]
+     [| 21 22 |]
+     [| 31 32 |]|]_i_j)
+;=>[| [| 111 112 |] [| 221 222 |] [| 331 332 |] |]_i_j
+
+;;
+;; Derivative
+;;
+(∂/∂ (f x y z) x)
+;=>(f_1 x y z)
+
+(∂/∂ [| (f x) (g x) |] x)
+;=>[| (f_1 x) (g_1 x) |]
+
+(∂/∂ (f x y z) [| x y z |])
+;=>[| (f_1 x y z) (f_2 x y z) (f_3 x y z) |]
+
+([| (∂/∂ $ x) (∂/∂ $ y) |] (f x y))
+;=>[| (f_1 x y) (f_2 x y) |]
+
+([| (∂/∂ $ x) (∂/∂ $ y) |] [| (f x y) (g x y) |])
+;=>[| [| (f_1 x y) (g_1 x y) |] [| (f_2 x y) (g_2 x y) |] |]
+
+;;
+;; Nabla
+;;
+(define $∇ ∂/∂)
+
+(∇ (f x y) [| x y |])
+;=>[| (f_1 x y) (f_2 x y) |]
+
+(∇ [| (f x y) (g x y) |] [| x y |])
+;=>[| [| (f_1 x y) (f_2 x y) |] [| (g_1 x y) (g_2 x y) |] |]
+
+;;
+;; Contraction
+;;
+(contract + (* [|1 2 3|]~i [|10 20 30|]_i))
+;=>
+140
+
+(define $trace (lambda [%t] (with-symbols {i} (contract + t~i_i))))
+
+(trace [|[|10 20 30|] [|20 40 60|] [|30 60 90|]|])
+;=>
+140
+
+;;
+;; Divergence
+;;
+(define $div (compose ∇ (trace $)))
+
+(div [| (f x y z) (g x y z) (h x y z) |] [| x y z |])
+;=>(+ (f_1 x y z) (g_2 x y z) (h_3 x y z))
+
+;;
+;; Taylor Expansion
+;;
+(define $taylor-expansion
+  (lambda [%f %xs %as]
+    (with-symbols {h}
+      (let {[$hs (generate-tensor 1#h_%1 (tensor-size xs))]}
+        (map2 *
+              (map 1#(/ 1 (fact %1)) nats0)
+              (map (compose (V.substitute xs as $)
+                            (V.substitute hs (with-symbols {i} (- xs_i as_i)) $))
+                   (iterate (compose (∇ $ xs) (V.* hs $)) f)))))))
+
+(take 3 (taylor-expansion (f x) x 0))
+;=>
+;{(f 0)
+; (* x (f_1 0))
+; (/ (* x^2 (f_1_1 0))
+;    2)}
+
+(take 3 (taylor-expansion (f x y) [| x y |] [| 0 0 |]))
+;=>
+;{(f 0 0)
+; (+ (* x (f_1 0 0))
+;    (* y (f_2 0 0)))
+; (/ (+ (* x^2 (f_1_1 0 0))
+;       (* x y (f_2_1 0 0))
+;       (* y x (f_1_2 0 0))
+;       (* y^2 (f_2_2 0 0)))
+;    2)}
diff --git a/sample/math/geometry/curvature-form.egi b/sample/math/geometry/curvature-form.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/curvature-form.egi
@@ -0,0 +1,54 @@
+;;; Parameters and Metric tensor
+
+(define $x [| θ φ |])
+
+(define $g__ [| [| r^2 0 |] [| 0 (* r^2 (sin θ)^2) |] |])
+(define $g~~ [| [| (/ 1 r^2) 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) |] |])
+
+;;; Christoffel symbols
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+;;; Riemann curvature tensor
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_1_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
+R~#_#_1_2;[| [| 0 (sin θ)^2 |] [| -1 0 |] |]~#_#
+R~#_#_2_1;[| [| 0 (* -1 (sin θ)^2) |] [| 1 0 |] |]~#_#
+R~#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]~#_#
+
+;;; Connection form
+
+(define $ω Γ~#_#_#)
+
+;;; Curvature form
+
+(define $d
+  (lambda [%A]
+    !((flip ∂/∂) x A)))
+
+(define $wedge
+  (lambda [%X %Y]
+    !(. X Y)))
+
+(define $Ω
+  (with-symbols {i j}
+    (df-normalize (+ (d ω~i_j)
+                     (wedge ω~i_k ω~k_j)))))
+
+Ω~#_#_1_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
+Ω~#_#_1_2;[| [| 0 (/ (sin θ)^2 2) |] [| (/ -1 2) 0 |] |]~#_#
+Ω~#_#_2_1;[| [| 0 (/ (* -1 (sin θ)^2) 2) |] [| (/ 1 2) 0 |] |]~#_#
+Ω~#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]~#_#
+
+
diff --git a/sample/math/geometry/curvature.egi b/sample/math/geometry/curvature.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/curvature.egi
@@ -0,0 +1,45 @@
+(define $d/dt (d/d $ t))
+
+(define $ds/dt (sqrt (+ (d/dt (x t))^2 (d/dt (y t))^2)))
+
+ds/dt;(sqrt (+ (x' t)^2 (y' t)^2))
+
+(define $dt/ds (/ 1 ds/dt))
+
+dt/ds;(/ 1 (sqrt (+ (x' t)^2 (y' t)^2)))
+
+(define $e1 [(* (d/dt (x t)) dt/ds)
+             (* (d/dt (y t)) dt/ds)])
+
+e1
+;[(/ (x' t)
+;    (sqrt (+ (x' t)^2 (y' t)^2)))
+; (/ (y' t)
+;    (sqrt (+ (x' t)^2 (y' t)^2)))]
+
+(define $e2 [(* -1 (d/dt (y t)) dt/ds)
+             (* (d/dt (x t)) dt/ds)])
+
+e2
+;[(/ (* -1 (y' t))
+;    (sqrt (+ (x' t)^2 (y' t)^2)))
+; (/ (x' t)
+;    (sqrt (+ (x' t)^2 (y' t)^2)))]
+
+(define $de1/ds [(* (d/dt (fst e1)) dt/ds)
+                 (* (d/dt (snd e1)) dt/ds)])
+
+de1/ds
+;[(/ (+ (* (y' t)^2 (x'' t))
+;       (* -1 (y' t) (y'' t) (x' t)))
+;    (+ (x' t)^4 (* 2 (y' t)^2 (x' t)^2) (y' t)^4))
+; (/ (+ (* (x' t)^2 (y'' t))
+;       (* -1 (x' t) (x'' t) (y' t)))
+;    (+ (x' t)^4 (* 2 (y' t)^2 (x' t)^2) (y' t)^4))]
+
+(define $K (/ (fst de1/ds) (fst e2)))
+
+K
+;(/ (+ (* (y' t) (x'' t) (sqrt (+ (x' t)^2 (y' t)^2)))
+;      (* -1 (y'' t) (x' t) (sqrt (+ (x' t)^2 (y' t)^2))))
+;   (+ (* -1 (x' t)^4) (* -2 (y' t)^2 (x' t)^2) (* -1 (y' t)^4)))
diff --git a/sample/math/geometry/euler-form-of-S2.egi b/sample/math/geometry/euler-form-of-S2.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/euler-form-of-S2.egi
@@ -0,0 +1,74 @@
+;;; Parameters
+
+(define $x [| θ φ |])
+
+(define $X [|(* r (sin θ) (cos φ)) ; = x
+             (* r (sin θ) (sin φ)) ; = y
+             (* r (cos θ))         ; = z
+             |])
+
+;;; Local basis
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[|(* r (cos θ) (cos φ)) (* r (cos θ) (sin φ)) (* -1 r (sin θ)) |]
+;  [|(* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ)) 0 |]
+;  |]_#~#
+
+;;; Metric tensor
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {2 2}))
+(define $g~~ (M.inverse g_#_#))
+
+g_#_#;[| [| r^2 0 |] [| 0 (* r^2 (sin θ)^2) |] |]_#_#
+g~#~#;[| [| (/ 1 r^2) 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) |] |]~#~#
+
+;;; Christoffel symbols
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+;;; Connection form
+
+(define $d
+  (lambda [%A]
+    !((flip ∂/∂) x A)))
+
+(define $ω0 Γ~#_#_#)
+ω0~#_#_1;[| [| 0 0 |] [| 0 (/ (cos θ) (sin θ)) |] |]~#_#
+ω0~#_#_2;[| [| 0 (* -1 (sin θ) (cos θ)) |] [| (/ (cos θ) (sin θ)) 0 |] |]~#_#
+
+(define $A [|[| (/ 1 r) 0 |] [| 0 (/ 1 (* r (sin θ))) |]|])
+
+(define $ω (+ (. (M.inverse A)~i_j ω0~j_k A~k_l) (. (M.inverse A)~i_j (d A~j_l))))
+ω~#_#_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
+ω~#_#_2;[| [| 0 (* -1 (cos θ)) |] [| (cos θ) 0 |] |]~#_#
+
+;;; Curvature form
+
+(define $wedge
+  (lambda [%X %Y]
+    !(. X Y)))
+
+(define $Ω
+  (with-symbols {i j}
+    (df-normalize (+ (d ω~i_j)
+                     (wedge ω~i_k ω~k_j)))))
+Ω~#_#_1_2;[| [| 0 (sin θ) |] [| (* -1 (sin θ)) 0 |] |]~#_#
+Ω~#_#_2_1;[| [| 0 (* -1 (sin θ)) |] [| (sin θ) 0 |] |]~#_#
+Ω~1_2;[| [| 0 (sin θ) |] [| (* -1 (sin θ)) 0 |] |]
+Ω~2_1;[| [| 0 (* -1 (sin θ)) |] [| (sin θ) 0 |] |]
+
+;;; Euler form
+
+(define $euler-form (* (/ 1 (* 2 π)) (- Ω~1_2 Ω~2_1)))
+
+euler-form;[| [| 0 (/ (sin θ) (* 2 π)) |] [| (/ (* -1 (sin θ)) (* 2 π)) 0 |] |]
+
+; χ(S^2) = ∫ dθ dφ (/ (sin θ) (* 2 π)) = ∫ dθ (sin θ)
+; = [ (* -1 (cos θ)) ] 0-π = (cos 0) - (cos π) = 2
diff --git a/sample/math/geometry/euler-form-of-T2.egi b/sample/math/geometry/euler-form-of-T2.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/euler-form-of-T2.egi
@@ -0,0 +1,74 @@
+;;; Parameters
+
+(define $x [| θ φ |])
+
+(define $X [|(* '(+ (* a (cos θ)) b) (cos φ)) ; = x
+             (* '(+ (* a (cos θ)) b) (sin φ)) ; = y
+             (* a (sin θ))                    ; = z
+             |])
+
+;;; Local basis
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[| (* -1 a (sin θ) (cos φ)) (* -1 a (sin θ) (sin φ)) (* a (cos θ)) |]
+;  [| (* -1 '(+ (* a (cos θ)) b) (sin φ)) (* '(+ (* a (cos θ)) b) (cos φ)) 0 |]
+;  |]~#~#
+
+;;; Metric tensor
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {2 2}))
+(define $g~~ (M.inverse g_#_#))
+
+g_#_#;[| [| a^2 0 |] [| 0 '(+ (* a (cos θ)) b)^2 |] |]_#_#
+g~#~#;[| [| (/ 1 a^2) 0 |] [| 0 (/ 1 '(+ (* a (cos θ)) b)^2) |] |]~#~#
+
+;;; Christoffel symbols
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+;;; Connection form
+
+(define $d
+  (lambda [%A]
+    !((flip ∂/∂) x A)))
+
+(define $ω0 Γ~#_#_#)
+ω0~#_#_1;[| [| 0 0 |] [| 0 (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) |] |]~#_#
+ω0~#_#_2;[| [| 0 (/ (* '(+ (* a (cos θ)) b) (sin θ)) a) |] [| (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) 0 |] |]~#_#
+
+(define $A [|[| (/ 1 a) 0 |] [| 0 (/ 1 '(+ (* a (cos θ)) b)) |]|])
+
+(define $ω (+ (. (M.inverse A)~i_j ω0~j_k A~k_l) (. (M.inverse A)~i_j (d A~j_l))))
+ω~#_#_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
+ω~#_#_2;[| [| 0 (sin θ) |] [| (* -1 (sin θ)) 0 |] |]~#_#
+
+;;; Curvature form
+
+(define $wedge
+  (lambda [%X %Y]
+    !(. X Y)))
+
+(define $Ω
+  (with-symbols {i j}
+    (df-normalize (+ (d ω~i_j)
+                     (wedge ω~i_k ω~k_j)))))
+Ω~#_#_1_2;[| [| 0 (cos θ) |] [| (* -1 (cos θ)) 0 |] |]~#_#
+Ω~#_#_2_1;[| [| 0 (* -1 (cos θ)) |] [| (cos θ) 0 |] |]~#_#
+Ω~1_2;[| [| 0 (cos θ) |] [| (* -1 (cos θ)) 0 |] |]
+Ω~2_1;[| [| 0 (* -1 (cos θ)) |] [| (cos θ) 0 |] |]
+
+;;; Euler form
+
+(define $euler-form (* (/ 1 (* 2 π)) (- Ω~1_2 Ω~2_1)))
+
+euler-form;[| [| 0 (/ (cos θ) (* 2 π)) |] [| (/ (* -1 (cos θ)) (* 2 π)) 0 |] |]
+
+; χ(T^2) = ∫ dθ dφ (/ (cos θ) (* 2 π)) = ∫ dθ (cos θ)
+; = [ (sin θ) ] 0-π = (sin π) - (sin 0) = 0
diff --git a/sample/math/geometry/exterior-derivative.egi b/sample/math/geometry/exterior-derivative.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/exterior-derivative.egi
@@ -0,0 +1,13 @@
+(define $N 3)
+(define $params [| x y z |])
+(define $g [| [| 1 0 0 |] [| 0 1 0 |] [| 0 0 1 |] |])
+
+(define $d
+  (lambda [%X]
+    !((flip ∂/∂) params X)))
+
+(d (f x y z))
+;[| (f|1 x y z) (f|2 x y z) (f|3 x y z) |]
+
+(df-normalize (d (d (f x y z))))
+;[| [| 0 0 0 |] [| 0 0 0 |] [| 0 0 0 |] |]
diff --git a/sample/math/geometry/hodge-E3.egi b/sample/math/geometry/hodge-E3.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/hodge-E3.egi
@@ -0,0 +1,22 @@
+(define $N 3)
+(define $params [| x y z |])
+(define $g [| [| 1 0 0 |] [| 0 1 0 |] [| 0 0 1 |] |])
+
+(define $hodge
+  (lambda [%A]
+    (let {[$k (df-order A)]}
+      (with-symbols {i j}
+        (* (sqrt (abs (M.det g_#_#)))
+           (foldl . (. (subrefs A (map 1#j_%1 (between 1 k)))
+                       (subrefs (ε' N k) (map 1#i_%1 (between 1 N))))
+                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))
+
+(define $dx [| 1 0 0 |])
+(define $dy [| 0 1 0 |])
+(define $dz [| 0 0 1 |])
+
+(hodge dx)
+;[| [| 0 0 0 |] [| 0 0 1 |] [| 0 0 0 |] |] = (wedge dy dz)
+
+(hodge (wedge dx dy))
+;[| 0 0 1 |] = dz
diff --git a/sample/math/geometry/hodge-Minkowski.egi b/sample/math/geometry/hodge-Minkowski.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/hodge-Minkowski.egi
@@ -0,0 +1,25 @@
+(define $N 4)
+(define $params [| t x y z |])
+(define $g [| [| -1 0 0 0 |] [| 0 1 0 0 |] [| 0 0 1 0 |] [| 0 0 0 1 |] |])
+
+(define $hodge
+  (lambda [%A]
+    (let {[$k (df-order A)]}
+      (with-symbols {i j}
+        (* (sqrt (abs (M.det g_#_#)))
+           (foldl . (. (subrefs A (map 1#j_%1 (between 1 k)))
+                       (subrefs (ε' N k) (map 1#i_%1 (between 1 N))))
+                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))
+
+(define $dt [| 1 0 0 0 |])
+(define $dx [| 0 1 0 0 |])
+(define $dy [| 0 0 1 0 |])
+(define $dz [| 0 0 0 1 |])
+
+(hodge (wedge dt dx))
+;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 -1 |] [| 0 0 0 0 |] |]
+;= (* -1 (wedge dy dz))
+
+(hodge (wedge dy dz))
+;[| [| 0 1 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]
+;= (wedge dt dx)
diff --git a/sample/math/geometry/hodge-laplacian-polar.egi b/sample/math/geometry/hodge-laplacian-polar.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/hodge-laplacian-polar.egi
@@ -0,0 +1,39 @@
+;;; Parameters and metrics
+
+(define $N 2)
+
+(define $x [|r θ|])
+
+(define $g__ [| [| 1 0 |] [| 0 r^2 |] |])
+(define $g~~ (M.inverse g_#_#))
+
+;;; Hodge Laplacian
+
+(define $d
+  (lambda [%X]
+    !((flip ∂/∂) x X)))
+
+(define $hodge
+  (lambda [%A]
+    (let {[$k (df-order A)]}
+      (with-symbols {i j}
+        (* (sqrt (abs (M.det g_#_#)))
+           (foldl . (. (subrefs A (map 1#j_%1 (between 1 k)))
+                       (subrefs (ε' N k) (map 1#i_%1 (between 1 N))))
+                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))
+
+(define $δ
+  (lambda [%A]
+    (let {[$r (df-order A)]}
+      (* (** -1 (+ (* N r) 1))
+         (hodge (d (hodge A)))))))
+
+(define $Δ
+  (lambda [%A]
+    (match (df-order A) integer
+      {[,0 (δ (d A))]
+       [,2 (d (δ A))]
+       [_ (+ (d (δ A)) (δ (d A)))]})))
+
+(Δ (f r θ))
+;(/ (+ (* -1 (f|2|2 r θ)) (* -1 r (f|1 r θ)) (* -1 r^2 (f|1|1 r θ))) r^2)
diff --git a/sample/math/geometry/hodge-laplacian-spherical.egi b/sample/math/geometry/hodge-laplacian-spherical.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/hodge-laplacian-spherical.egi
@@ -0,0 +1,46 @@
+;;; Parameters and metrics
+
+(define $N 3)
+
+(define $x [|r θ φ|])
+
+(define $g__ [| [| 1 0 0 |] [| 0 r^2 0 |] [| 0 0 (* r^2 (sin θ)^2) |] |])
+(define $g~~ (M.inverse g_#_#))
+
+;;; Hodge Laplacian
+
+(define $d
+  (lambda [%X]
+    !((flip ∂/∂) x X)))
+
+(define $hodge
+  (lambda [%A]
+    (let {[$k (df-order A)]}
+      (with-symbols {i j}
+        (* (sqrt (abs (M.det g_#_#)))
+           (foldl . (. (subrefs A (map 1#j_%1 (between 1 k)))
+                       (subrefs (ε' N k) (map 1#i_%1 (between 1 N))))
+                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))
+
+(define $δ
+  (lambda [%A]
+    (let {[$r (df-order A)]}
+      (* (** -1 (+ (* N r) 1))
+         (hodge (d (hodge A)))))))
+
+(define $Δ
+  (lambda [%A]
+    (match (df-order A) integer
+      {[,0 (δ (d A))]
+       [,N (d (δ A))]
+       [_ (+ (d (δ A)) (δ (d A)))]})))
+
+(Δ (f r θ φ))
+;(/ (+ (f|3|3 r θ φ) (* (sin θ) (cos θ) (f|2 r θ φ)) (* (sin θ)^2 (f|2|2 r θ φ)) (* 2 r (sin θ)^2 (f|1 r θ φ)) (* r^2 (sin θ)^2 (f|1|1 r θ φ))) (* (sin θ)^2 r^2))
+;=
+;(/ (+ (* r^2 (sin θ)^2 (f|1|1 r θ φ))
+;      (* 2 r (sin θ)^2 (f|1 r θ φ))
+;      (* (sin θ) (cos θ) (f|2 r θ φ))
+;      (* (sin θ)^2 (f|2|2 r θ φ))
+;      (f|3|3 r θ φ))
+;   (* (sin θ)^2 r^2))
diff --git a/sample/math/geometry/hodge-laplacian.egi b/sample/math/geometry/hodge-laplacian.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/hodge-laplacian.egi
@@ -0,0 +1,43 @@
+;;; Parameters and metrics
+
+(define $N 2)
+
+(define $params [|x y|])
+
+(define $g__ [| [| (G_1_1 x y) (G_1_2 x y) |] [| (G_2_1 x y) (G_2_2 x y) |] |])
+(define $g~~ [| [| (G~1~1 x y) (G~1~2 x y) |] [| (G~2~1 x y) (G~2~2 x y) |] |])
+
+;;; Hodge Laplacian
+
+(define $d
+  (lambda [%X]
+    !((flip ∂/∂) params X)))
+
+(define $hodge
+  (lambda [%A]
+    (let {[$k (df-order A)]}
+      (with-symbols {i j}
+        (* (sqrt (abs (M.det g_#_#)))
+           (foldl . (. (subrefs A (map 1#j_%1 (between 1 k)))
+                       (subrefs (ε' N k) (map 1#i_%1 (between 1 N))))
+                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))
+
+(define $δ
+  (lambda [%A]
+    (let {[$r (df-order A)]}
+      (* (** -1 (+ (* N r) 1))
+         (hodge (d (hodge A)))))))
+
+(define $Δ
+  (lambda [%A]
+    (match (df-order A) integer
+      {[,0 (δ (d A))]
+       [,2 (d (δ A))]
+       [_ (+ (d (δ A)) (δ (d A)))]})))
+
+(d (f x y))
+(hodge (d (f x y)))
+(d (hodge (d (f x y))))
+(δ (d (f x y)))
+(Δ (f x y))
+;
diff --git a/sample/math/geometry/polar-laplacian-2d-2.egi b/sample/math/geometry/polar-laplacian-2d-2.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/polar-laplacian-2d-2.egi
@@ -0,0 +1,68 @@
+;;;
+;;; Polar coordinates
+;;;
+
+(define $x [|r θ|])
+
+(define $X [|(* r (cos θ)) ; = x
+             (* r (sin θ)) ; = y
+             |])
+
+;;
+;; Local coordinates
+;;
+
+(define $e ((∂/∂ X~# $) x_#))
+e
+;[| [| (cos θ) (sin θ) |] [| (* -1 r (sin θ)) (* r (cos θ)) |] |]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {2 2}))
+(define $g~~ (with-symbols {i j} (/ (unit-tensor {2 2})_i_j g_i_j)))
+
+g_#_#;[| [| 1 0 |] [| 0 r^2 |] |]_#_#
+g~#~#;[| [| 1 0 |] [| 0 (/ 1 r^2) |] |]~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ___
+  (with-symbols {j k l}
+    (* (/ 1 2)
+       (+ (∂/∂ g_j_l x_k)
+          (∂/∂ g_j_k x_l)
+          (* -1 (∂/∂ g_k_l x_j))))))
+
+Γ_#_#_#;(tensor {2 2 2} {0 0 0 (* -1 r) 0 r r 0} )_#_#_#
+Γ_1_#_#;[| [| 0 0 |] [| 0 (* -1 r) |] |]_#_#
+Γ_2_#_#;[| [| 0 r |] [| r 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__
+  (with-symbols {i j k l}
+    (. g~i~j Γ_j_k_l)))
+
+Γ~#_#_#;(tensor {2 2 2} {0 0 0 (* -1 r) 0 (/ 1 r) (/ 1 r) 0} )~#_#_#
+Γ~1_#_#;[| [| 0 0 |] [| 0 (* -1 r) |] |]_#_#
+Γ~2_#_#;[| [| 0 (/ 1 r) |] [| (/ 1 r) 0 |] |]_#_#
+
+;;
+;; Derive Laplacian
+;;
+
+(. g~i~j (∂/∂ (∂/∂ (f r θ) x_j) x_i))
+;(/ (+ (* (f|1|1 r θ) r^2) (f|2|2 r θ)) r^2)
+(. (. g~i~j Γ~k_i_j) (∂/∂ (f r θ) x_k))
+;(/ (* -1 (f|1 r θ)) r)
+
+(define $Laplacian (- (. g~i~j (∂/∂ (∂/∂ (f r θ) x_j) x_i))
+                        (. (. g~i~j Γ~k_i_j) (∂/∂ (f r θ) x_k))))
+Laplacian
+;(/ (+ (* (f|1|1 r θ) r^2) (f|2|2 r θ) (* (f|1 r θ) r)) r^2)
diff --git a/sample/math/geometry/polar-laplacian-2d-3.egi b/sample/math/geometry/polar-laplacian-2d-3.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/polar-laplacian-2d-3.egi
@@ -0,0 +1,38 @@
+;;;
+;;; Polar coordinates
+;;;
+
+(define $x [|r θ|])
+
+(define $X [|(* r (cos θ)) ; = x
+             (* r (sin θ)) ; = y
+             |])
+
+;;
+;; Local coordinates
+;;
+
+(define $e ((∂/∂ X~# $) x_#))
+e
+;[| [| (cos θ) (sin θ) |] [| (* -1 r (sin θ)) (* r (cos θ)) |] |]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {2 2}))
+(define $g~~ (with-symbols {i j} (/ (unit-tensor {2 2})_i_j g_i_j)))
+
+g_#_#;[| [| 1 0 |] [| 0 r^2 |] |]_#_#
+g~#~#;[| [| 1 0 |] [| 0 (/ 1 r^2) |] |]~#~#
+
+;;
+;; Derive Laplacian
+;;
+
+(define $sqrt-g (sqrt (M.det g_#_#)))
+sqrt-g;r
+
+(define $Laplacian (/ (contract + (∂/∂ (* sqrt-g (. g~i~j (∂/∂ (f r θ) x_j))) x_i)) sqrt-g))
+Laplacian
+;(/ (+ (* (f|1 r θ) r) (* r^2 (f|1|1 r θ)) (f|2|2 r θ)) r^2)
diff --git a/sample/math/geometry/polar-laplacian-2d.egi b/sample/math/geometry/polar-laplacian-2d.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/polar-laplacian-2d.egi
@@ -0,0 +1,39 @@
+(define $x (* r (cos θ)))
+(define $y (* r (sin θ)))
+
+(define $u-r (∂/∂ (u x y) r))
+u-r
+;(+ (* (u|1 (* r (cos θ)) (* r (sin θ))) (cos θ))
+;   (* (u|2 (* r (cos θ)) (* r (sin θ))) (sin θ)))
+
+(define $u-r-r (∂/∂ (∂/∂ (u x y) r) r))
+u-r-r
+;(+ (* (u|1|1 (* r (cos θ)) (* r (sin θ))) (cos θ)^2)
+;   (* (u|1|2 (* r (cos θ)) (* r (sin θ))) (sin θ) (cos θ))
+;   (* (u|2|1 (* r (cos θ)) (* r (sin θ))) (cos θ) (sin θ))
+;   (* (u|2|2 (* r (cos θ)) (* r (sin θ))) (sin θ)^2))
+
+(define $u-θ (∂/∂ (u x y) θ))
+u-θ
+;(+ (* -1 (u|1 (* r (cos θ)) (* r (sin θ))) r (sin θ))
+;   (* (u|2 (* r (cos θ)) (* r (sin θ))) r (cos θ)))
+
+(define $u-θ-θ (∂/∂ (∂/∂ (u x y) θ) θ))
+u-θ-θ
+;(+ (* (u|1|1 (* r (cos θ)) (* r (sin θ))) r^2 (sin θ)^2)
+;   (* -1 (u|1|2 (* r (cos θ)) (* r (sin θ))) r^2 (cos θ) (sin θ))
+;   (* -1 (u|1 (* r (cos θ)) (* r (sin θ))) r (cos θ))
+;   (* -1 (u|2|1 (* r (cos θ)) (* r (sin θ))) r^2 (sin θ) (cos θ))
+;   (* (u|2|2 (* r (cos θ)) (* r (sin θ))) r^2 (cos θ)^2)
+;   (* -1 (u|2 (* r (cos θ)) (* r (sin θ))) r (sin θ)))
+
+(+ u-r-r (* (/ 1 (** r 2)) u-θ-θ))
+;(/ (+ (* -1 (u|1 (* r (cos θ)) (* r (sin θ))) (cos θ))
+;      (* -1 (u|2 (* r (cos θ)) (* r (sin θ))) (sin θ))
+;      (* (u|1|1 (* r (cos θ)) (* r (sin θ))) r)
+;      (* (u|2|2 (* r (cos θ)) (* r (sin θ))) r))
+;   r)
+
+(+ u-r-r (* (/ 1 r) u-r) (* (/ 1 (** r 2)) u-θ-θ))
+;(+ (u|1|1 (* r (cos θ)) (* r (sin θ)))
+;   (u|2|2 (* r (cos θ)) (* r (sin θ))))
diff --git a/sample/math/geometry/polar-laplacian-3d-2.egi b/sample/math/geometry/polar-laplacian-3d-2.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/polar-laplacian-3d-2.egi
@@ -0,0 +1,73 @@
+;;;
+;;; Spherical coordinates
+;;;
+
+(define $x [|r θ φ|])
+
+(define $X [|(* r (sin θ) (cos φ)) ; = x
+             (* r (sin θ) (sin φ)) ; = y
+             (* r (cos θ))         ; = z
+             |])
+
+;;
+;; Local coordinates
+;;
+
+(define $e ((∂/∂ X~# $) x_#))
+e
+;[|[| (* (sin θ) (cos φ)) (* (sin θ) (sin φ)) (cos θ) |]
+;  [| (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ)) (* -1 r (sin θ)) |]
+;  [| (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ)) 0 |]|]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {3 3}))
+(define $g~~ (with-symbols {i j} (/ (unit-tensor {3 3})_i_j g_i_j)))
+
+g_#_#;[| [| 1 0 0 |] [| 0 r^2 0 |] [| 0 0 (* r^2 (sin θ)^2) |] |]_#_#
+g~#~#;[| [| 1 0 0 |] [| 0 (/ 1 r^2) 0 |] [| 0 0 (/ 1 (* r^2 (sin θ)^2)) |] |]~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ___
+  (with-symbols {j k l}
+    (* (/ 1 2)
+       (+ (∂/∂ g_j_l x_k)
+          (∂/∂ g_j_k x_l)
+          (* -1 (∂/∂ g_k_l x_j))))))
+
+Γ_#_#_#;(tensor {3 3 3} {0 0 0 0 (* -1 r) 0 0 0 (* -1 r (sin θ)^2) 0 r 0 r 0 0 0 0 (* -1 r^2 (sin θ) (cos θ)) 0 0 (* r (sin θ)^2) 0 0 (* r^2 (sin θ) (cos θ)) (* r (sin θ)^2) (* r^2 (sin θ) (cos θ)) 0} )_#_#_#
+Γ_1_#_#;[| [| 0 0 0 |] [| 0 (* -1 r) 0 |] [| 0 0 (* -1 r (sin θ)^2) |] |]_#_#
+Γ_2_#_#;[| [| 0 r 0 |] [| r 0 0 |] [| 0 0 (* -1 r^2 (sin θ) (cos θ)) |] |]_#_#
+Γ_3_#_#;[| [| 0 0 (* r (sin θ)^2) |] [| 0 0 (* r^2 (sin θ) (cos θ)) |] [| (* r (sin θ)^2) (* r^2 (sin θ) (cos θ)) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__
+  (with-symbols {i j k l}
+    (. g~i~j Γ_j_k_l)))
+
+Γ~#_#_#;(tensor {3 3 3} {0 0 0 0 (* -1 r) 0 0 0 (* -1 r (sin θ)^2) 0 (/ 1 r) 0 (/ 1 r) 0 0 0 0 (* -1 (sin θ) (cos θ)) 0 0 (/ 1 r) 0 0 (/ (cos θ) (sin θ)) (/ 1 r) (/ (cos θ) (sin θ)) 0} )~#_#_#
+Γ~1_#_#;[| [| 0 0 0 |] [| 0 (* -1 r) 0 |] [| 0 0 (* -1 r (sin θ)^2) |] |]_#_#
+Γ~2_#_#;[| [| 0 (/ 1 r) 0 |] [| (/ 1 r) 0 0 |] [| 0 0 (* -1 (sin θ) (cos θ)) |] |]_#_#
+Γ~3_#_#;[| [| 0 0 (/ 1 r) |] [| 0 0 (/ (cos θ) (sin θ)) |] [| (/ 1 r) (/ (cos θ) (sin θ)) 0 |] |]_#_#
+
+;;
+;; Laplacian
+;;
+
+(. g~i~j (∂/∂ (∂/∂ (f r θ φ) x_j) x_i))
+;(/ (+ (* (f|1|1 r θ φ) r^2 (sin θ)^2) (* (f|2|2 r θ φ) (sin θ)^2) (f|3|3 r θ φ)) (* r^2 (sin θ)^2))
+(. (. g~i~j Γ~k_i_j) (∂/∂ (f r θ φ) x_k))
+;(/ (+ (* -2 (f|1 r θ φ) r (sin θ)) (* -1 (cos θ) (f|2 r θ φ))) (* r^2 (sin θ)))
+
+(define $Laplacian (- (. g~i~j (∂/∂ (∂/∂ (f r θ φ) x_j) x_i))
+                        (. (. g~i~j Γ~k_i_j) (∂/∂ (f r θ φ) x_k))))
+Laplacian
+;(/ (+ (* (f|1|1 r θ φ) r^2 (sin θ)^2) (* (f|2|2 r θ φ) (sin θ)^2) (f|3|3 r θ φ) (* 2 (f|1 r θ φ) r (sin θ)^2) (* (cos θ) (f|2 r θ φ) (sin θ))) (* r^2 (sin θ)^2))
diff --git a/sample/math/geometry/polar-laplacian-3d-3.egi b/sample/math/geometry/polar-laplacian-3d-3.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/polar-laplacian-3d-3.egi
@@ -0,0 +1,41 @@
+;;;
+;;; Spherical coordinates
+;;;
+
+(define $x [|r θ φ|])
+
+(define $X [|(* r (sin θ) (cos φ)) ; = x
+             (* r (sin θ) (sin φ)) ; = y
+             (* r (cos θ))         ; = z
+             |])
+
+;;
+;; Local coordinates
+;;
+
+(define $e ((∂/∂ X~# $) x_#))
+e
+;[|[| (* (sin θ) (cos φ)) (* (sin θ) (sin φ)) (cos θ) |]
+;  [| (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ)) (* -1 r (sin θ)) |]
+;  [| (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ)) 0 |]|]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {3 3}))
+(define $g~~ (with-symbols {i j} (/ (unit-tensor {3 3})_i_j g_i_j)))
+
+g_#_#;[| [| 1 0 0 |] [| 0 r^2 0 |] [| 0 0 (* r^2 (sin θ)^2) |] |]_#_#
+g~#~#;[| [| 1 0 0 |] [| 0 (/ 1 r^2) 0 |] [| 0 0 (/ 1 (* r^2 (sin θ)^2)) |] |]~#~#
+
+;;
+;; Laplacian
+;;
+
+(define $sqrt-g (sqrt (M.det g_#_#)))
+sqrt-g;(* r^2 (sin θ))
+
+(define $Laplacian (/ (contract + (∂/∂ (* sqrt-g (. g~i~j (∂/∂ (f r θ φ) x_j))) x_i)) sqrt-g))
+Laplacian
+;(/ (+ (* 2 r (sin θ)^2 (f|1 r θ φ)) (* r^2 (sin θ)^2 (f|1|1 r θ φ)) (* (cos θ) (f|2 r θ φ) (sin θ)) (* (sin θ)^2 (f|2|2 r θ φ)) (f|3|3 r θ φ)) (* (sin θ)^2 r^2))
diff --git a/sample/math/geometry/polar-laplacian-3d.egi b/sample/math/geometry/polar-laplacian-3d.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/polar-laplacian-3d.egi
@@ -0,0 +1,61 @@
+(define $x (* r (sin θ) (cos φ)))
+(define $y (* r (sin θ) (sin φ)))
+(define $z (* r (cos θ)))
+
+(define $u-r (∂/∂ (u x y z) r))
+u-r
+;(+ (* (u|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (sin θ) (cos φ))
+;   (* (u|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (sin θ) (sin φ))
+;   (* (u|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (cos θ)))
+
+(define $u-r-r (∂/∂ (∂/∂ (u x y z) r) r))
+u-r-r
+;(+ (* (u|1|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (sin θ)^2 (cos φ)^2)
+;   (* (u|1|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (sin θ)^2 (sin φ) (cos φ))
+;   (* (u|1|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (cos θ) (sin θ) (cos φ))
+;   (* (u|2|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (sin θ)^2 (cos φ) (sin φ))
+;   (* (u|2|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (sin θ)^2 (sin φ)^2)
+;   (* (u|2|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (cos θ) (sin θ) (sin φ))
+;   (* (u|3|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (sin θ) (cos φ) (cos θ))
+;   (* (u|3|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (sin θ) (sin φ) (cos θ))
+;   (* (u|3|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) (cos θ)^2))
+
+(define $u-θ (∂/∂ (u x y z) θ))
+u-θ
+;(+ (* (u|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (cos θ) (cos φ))
+;   (* (u|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (cos θ) (sin φ))
+;   (* -1 (u|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (sin θ)))
+
+(define $u-θ-θ (∂/∂ (∂/∂ (u x y z) θ) θ))
+u-θ-θ
+;(+ (* (u|1|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (cos θ)^2 (cos φ)^2)
+;   (* (u|1|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (cos θ)^2 (sin φ) (cos φ))
+;   (* -1 (u|1|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (sin θ) (cos θ) (cos φ))
+;   (* -1 (u|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (sin θ) (cos φ))
+;   (* (u|2|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (cos θ)^2 (cos φ) (sin φ))
+;   (* (u|2|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (cos θ)^2 (sin φ)^2)
+;   (* -1 (u|2|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (sin θ) (cos θ) (sin φ))
+;   (* -1 (u|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (sin θ) (sin φ))
+;   (* -1 (u|3|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (cos θ) (cos φ) (sin θ))
+;   (* -1 (u|3|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (cos θ) (sin φ) (sin θ))
+;   (* (u|3|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (sin θ)^2)
+;   (* -1 (u|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (cos θ)))
+
+(define $u-φ (∂/∂ (u x y z) φ))
+u-φ
+;(+ (* -1 (u|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (sin θ) (sin φ))
+;   (* (u|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (sin θ) (cos φ)))
+
+(define $u-φ-φ (∂/∂ (∂/∂ (u x y z) φ) φ))
+u-φ-φ
+;(+ (* (u|1|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (sin θ)^2 (sin φ)^2)
+;   (* -1 (u|1|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (sin θ)^2 (cos φ) (sin φ))
+;   (* -1 (u|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (sin θ) (cos φ))
+;   (* -1 (u|2|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (sin θ)^2 (sin φ) (cos φ))
+;   (* (u|2|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r^2 (sin θ)^2 (cos φ)^2)
+;   (* -1 (u|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))) r (sin θ) (sin φ)))
+
+(+ u-r-r (* (/ 2 r) u-r) (* (/ 1 (** r 2)) u-θ-θ) (* (/ (cos θ) (* (** r 2) (sin θ))) u-θ) (* (/ 1 (** (* r (sin θ)) 2)) u-φ-φ))
+;(+ (u|3|3 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ)))
+;   (u|1|1 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ)))
+;   (u|2|2 (* r (sin θ) (cos φ)) (* r (sin θ) (sin φ)) (* r (cos θ))))
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-FLRW-metric.egi b/sample/math/geometry/riemann-curvature-tensor-of-FLRW-metric.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-FLRW-metric.egi
@@ -0,0 +1,101 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|w r θ φ|])
+
+;;
+;; Metric tensor
+;;
+
+(define $W (lambda [$r] (/ 1 '(- 1 (* K r^2)))))
+
+(define $g__
+  [|[| -1 0 0 0 |]
+    [| 0 (* (`a w)^2 (W r)) 0 0 |]
+    [| 0 0 (* (`a w)^2 r^2) 0 |]
+    [| 0 0 0 (* (`a w)^2 r^2 (sin θ)^2) |]
+    |])
+
+(define $g~~ (M.inverse g_#_#))
+g~#~#
+;[|[| -1 0 0 0 |]
+;  [| 0 (/ (* -1 '(+ 1 (* -1 K r^2))) (* -1 (a w)^2)) 0 0 |]
+;  [| 0 0 (/ -1 (* -1 (a w)^2 r^2)) 0 |]
+;  [| 0 0 0 (/ -1 (* -1 (a w)^2 r^2 (sin θ)^2)) |]|]~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;[| [| 1 0 0 0 |] [| 0 1 0 0 |] [| 0 0 1 0 |] [| 0 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_k x_l)
+        (∂/∂ g_j_l x_k)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_1_#_#;[| [| 0 0 0 0 |] [| 0 (/ (* -1 (a w) (a|1 w)) '(+ 1 (* -1 K r^2))) 0 0 |] [| 0 0 (* -1 (a w) (a|1 w) r^2) 0 |] [| 0 0 0 (* -1 (a w) (a|1 w) r^2 (sin θ)^2) |] |]_#_#
+Γ_2_#_#;[| [| 0 (/ (* (a w) (a|1 w)) '(+ 1 (* -1 K r^2))) 0 0 |] [| (/ (* (a w) (a|1 w)) '(+ 1 (* -1 K r^2))) (/ (* K r (a w)^2) '(+ 1 (* -1 K r^2))^2) 0 0 |] [| 0 0 (* -1 (a w)^2 r) 0 |] [| 0 0 0 (* -1 (a w)^2 r (sin θ)^2) |] |]_#_#
+Γ_3_#_#;[| [| 0 0 (* (a w) (a|1 w) r^2) 0 |] [| 0 0 (* (a w)^2 r) 0 |] [| (* (a w) (a|1 w) r^2) (* (a w)^2 r) 0 0 |] [| 0 0 0 (* -1 (a w)^2 r^2 (sin θ) (cos θ)) |] |]_#_#
+Γ_4_#_#;[| [| 0 0 0 (* (a w) (a|1 w) r^2 (sin θ)^2) |] [| 0 0 0 (* (a w)^2 r (sin θ)^2) |] [| 0 0 0 (* (a w)^2 r^2 (sin θ) (cos θ)) |] [| (* (a w) (a|1 w) r^2 (sin θ)^2) (* (a w)^2 r (sin θ)^2) (* (a w)^2 r^2 (sin θ) (cos θ)) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~1_#_#;[| [| 0 0 0 0 |] [| 0 (/ (* (a w) (a|1 w)) '(+ 1 (* -1 K r^2))) 0 0 |] [| 0 0 (* (a w) (a|1 w) r^2) 0 |] [| 0 0 0 (* (a w) (a|1 w) r^2 (sin θ)^2) |] |]_#_#
+Γ~2_#_#;[| [| 0 (/ (* -1 (a|1 w)) (* -1 (a w))) 0 0 |] [| (/ (* -1 (a|1 w)) (* -1 (a w))) (/ (* -1 K r) (* -1 '(+ 1 (* -1 K r^2)))) 0 0 |] [| 0 0 (* -1 '(+ 1 (* -1 K r^2)) r) 0 |] [| 0 0 0 (* -1 '(+ 1 (* -1 K r^2)) r (sin θ)^2) |] |]_#_#
+Γ~3_#_#;[| [| 0 0 (/ (* -1 (a|1 w)) (* -1 (a w))) 0 |] [| 0 0 (/ -1 (* -1 r)) 0 |] [| (/ (* -1 (a|1 w)) (* -1 (a w))) (/ -1 (* -1 r)) 0 0 |] [| 0 0 0 (* -1 (sin θ) (cos θ)) |] |]_#_#
+Γ~4_#_#;[| [| 0 0 0 (/ (* -1 (a|1 w)) (* -1 (a w))) |] [| 0 0 0 (/ -1 (* -1 r)) |] [| 0 0 0 (/ (* -1 (cos θ)) (* -1 (sin θ))) |] [| (/ (* -1 (a|1 w)) (* -1 (a w))) (/ -1 (* -1 r)) (/ (* -1 (cos θ)) (* -1 (sin θ))) 0 |] |]_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_1_1;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_2;[| [| 0 (/ (* (a w) (a|1|1 w)) (+ -1 (* K r^2))) 0 0 |] [| (/ (* -1 (a|1|1 w)) (a w)) 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_3;[| [| 0 0 (* -1 (a w) (a|1|1 w) r^2) 0 |] [| 0 0 0 0 |] [| (/ (* -1 (a|1|1 w)) (a w)) 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_4;[| [| 0 0 0 (* -1 (a w) (a|1|1 w) r^2 (sin θ)^2) |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| (/ (* -1 (a|1|1 w)) (a w)) 0 0 0 |] |]~#_#
+R~#_#_2_1;[| [| 0 (/ (* -1 (a w) (a|1|1 w)) (+ -1 (* K r^2))) 0 0 |] [| (/ (a|1|1 w) (a w)) 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_2;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_3;[| [| 0 0 0 0 |] [| 0 0 (+ (* -1 K r^2) (* -1 (a|1 w)^2 r^2)) 0 |] [| 0 (/ (+ (* -1 (a|1 w)^2) (* -1 K)) (+ -1 (* K r^2))) 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_4;[| [| 0 0 0 0 |] [| 0 0 0 (+ (* -1 K r^2 (sin θ)^2) (* -1 (a|1 w)^2 r^2 (sin θ)^2)) |] [| 0 0 0 0 |] [| 0 (/ (+ (* -1 (a|1 w)^2) (* -1 K)) (+ -1 (* K r^2))) 0 0 |] |]~#_#
+R~#_#_3_1;[| [| 0 0 (* (a w) (a|1|1 w) r^2) 0 |] [| 0 0 0 0 |] [| (/ (a|1|1 w) (a w)) 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_2;[| [| 0 0 0 0 |] [| 0 0 (+ (* K r^2) (* (a|1 w)^2 r^2)) 0 |] [| 0 (/ (+ (a|1 w)^2 K) (+ -1 (* K r^2))) 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_3;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_4;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 (+ (* -1 (a|1 w)^2 r^2 (sin θ)^2) (* -1 K r^2 (sin θ)^2)) |] [| 0 0 (+ (* (a|1 w)^2 r^2) (* K r^2)) 0 |] |]~#_#
+R~#_#_4_1;[| [| 0 0 0 (* (a w) (a|1|1 w) r^2 (sin θ)^2) |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| (/ (a|1|1 w) (a w)) 0 0 0 |] |]~#_#
+R~#_#_4_2;[| [| 0 0 0 0 |] [| 0 0 0 (+ (* K r^2 (sin θ)^2) (* (a|1 w)^2 r^2 (sin θ)^2)) |] [| 0 0 0 0 |] [| 0 (/ (+ (a|1 w)^2 K) (+ -1 (* K r^2))) 0 0 |] |]~#_#
+R~#_#_4_3;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 (+ (* (a|1 w)^2 r^2 (sin θ)^2) (* K r^2 (sin θ)^2)) |] [| 0 0 (+ (* -1 (a|1 w)^2 r^2) (* -1 K r^2)) 0 |] |]~#_#
+R~#_#_4_4;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_1_#;[| (/ (* -3 (a|1|1 w)) (a w)) 0 0 0 |]_#
+Ric_2_#;[| 0 (/ (+ (* -1 (a w) (a|1|1 w)) (* -2 (a|1 w)^2) (* -2 K)) (+ -1 (* K r^2))) 0 0 |]_#
+Ric_3_#;[| 0 0 (+ (* (a w) (a|1|1 w) r^2) (* 2 K r^2) (* 2 (a|1 w)^2 r^2)) 0 |]_#
+Ric_4_#;[| 0 0 0 (+ (* (a w) (a|1|1 w) r^2 (sin θ)^2) (* 2 K r^2 (sin θ)^2) (* 2 (a|1 w)^2 r^2 (sin θ)^2)) |]_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (expand-all' (. g~j~k Ric_j_k))))
+
+scalar-curvature
+;(/ (+ (* 6 (a|1|1 w) (a w)) (* 6 (a|1 w)^2) (* 6 K))
+;   (a w)^2)
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-M3-conformal.egi b/sample/math/geometry/riemann-curvature-tensor-of-M3-conformal.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-M3-conformal.egi
@@ -0,0 +1,72 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|α β γ|])
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(* (a α β γ) (G_%1_%2 α β γ)) {3 3}))
+(define $g~~ (generate-tensor 2#(* (/ 1 (a α β γ)) (G~%1~%2 α β γ)) {3 3}))
+g_#_#
+g~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ___
+  (with-symbols {j k l}
+    (* (/ 1 2)
+       (+ (∂/∂ g_j_l x_k)
+          (∂/∂ g_j_k x_l)
+          (* -1 (∂/∂ g_k_l x_j))))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__
+  (with-symbols {i j k l}
+    (. g~i~j Γ_j_k_l)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i j k} (contract + R~i_j_k_i)))
+
+Ric_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(let {[[$es $os] (even-and-odd-permutations 3)]}
+  (- (sum (map (lambda [$σ] (. R~u_1_s_(σ 1) R~s_u_(σ 3)_(σ 2))) es))
+     (sum (map (lambda [$σ] (. R~u_1_s_(σ 1) R~s_t_(σ 3)_(σ 2))) os))))
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-M5-conformal.egi b/sample/math/geometry/riemann-curvature-tensor-of-M5-conformal.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-M5-conformal.egi
@@ -0,0 +1,56 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|α β γ δ ε|])
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(* (a α β γ δ ε) (G_%1_%2 α β γ δ ε)) {5 5}))
+(define $g~~ (generate-tensor 2#(* (/ 1 (a α β γ δ ε)) (G~%1~%2 α β γ δ ε)) {5 5}))
+g_#_#
+g~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ___
+  (with-symbols {j k l}
+    (* (/ 1 2)
+       (+ (∂/∂ g_j_l x_k)
+          (∂/∂ g_j_k x_l)
+          (* -1 (∂/∂ g_k_l x_j))))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__
+  (with-symbols {i j k l}
+    (. g~i~j Γ_j_k_l)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(let {[[$es $os] (even-and-odd-permutations 5)]}
+  (- (sum (map (lambda [$σ] (. R~u_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4))) es))
+     (sum (map (lambda [$σ] (. R~u_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4))) os))))
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S1.egi b/sample/math/geometry/riemann-curvature-tensor-of-S1.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S1.egi
@@ -0,0 +1,80 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|θ|])
+
+(define $X [|(* r (sin θ)) ; = x
+             (* r (cos θ)) ; = y
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e;[| [| (* r (cos θ)) (* -1 r (sin θ)) |] |]_#~#
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {1 1}))
+(define $g~~ (M.inverse g_#_#))
+
+g_#_#;[| [| r^2 |] |]_#_#
+g~#~#;[| [| (/ 1 r^2) |] |]~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ___
+  (with-symbols {j k l}
+    (* (/ 1 2)
+       (+ (∂/∂ g_j_k x_l)
+          (∂/∂ g_j_l x_k)
+          (* -1 (∂/∂ g_k_l x_j))))))
+
+Γ_#_#_#;(tensor {1 1 1} {0} )_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__
+  (with-symbols {i j k l}
+    (. g~i~j Γ_j_k_l)))
+
+Γ~#_#_#;(tensor {1 1 1} {0} )~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#;(tensor {1 1 1 1} {0} )~#_#_#_#
+
+(define $R____ (with-symbols {i} (. g_i_# R~i_#_#_#)))
+
+R_#_#_#_#;(tensor {1 1 1 1} {0} )_#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i j k} (contract + R~i_j_k_i)))
+
+Ric_#_#;[| [| 0 |] |]_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature;0
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S2.egi b/sample/math/geometry/riemann-curvature-tensor-of-S2.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S2.egi
@@ -0,0 +1,119 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|θ φ|])
+
+(define $X [|(* r (sin θ) (cos φ)) ; = x
+             (* r (sin θ) (sin φ)) ; = y
+             (* r (cos θ))         ; = z
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[|(* r (cos θ) (cos φ)) (* r (cos θ) (sin φ)) (* -1 r (sin θ)) |]
+;  [|(* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ)) 0 |]
+;  |]_#~#
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {2 2}))
+(define $g~~ (M.inverse g_#_#))
+
+g_#_#;[| [| r^2 0 |] [| 0 (* r^2 (sin θ)^2) |] |]_#_#
+g~#~#;[| [| (/ 1 r^2) 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) |] |]~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_#_#_#;(tensor {2 2 2} {0 0 0 (* -1 r^2 (sin θ) (cos θ)) 0 (* r^2 (sin θ) (cos θ)) (* r^2 (sin θ) (cos θ)) 0} )_#_#_#
+Γ_1_#_#;[| [| 0 0 |] [| 0 (* -1 r^2 (sin θ) (cos θ)) |] |]_#_#
+Γ_2_#_#;[| [| 0 (* r^2 (sin θ) (cos θ)) |] [| (* r^2 (sin θ) (cos θ)) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~#_#_#;(tensor {2 2 2} {0 0 0 (* -1 (sin θ) (cos θ)) 0 (/ (cos θ) (sin θ)) (/ (cos θ) (sin θ)) 0} )~#_#_#
+Γ~1_#_#;[| [| 0 0 |] [| 0 (* -1 (sin θ) (cos θ)) |] |]_#_#
+Γ~2_#_#;[| [| 0 (/ (cos θ) (sin θ)) |] [| (/ (cos θ) (sin θ)) 0 |] |]_#_#
+
+;;
+;; Covariant derivative of metric tensor
+;;
+(define $∇g___
+  (with-symbols {i j m n}
+    (- (∂/∂ g_i_j x_m)
+       (. Γ~n_m_i g_n_j)
+       (. Γ~n_m_j g_i_n))))
+
+∇g_#_#_#;=>(tensor {2 2 2} {0 0 0 0 0 0 0 0} )
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#;(tensor {2 2 2 2} {0 0 0 0 0 (sin θ)^2 (* -1 (sin θ)^2) 0 0 -1 1 0 0 0 0 0} )~#_#_#_#
+R~#_#_1_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
+R~#_#_1_2;[| [| 0 (sin θ)^2 |] [| -1 0 |] |]~#_#
+R~#_#_2_1;[| [| 0 (* -1 (sin θ)^2) |] [| 1 0 |] |]~#_#
+R~#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]~#_#
+
+(define $R____ (with-symbols {i} (. g_i_# R~i_#_#_#)))
+
+R_#_#_#_#;(tensor {2 2 2 2} {0 0 0 0 0 (* r^2 (sin θ)^2) (* -1 r^2 (sin θ)^2) 0 0 (* -1 r^2 (sin θ)^2) (* r^2 (sin θ)^2) 0 0 0 0 0} )_#_#_#_#
+R_#_#_1_1;[| [| 0 0 |] [| 0 0 |] |]_#_#
+R_#_#_1_2;[| [| 0 (* r^2 (sin θ)^2) |] [| (* -1 r^2 (sin θ)^2) 0 |] |]_#_#
+R_#_#_2_1;[| [| 0 (* -1 r^2 (sin θ)^2) |] [| (* r^2 (sin θ)^2) 0 |] |]_#_#
+R_#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#;[| [| 1 0 |] [| 0 (sin θ)^2 |] |]_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature;(/ 2 r^2)
+
+;;
+;; Covariant derivative of Riemann curvature tensor
+;;
+
+(define $∇R_____
+  (with-symbols {i j k l m n}
+    (- (∂/∂ R_i_j_k_l x_m)
+       (. Γ~n_m_i R_n_j_k_l)
+       (. Γ~n_m_j R_i_n_k_l)
+       (. Γ~n_m_k R_i_j_n_l)
+       (. Γ~n_m_l R_i_j_k_n))))
+
+∇R_#_#_#_#_#
+;(tensor {2 2 2 2 2} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )_#_#_#_#_#
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-conformal-fast.egi b/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-conformal-fast.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-conformal-fast.egi
@@ -0,0 +1,78 @@
+;;
+;; Parameters
+;;
+
+(define $x [| φ θ ψ y α |])
+
+;;
+;; Riemann metric of S2 x S3
+;;
+
+(define $g__
+  (* (a φ θ ψ y α)^2
+     [|[| (/ (+ (* 3 '(+ 1 (* -1 y))^2 (sin θ)^2 '(+ a (* -1 y^2))) (* 2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ)^2 '(+ 1 (* -1 y))) (* '(+ a (* -2 y) y^2)^2 (cos θ)^2)) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (+ (* -2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ) '(+ 1 (* -1 y))) (* -1 '(+ a (* -2 y) y^2)^2 (cos θ))) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (* -1 '(+ a (* -2 y) y^2) (cos θ)) (* 3 '(+ 1 (* -1 y)))) |]
+       [| 0 (/ '(+ 1 (* -1 y)) 6) 0 0 0 |]
+       [| (/ (+ (* -2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ) '(+ 1 (* -1 y))) (* -1 '(+ a (* -2 y) y^2)^2 (cos θ))) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (+ (* 2 '(+ a (* -3 y^2) (* 2 y^3)) '(+ 1 (* -1 y))) '(+ a (* -2 y) y^2)^2) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (* 1 '(+ a (* -2 y) y^2)) (* 3 '(+ 1 (* -1 y)))) |]
+       [| 0 0 0 (/ '(+ 1 (* -1 y)) (* 2 '(+ a (* -3 y^2) (* 2 y^3)))) 0 |]
+       [| (/ (* -1 '(+ a (* -2 y) y^2) (cos θ)) (* 3 '(+ 1 (* -1 y)))) 0 (/ (* 1 '(+ a (* -2 y) y^2)) (* 3 '(+ 1 (* -1 y)))) 0 (/ (* 2 '(+ a (* -1 y^2))) '(+ 1 (* -1 y))) |]
+       |]_#_#))
+
+(define $g~~ (M.inverse g_#_#))
+g~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+Ric_#_#
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(define $ret (let {[[$es $os] (even-and-odd-permutations 5)]}
+               (- (sum' (map (lambda [$σ] (debug (.' R~u_5_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4)))) es))
+                  (sum' (map (lambda [$σ] (debug (.' R~u_5_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4)))) os)))))
+
+ret
+
+(define $ret2 (/ (expand-all' (numerator ret)) (denominator ret)))
+
+ret2
+
+(define $ret3 (/ (2#%1 (P./ (numerator ret2) (* (+ 1 (* -1 y))^3 (+ a (* -1 y^2))^5) y))
+                 (/ (denominator ret2) (* '(+ 1 (* -1 y))^3 '(+ a (* -1 y^2))^5))))
+
+ret3
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-fast.egi b/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-fast.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-fast.egi
@@ -0,0 +1,80 @@
+;;
+;; Parameters
+;;
+
+(define $x [| φ θ ψ y α |])
+
+;;
+;; Riemann metric of S2 x S3
+;;
+
+(define $g__
+  [|[| (/ (+ (* 3 '(+ 1 (* -1 y))^2 (sin θ)^2 '(+ a (* -1 y^2))) (* 2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ)^2 '(+ 1 (* -1 y))) (* '(+ a (* -2 y) y^2)^2 (cos θ)^2)) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (+ (* -2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ) '(+ 1 (* -1 y))) (* -1 '(+ a (* -2 y) y^2)^2 (cos θ))) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (* -1 '(+ a (* -2 y) y^2) (cos θ)) (* 3 '(+ 1 (* -1 y)))) |]
+    [| 0 (/ '(+ 1 (* -1 y)) 6) 0 0 0 |]
+    [| (/ (+ (* -2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ) '(+ 1 (* -1 y))) (* -1 '(+ a (* -2 y) y^2)^2 (cos θ))) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (+ (* 2 '(+ a (* -3 y^2) (* 2 y^3)) '(+ 1 (* -1 y))) '(+ a (* -2 y) y^2)^2) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (* 1 '(+ a (* -2 y) y^2)) (* 3 '(+ 1 (* -1 y)))) |]
+    [| 0 0 0 (/ '(+ 1 (* -1 y)) (* 2 '(+ a (* -3 y^2) (* 2 y^3)))) 0 |]
+    [| (/ (* -1 '(+ a (* -2 y) y^2) (cos θ)) (* 3 '(+ 1 (* -1 y)))) 0 (/ (* 1 '(+ a (* -2 y) y^2)) (* 3 '(+ 1 (* -1 y)))) 0 (/ (* 2 '(+ a (* -1 y^2))) '(+ 1 (* -1 y))) |]
+    |]_#_#)
+
+(define $g~~ (M.inverse g_#_#))
+g~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+Ric_#_#
+
+(expand-all' (with-symbols {i j} (-' Ric_i_j (*' 4 g_i_j))))
+;[| [| 0 0 0 0 0 |] [| 0 0 0 0 0 |] [| 0 0 0 0 0 |] [| 0 0 0 0 0 |] [| 0 0 0 0 0 |] |]
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(define $ret (let {[[$es $os] (even-and-odd-permutations 5)]}
+               (- (sum' (map (lambda [$σ] (.' R~u_5_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4))) es))
+                  (sum' (map (lambda [$σ] (.' R~u_5_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4))) os)))))
+
+(define $ret2 (/ (expand-all' (numerator ret)) (denominator ret)))
+
+ret2
+;(/ (+ (* -128 a^6 y (sin θ)) (* 832 a^5 y^3 (sin θ)) (* -2240 a^4 y^5 (sin θ)) (* 3200 a^3 y^7 (sin θ)) (* -2560 a^2 y^9 (sin θ)) (* 1088 a y^11 (sin θ)) (* 384 a^6 y^2 (sin θ)) (* -1984 a^5 y^4 (sin θ)) (* 4160 a^4 y^6 (sin θ)) (* -4480 a^3 y^8 (sin θ)) (* 2560 a^2 y^10 (sin θ)) (* -704 a y^12 (sin θ)) (* -704 a^6 y^3 (sin θ)) (* 2560 a^5 y^5 (sin θ)) (* -4480 a^4 y^7 (sin θ)) (* 4160 a^3 y^9 (sin θ)) (* -1984 a^2 y^11 (sin θ)) (* 384 a y^13 (sin θ)) (* 1088 a^6 y^4 (sin θ)) (* -2560 a^5 y^6 (sin θ)) (* 3200 a^4 y^8 (sin θ)) (* -2240 a^3 y^10 (sin θ)) (* 832 a^2 y^12 (sin θ)) (* -128 a y^14 (sin θ)) (* -960 a^6 y^5 (sin θ)) (* 1920 a^5 y^7 (sin θ)) (* -1920 a^4 y^9 (sin θ)) (* 960 a^3 y^11 (sin θ)) (* -192 a^2 y^13 (sin θ)) (* 320 a^6 y^6 (sin θ)) (* -640 a^5 y^8 (sin θ)) (* 640 a^4 y^10 (sin θ)) (* -320 a^3 y^12 (sin θ)) (* 64 a^2 y^14 (sin θ)) (* 64 y^14 (sin θ)) (* 64 a^7 y (sin θ)) (* -192 a^7 y^2 (sin θ)) (* 192 a^7 y^3 (sin θ)) (* -64 a^7 y^4 (sin θ)) (* -192 a^5 (sin θ) y^2) (* 960 a^4 (sin θ) y^4) (* -1920 a^3 (sin θ) y^6) (* 1920 a^2 y^8 (sin θ)) (* -960 a y^10 (sin θ)) (* -320 y^3 a^4 (sin θ)) (* 640 y^5 a^3 (sin θ)) (* -640 y^7 a^2 (sin θ)) (* 320 y^9 a (sin θ)) (* -64 y^11 (sin θ)) (* 192 y^12 (sin θ)) (* 64 a^5 y (sin θ)) (* -192 y^13 (sin θ))) (* 3 '(+ 1 (* -1 y))^8 '(+ a (* -1 y^2))^5))
+
+(define $ret3 (/ (2#%1 (P./ (numerator ret2) (* (+ 1 (* -1 y))^3 (+ a (* -1 y^2))^5) y))
+                 (/ (denominator ret2) (* '(+ 1 (* -1 y))^3 '(+ a (* -1 y^2))^5))))
+
+ret3
+;(/ (+ (* 128 a (sin θ) y) (* -64 a^2 (sin θ) y) (* -64 (sin θ) y)) (* 3 '(+ 1 (* -1 y))^5))
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-integral.egi b/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-integral.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S2xS3-integral.egi
@@ -0,0 +1,57 @@
+(define $ret3 (/ (+ (* 8 a (sin θ) y) (* -4 a^2 (sin θ) y) (* -4 (sin θ) y)) (* 45 '(+ 1 (* -1 y))^5)))
+
+(define $ret4 (- (let {[$θ π]} (/ (+ (* 8 a (sin θ) y) (* -4 a^2 (sin θ) y) (* -4 (sin θ) y)) (* 45 '(+ 1 (* -1 y))^5)))
+                 (let {[$θ 0]} (/ (+ (* 8 a (sin θ) y) (* -4 a^2 (sin θ) y) (* -4 (sin θ) y)) (* 45 '(+ 1 (* -1 y))^5)))))
+
+"ret4"
+ret4
+;(/ (+ (* 16 a y) (* -8 a^2 y) (* -8 y)) (* 45 '(+ 1 (* -1 y))^5))
+
+(define $ret5 (d/d (/ (* 2 (+ 1 (* -1 a))^2 (- 1 (* 4 y))) (* 135 '(+ 1 (* -1 y))^4)) y))
+
+"ret5"
+ret5
+
+(define $ret6 (/ (expand-all' (numerator ret5)) (denominator ret5)))
+
+"ret6"
+ret6
+
+(define $ret7 (/ (* 2 (+ 1 (* -1 a))^2 (- 1 (* 4 y))) (* 135 '(+ 1 (* -1 y))^4)))
+
+(define $y1 (* (/ 1 2) (+ 1 (* -1 λ) (* -1 (sqrt (- 1 (/ λ^2 3)))))))
+(define $y2 (+ y1 λ))
+
+
+(let {[$y y2]} ret7)
+(let {[$y y1]} ret7)
+
+(define $ret8 (- (let {[$y y2]} (/ (* 2 (+ 1 (* -1 a))^2 (- 1 (* 4 y))) (* 135 '(+ 1 (* -1 y))^4)))
+                 (let {[$y y1]} (/ (* 2 (+ 1 (* -1 a))^2 (- 1 (* 4 y))) (* 135 '(+ 1 (* -1 y))^4)))))
+
+"ret8"
+ret8
+;(/ (+ (* -6 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 12 a '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 24 a λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -8 a (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -6 a^2 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 a^2 λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 4 a^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 6 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 a '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 24 a λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 8 a (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 6 a^2 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 a^2 λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -4 a^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4)) (* 405 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4))
+
+(define $ret9 (let {[$a (- (* 3 y1^2) (* 2 y2^3))]}
+                (/ (+ (* -6 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 12 a '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 24 a λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -8 a (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -6 a^2 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 a^2 λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 4 a^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 6 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 a '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 24 a λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 8 a (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 6 a^2 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -12 a^2 λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -4 a^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4)) (* 405 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4))))
+
+"ret9"
+ret9
+;(/ (+ (* -324 λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 54 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -108 λ (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 5742 λ^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -17793 λ^2 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 5544 λ^3 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 162 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -15390 λ^3 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 1548 λ^4 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 2808 λ^5 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 912 λ^6 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 360 λ^4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 96 λ^7 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -288 λ^5 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -32 λ^6 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -324 λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -54 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -108 λ (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -5742 λ^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 17793 λ^2 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 2520 λ^3 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -162 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -6966 λ^3 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -8028 λ^4 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 216 λ^5 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 816 λ^6 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 1368 λ^4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 96 λ^7 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 288 λ^5 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 32 λ^6 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4)) (* 21870 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4))
+
+(define $ret10 (let {[$λ (/ (* 3 q) (* 2 p))]}
+                 (/ (+ (* -324 λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 54 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -108 λ (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 5742 λ^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -17793 λ^2 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 5544 λ^3 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 162 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -15390 λ^3 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 1548 λ^4 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 2808 λ^5 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 912 λ^6 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 360 λ^4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 96 λ^7 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -288 λ^5 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -32 λ^6 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -324 λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -54 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -108 λ (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -5742 λ^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 17793 λ^2 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 2520 λ^3 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -162 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -6966 λ^3 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -8028 λ^4 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 216 λ^5 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 816 λ^6 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 1368 λ^4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 96 λ^7 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 288 λ^5 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 32 λ^6 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4)) (* 21870 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4))))
+
+"ret10"
+ret10
+
+(define $ret11 (let* {[$p 7]
+                      [$q 3]
+                      [$λ (/ (* 3 q) (* 2 p))]}
+                 (* (/ (+ (* -324 λ '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 54 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -108 λ (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 5742 λ^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -17793 λ^2 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 5544 λ^3 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 162 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -15390 λ^3 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 1548 λ^4 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 2808 λ^5 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 912 λ^6 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 360 λ^4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 96 λ^7 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -288 λ^5 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -32 λ^6 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -324 λ '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -54 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -108 λ (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -5742 λ^2 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 17793 λ^2 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 2520 λ^3 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -162 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -6966 λ^3 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* -8028 λ^4 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 216 λ^5 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 816 λ^6 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 1368 λ^4 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 96 λ^7 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 288 λ^5 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4) (* 32 λ^6 (sqrt (+ 9 (* -3 λ^2))) '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4)) (* 21870 '(/ (+ 3 (* -3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4 '(/ (+ 3 (* 3 λ) (sqrt (+ 9 (* -3 λ^2)))) 6)^4))
+                    (* 2^4 π^4 (/ q (+ (* 3 q^2) (* -2 p^2) (* p (sqrt (+ (* 4 p^2) (* -3 q^2))))))))))
+
+
+(expand-all ret11)
+;(/ (* -1849 π^4) 22050)
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S2xS3.egi b/sample/math/geometry/riemann-curvature-tensor-of-S2xS3.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S2xS3.egi
@@ -0,0 +1,81 @@
+;;
+;; Parameters
+;;
+
+(define $x [| φ θ ψ y α |])
+
+;;
+;; Riemann metric of S2 x S3
+;;
+
+(define $g__
+  [|[| (/ (+ (* 3 '(+ 1 (* -1 y))^2 (sin θ)^2 '(+ a (* -1 y^2))) (* 2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ)^2 '(+ 1 (* -1 y))) (* '(+ a (* -2 y) y^2)^2 (cos θ)^2)) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (+ (* -2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ) '(+ 1 (* -1 y))) (* -1 '(+ a (* -2 y) y^2)^2 (cos θ))) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (* -1 '(+ a (* -2 y) y^2) (cos θ)) (* 3 '(+ 1 (* -1 y)))) |]
+    [| 0 (/ '(+ 1 (* -1 y)) 6) 0 0 0 |]
+    [| (/ (+ (* -2 '(+ a (* -3 y^2) (* 2 y^3)) (cos θ) '(+ 1 (* -1 y))) (* -1 '(+ a (* -2 y) y^2)^2 (cos θ))) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (+ (* 2 '(+ a (* -3 y^2) (* 2 y^3)) '(+ 1 (* -1 y))) '(+ a (* -2 y) y^2)^2) (* 18 '(+ a (* -1 y^2)) '(+ 1 (* -1 y)))) 0 (/ (* 1 '(+ a (* -2 y) y^2)) (* 3 '(+ 1 (* -1 y)))) |]
+    [| 0 0 0 (/ '(+ 1 (* -1 y)) (* 2 '(+ a (* -3 y^2) (* 2 y^3)))) 0 |]
+    [| (/ (* -1 '(+ a (* -2 y) y^2) (cos θ)) (* 3 '(+ 1 (* -1 y)))) 0 (/ (* 1 '(+ a (* -2 y) y^2)) (* 3 '(+ 1 (* -1 y)))) 0 (/ (* 2 '(+ a (* -1 y^2))) '(+ 1 (* -1 y))) |]
+    |]_#_#)
+
+(define $g~~ (M.inverse g_#_#))
+g~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+Ric_#_#
+
+(expand-all' (with-symbols {i j} (-' Ric_i_j (*' 4 g_i_j))))
+;[| [| 0 0 0 0 0 |] [| 0 0 0 0 0 |] [| 0 0 0 0 0 |] [| 0 0 0 0 0 |] [| 0 0 0 0 0 |] |]
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(define $ret (let {[[$es $os] (even-and-odd-permutations 5)]}
+               (/ (- (sum (map (lambda [$σ] (. R~u_5_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4))) es))
+                     (sum (map (lambda [$σ] (. R~u_5_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4))) os)))
+                  (* 2 (fact 5)))))
+
+(define $ret2 (/ (expand-all' (numerator ret)) (denominator ret)))
+
+ret2
+;
+
+(define $ret3 (/ (2#%1 (P./ (numerator ret2) (* (+ 1 (* -1 y))^3 (+ a (* -1 y^2))^5) y))
+                 (/ (denominator ret2) (* '(+ 1 (* -1 y))^3 '(+ a (* -1 y^2))^5))))
+
+ret3
+;(/ (+ (* 8 a (sin θ) y) (* -4 a^2 (sin θ) y) (* -4 (sin θ) y)) (* 45 '(+ 1 (* -1 y))^5))
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S3.egi b/sample/math/geometry/riemann-curvature-tensor-of-S3.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S3.egi
@@ -0,0 +1,108 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|θ φ ψ|])
+
+(define $X [|(* r (cos θ))
+             (* r (sin θ) (cos φ))
+             (* r (sin θ) (sin φ) (cos ψ))
+             (* r (sin θ) (sin φ) (sin ψ))
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[| (* -1 r (sin θ)) (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ) (cos ψ)) (* r (cos θ) (sin φ) (sin ψ)) |]
+;  [| 0 (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ) (cos ψ)) (* r (sin θ) (cos φ) (sin ψ)) |]
+;  [| 0 0 (* -1 r (sin θ) (sin φ) (sin ψ)) (* r (sin θ) (sin φ) (cos ψ)) |]|]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {3 3}))
+(define $g~~ (M.inverse g_#_#))
+g_#_#;[| [| r^2 0 0 |] [| 0 (* r^2 (sin θ)^2) 0 |] [| 0 0 (* r^2 (sin θ)^2 (sin φ)^2) |] |]_#_#
+g~#~#;[| [| (/ 1 r^2) 0 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) 0 |] [| 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2)) |] |]~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k));[| [| 1 0 0 |] [| 0 1 0 |] [| 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_k x_l)
+        (∂/∂ g_j_l x_k)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_1_#_#;[| [| 0 0 0 |] [| 0 (* -1 r^2 (sin θ) (cos θ)) 0 |] [| 0 0 (* -1 r^2 (sin θ) (cos θ) (sin φ)^2) |] |]_#_#
+Γ_2_#_#;[| [| 0 (* r^2 (sin θ) (cos θ)) 0 |] [| (* r^2 (sin θ) (cos θ)) 0 0 |] [| 0 0 (* -1 r^2 (sin θ)^2 (sin φ) (cos φ)) |] |]_#_#
+Γ_3_#_#;[| [| 0 0 (* r^2 (sin θ) (cos θ) (sin φ)^2) |] [| 0 0 (* r^2 (sin θ)^2 (sin φ) (cos φ)) |] [| (* r^2 (sin θ) (cos θ) (sin φ)^2) (* r^2 (sin θ)^2 (sin φ) (cos φ)) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~1_#_#;[| [| 0 0 0 |] [| 0 (* -1 (sin θ) (cos θ)) 0 |] [| 0 0 (* -1 (sin θ) (cos θ) (sin φ)^2) |] |]_#_#
+Γ~2_#_#;[| [| 0 (/ (cos θ) (sin θ)) 0 |] [| (/ (cos θ) (sin θ)) 0 0 |] [| 0 0 (* -1 (sin φ) (cos φ)) |] |]_#_#
+Γ~3_#_#;[| [| 0 0 (/ (cos θ) (sin θ)) |] [| 0 0 (/ (cos φ) (sin φ)) |] [| (/ (cos θ) (sin θ)) (/ (cos φ) (sin φ)) 0 |] |]_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_1_1;[| [| 0 0 0 |] [| 0 0 0 |] [| 0 0 0 |] |]~#_#
+R~#_#_1_2;[| [| 0 (sin θ)^2 0 |] [| -1 0 0 |] [| 0 0 0 |] |]~#_#
+R~#_#_1_3;[| [| 0 0 (* (sin θ)^2 (sin φ)^2) |] [| 0 0 0 |] [| -1 0 0 |] |]~#_#
+R~#_#_2_1;[| [| 0 (* -1 (sin θ)^2) 0 |] [| 1 0 0 |] [| 0 0 0 |] |]~#_#
+R~#_#_2_2;[| [| 0 0 0 |] [| 0 0 0 |] [| 0 0 0 |] |]~#_#
+R~#_#_2_3;[| [| 0 0 0 |] [| 0 0 (* (sin θ)^2 (sin φ)^2) |] [| 0 (* -1 (sin θ)^2) 0 |] |]~#_#
+R~#_#_3_1;[| [| 0 0 (* -1 (sin θ)^2 (sin φ)^2) |] [| 0 0 0 |] [| 1 0 0 |] |]~#_#
+R~#_#_3_2;[| [| 0 0 0 |] [| 0 0 (* -1 (sin θ)^2 (sin φ)^2) |] [| 0 (sin θ)^2 0 |] |]~#_#
+R~#_#_3_3;[| [| 0 0 0 |] [| 0 0 0 |] [| 0 0 0 |] |]~#_#
+
+(define $R____ (with-symbols {i} (. g_i_# R~i_#_#_#)))
+
+R_#_#_#_#;(tensor {3 3 3 3} {0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^2) 0 (* -1 r^2 (sin θ)^2) 0 0 0 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2) 0 0 0 (* -1 r^2 (sin θ)^2) 0 (* r^2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^4 (sin φ)^2) 0 (* -1 r^2 (sin θ)^4 (sin φ)^2) 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2) 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^2) 0 (* r^2 (sin θ)^4 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0} )_#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#;[| [| 2 0 0 |] [| 0 (* 2 (sin θ)^2) 0 |] [| 0 0 (* 2 (sin θ)^2 (sin φ)^2) |] |]_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature;(/ 6 r^2)
+
+;;
+;; Conformal curvature tensor
+;;
+
+(define $C_i_k_l_m
+  (+ (. R_i_k_l_m)
+     (+ (- (. Ric_i_m g_k_l) (. Ric_i_l g_k_m))
+        (- (. Ric_k_l g_i_m) (. Ric_k_m g_i_l)))
+     (* (/ scalar-curvature 2) (- (. g_i_l g_k_m) (. g_i_m g_k_l)))))
+
+C_#_#_#_#
+;(tensor {3 3 3 3} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )_#_#_#_#
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S4.egi b/sample/math/geometry/riemann-curvature-tensor-of-S4.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S4.egi
@@ -0,0 +1,145 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|θ φ ψ η|])
+
+(define $X [|(* r (cos θ))
+             (* r (sin θ) (cos φ))
+             (* r (sin θ) (sin φ) (cos ψ))
+             (* r (sin θ) (sin φ) (sin ψ) (cos η))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η))
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[| (* -1 r (sin θ)) (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ) (cos ψ)) (* r (cos θ) (sin φ) (sin ψ) (cos η)) (* r (cos θ) (sin φ) (sin ψ) (sin η)) |]
+;  [| 0 (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ) (cos ψ)) (* r (sin θ) (cos φ) (sin ψ) (cos η)) (* r (sin θ) (cos φ) (sin ψ) (sin η)) |]
+;  [| 0 0 (* -1 r (sin θ) (sin φ) (sin ψ)) (* r (sin θ) (sin φ) (cos ψ) (cos η)) (* r (sin θ) (sin φ) (cos ψ) (sin η)) |]
+;  [| 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η)) (* r (sin θ) (sin φ) (sin ψ) (cos η)) |] |]_#~#
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {4 4}))
+(define $g~~ (M.inverse g_#_#))
+g_#_#;[| [| r^2 0 0 0 |] [| 0 (* r^2 (sin θ)^2) 0 0 |] [| 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 |] [| 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) |] |]_#_#
+g~#~#;[| [| (/ 1 r^2) 0 0 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) 0 0 |] [| 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2)) 0 |] [| 0 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2)) |] |]~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;[| [| 1 0 0 0 |] [| 0 1 0 0 |] [| 0 0 1 0 |] [| 0 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_k x_l)
+        (∂/∂ g_j_l x_k)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_1_#_#;[| [| 0 0 0 0 |] [| 0 (/ (* -1 r^2 (sin (* 2 θ))) 2) 0 0 |] [| 0 0 (/ (* -1 r^2 (sin (* 2 θ)) (sin φ)^2) 2) 0 |] [| 0 0 0 (/ (* -1 r^2 (sin (* 2 θ)) (sin φ)^2 (sin ψ)^2) 2) |] |]_#_#
+Γ_2_#_#;[| [| 0 (/ (* r^2 (sin (* 2 θ))) 2) 0 0 |] [| (/ (* r^2 (sin (* 2 θ))) 2) 0 0 0 |] [| 0 0 (/ (* -1 r^2 (sin θ)^2 (sin (* 2 φ))) 2) 0 |] [| 0 0 0 (/ (* -1 r^2 (sin θ)^2 (sin (* 2 φ)) (sin ψ)^2) 2) |] |]_#_#
+Γ_3_#_#;[| [| 0 0 (/ (* r^2 (sin (* 2 θ)) (sin φ)^2) 2) 0 |] [| 0 0 (/ (* r^2 (sin θ)^2 (sin (* 2 φ))) 2) 0 |] [| (/ (* r^2 (sin (* 2 θ)) (sin φ)^2) 2) (/ (* r^2 (sin θ)^2 (sin (* 2 φ))) 2) 0 0 |] [| 0 0 0 (/ (* -1 r^2 (sin θ)^2 (sin φ)^2 (sin (* 2 ψ))) 2) |] |]_#_#
+Γ_4_#_#;[| [| 0 0 0 (/ (* r^2 (sin (* 2 θ)) (sin φ)^2 (sin ψ)^2) 2) |] [| 0 0 0 (/ (* r^2 (sin θ)^2 (sin (* 2 φ)) (sin ψ)^2) 2) |] [| 0 0 0 (/ (* r^2 (sin θ)^2 (sin φ)^2 (sin (* 2 ψ))) 2) |] [| (/ (* r^2 (sin (* 2 θ)) (sin φ)^2 (sin ψ)^2) 2) (/ (* r^2 (sin θ)^2 (sin (* 2 φ)) (sin ψ)^2) 2) (/ (* r^2 (sin θ)^2 (sin φ)^2 (sin (* 2 ψ))) 2) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~1_#_#;[| [| 0 0 0 0 |] [| 0 (/ (* -1 (sin (* 2 θ))) 2) 0 0 |] [| 0 0 (/ (* -1 (sin (* 2 θ)) (sin φ)^2) 2) 0 |] [| 0 0 0 (/ (* -1 (sin (* 2 θ)) (sin φ)^2 (sin ψ)^2) 2) |] |]_#_#
+Γ~2_#_#;[| [| 0 (/ (cos θ) (sin θ)) 0 0 |] [| (/ (cos θ) (sin θ)) 0 0 0 |] [| 0 0 (/ (* -1 (sin (* 2 φ))) 2) 0 |] [| 0 0 0 (/ (* -1 (sin (* 2 φ)) (sin ψ)^2) 2) |] |]_#_#
+Γ~3_#_#;[| [| 0 0 (/ (cos θ) (sin θ)) 0 |] [| 0 0 (/ (cos φ) (sin φ)) 0 |] [| (/ (cos θ) (sin θ)) (/ (cos φ) (sin φ)) 0 0 |] [| 0 0 0 (/ (* -1 (sin (* 2 ψ))) 2) |] |]_#_#
+Γ~4_#_#;[| [| 0 0 0 (/ (cos θ) (sin θ)) |] [| 0 0 0 (/ (cos φ) (sin φ)) |] [| 0 0 0 (/ (cos ψ) (sin ψ)) |] [| (/ (cos θ) (sin θ)) (/ (cos φ) (sin φ)) (/ (cos ψ) (sin ψ)) 0 |] |]_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_1_1;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_2;[| [| 0 (* -1 (sin θ)^2) 0 0 |] [| 1 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_3;[| [| 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 |] [| 0 0 0 0 |] [| 1 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_4;[| [| 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2) |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 1 0 0 0 |] |]~#_#
+R~#_#_2_1;[| [| 0 (sin θ)^2 0 0 |] [| -1 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_2;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_3;[| [| 0 0 0 0 |] [| 0 0 (+ (* -1 (sin φ)^2) (* (cos θ)^2 (sin φ)^2)) 0 |] [| 0 (sin θ)^2 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_4;[| [| 0 0 0 0 |] [| 0 0 0 (+ (* -1 (sin φ)^2 (sin ψ)^2) (* (cos θ)^2 (sin φ)^2 (sin ψ)^2)) |] [| 0 0 0 0 |] [| 0 (sin θ)^2 0 0 |] |]~#_#
+R~#_#_3_1;[| [| 0 0 (* (sin θ)^2 (sin φ)^2) 0 |] [| 0 0 0 0 |] [| -1 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_2;[| [| 0 0 0 0 |] [| 0 0 (+ (sin φ)^2 (* -1 (cos θ)^2 (sin φ)^2)) 0 |] [| 0 (* -1 (sin θ)^2) 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_3;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_4;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 (+ (* -1 (sin ψ)^2) (* (cos θ)^2 (sin φ)^2 (sin ψ)^2) (* (cos φ)^2 (sin ψ)^2)) |] [| 0 0 (* (sin θ)^2 (sin φ)^2) 0 |] |]~#_#
+R~#_#_4_1;[| [| 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| -1 0 0 0 |] |]~#_#
+R~#_#_4_2;[| [| 0 0 0 0 |] [| 0 0 0 (+ (* (sin φ)^2 (sin ψ)^2) (* -1 (cos θ)^2 (sin φ)^2 (sin ψ)^2)) |] [| 0 0 0 0 |] [| 0 (* -1 (sin θ)^2) 0 0 |] |]~#_#
+R~#_#_4_3;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 (+ (sin ψ)^2 (* -1 (cos θ)^2 (sin φ)^2 (sin ψ)^2) (* -1 (cos φ)^2 (sin ψ)^2)) |] [| 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 |] |]~#_#
+R~#_#_4_4;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+
+(define $R____ (with-symbols {i} (. g_i_# R~i_#_#_#)))
+
+R_#_#_#_#;(tensor {4 4 4 4} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^2) 0 0 (* r^2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 (* r^2 (sin θ)^2) 0 0 (* -1 r^2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (+ (* (cos θ)^2 (sin θ)^2 r^2 (sin φ)^2) (* -1 r^2 (sin θ)^2 (sin φ)^2)) 0 0 (+ (* -1 (cos θ)^2 (sin θ)^2 r^2 (sin φ)^2) (* r^2 (sin θ)^2 (sin φ)^2)) 0 0 0 0 0 0 0 0 0 0 0 0 0 (+ (* (cos θ)^2 (sin θ)^2 r^2 (sin φ)^2 (sin ψ)^2) (* -1 r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2)) 0 0 0 0 0 (+ (* -1 (cos θ)^2 (sin θ)^2 r^2 (sin φ)^2 (sin ψ)^2) (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2)) 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^4 (sin φ)^2) 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (+ (* (cos θ)^2 (sin θ)^2 r^2 (sin φ)^4 (sin ψ)^2) (* -1 r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) (* r^2 (sin θ)^2 (sin φ)^2 (cos φ)^2 (sin ψ)^2)) 0 0 (+ (* -1 (cos θ)^2 (sin θ)^2 r^2 (sin φ)^4 (sin ψ)^2) (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) (* -1 r^2 (sin θ)^2 (sin φ)^2 (cos φ)^2 (sin ψ)^2)) 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^4 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^4 (sin φ)^4 (sin ψ)^2) 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^4 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )_#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#;[| [| 3 0 0 0 |] [| 0 (* 3 (sin θ)^2) 0 0 |] [| 0 0 (* 3 (sin θ)^2 (sin φ)^2) 0 |] [| 0 0 0 (* 3 (sin θ)^2 (sin φ)^2 (sin ψ)^2) |] |]_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature;(/ 12 r^2)
+
+;;
+;; Covariant derivative of Ricci curvature
+;;
+
+(define $∇Ric___
+  (with-symbols {i j k l m n}
+    (- (∂/∂ Ric_i_j x_m)
+       (. Γ~n_m_i Ric_n_j)
+       (. Γ~n_m_j Ric_i_n)
+       )))
+
+∇Ric_#_#_#
+;(tensor {4 4 4} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )_#_#_#
+
+;;
+;; Conformal curvature tensor
+;;
+
+(define $C_i_k_l_m
+  (+ (. R_i_k_l_m)
+     (+ (- (. Ric_i_m g_k_l) (. Ric_i_l g_k_m))
+        (- (. Ric_k_l g_i_m) (. Ric_k_m g_i_l)))
+     (* (/ scalar-curvature 2) (- (. g_i_l g_k_m) (. g_i_m g_k_l)))))
+
+C_#_#_#_#
+;;(tensor {4 4 4 4} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^2) 0 0 (* -1 r^2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 (* -1 r^2 (sin θ)^2) 0 0 (* r^2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^4 (sin φ)^2) 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^4 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^2 (sin ψ)^2) 0 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^2) 0 0 (* r^2 (sin θ)^4 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^4 (sin φ)^4 (sin ψ)^2) 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^4 (sin ψ)^2) 0 0 0 0 (* -1 r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 (* r^2 (sin θ)^4 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 r^2 (sin θ)^4 (sin φ)^4 (sin ψ)^2) 0 0 (* r^2 (sin θ)^4 (sin φ)^4 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )_#_#_#_#
+
+;;
+;; Pontryagin Class
+;;
+
+(define $P
+  (let {[[$es $os] (even-and-odd-permutations 4)]}
+    (- (sum (map (lambda [$σ] (. R~s_t_(σ 2)_(σ 1) R~t_s_(σ 4)_(σ 3))) es))
+       (sum (map (lambda [$σ] (. R~s_t_(σ 2)_(σ 1) R~t_s_(σ 4)_(σ 3))) os)))))
+
+P;0
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S5-conformal-weyl.egi b/sample/math/geometry/riemann-curvature-tensor-of-S5-conformal-weyl.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S5-conformal-weyl.egi
@@ -0,0 +1,126 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|θ φ ψ η δ|])
+
+(define $X [|(* r (cos θ))
+             (* r (sin θ) (cos φ))
+             (* r (sin θ) (sin φ) (cos ψ))
+             (* r (sin θ) (sin φ) (sin ψ) (cos η))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos δ))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η) (sin δ))
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[| (* -1 r (sin θ)) (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ) (cos ψ)) (* r (cos θ) (sin φ) (sin ψ) (cos η)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (cos δ)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (sin δ)) |]
+;  [| 0 (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ) (cos ψ)) (* r (sin θ) (cos φ) (sin ψ) (cos η)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (cos δ)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (sin δ)) |]
+;  [| 0 0 (* -1 r (sin θ) (sin φ) (sin ψ)) (* r (sin θ) (sin φ) (cos ψ) (cos η)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (cos δ)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (sin δ)) |]
+;  [| 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (cos δ)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (sin δ)) |]
+;  [| 0 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η) (sin δ)) (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos δ)) |] |]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(* (a θ φ ψ η δ)^2 (V.* e_%1 e_%2)) {5 5}))
+(define $g~~ (M.inverse g_#_#))
+g_#_#
+g~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+
+(define $R____ (with-symbols {i} (. g_i_# R~i_#_#_#)))
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature
+;(/ (+ (* 20 (a θ φ ψ η δ)^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -8 (a|1|1 θ φ ψ η δ) (a θ φ ψ η δ) (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -8 (a|2|2 θ φ ψ η δ) (a θ φ ψ η δ) (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -8 (a|3|3 θ φ ψ η δ) (a θ φ ψ η δ) (sin ψ)^2 (sin η)^2)
+;      (* -8 (a|4|4 θ φ ψ η δ) (a θ φ ψ η δ) (sin η)^2)
+;      (* -8 (a|5|5 θ φ ψ η δ) (a θ φ ψ η δ))
+;      (* -4 (a|1 θ φ ψ η δ)^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -4 (a|2 θ φ ψ η δ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -4 (a|3 θ φ ψ η δ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -4 (a|4 θ φ ψ η δ)^2 (sin η)^2)
+;      (* -4 (a|5 θ φ ψ η δ)^2)
+;      (* -32 (a|1 θ φ ψ η δ) (a θ φ ψ η δ) (cos θ) (sin θ) (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -24 (a|2 θ φ ψ η δ) (a θ φ ψ η δ) (cos φ) (sin φ) (sin ψ)^2 (sin η)^2)
+;      (* -16 (a|3 θ φ ψ η δ) (a θ φ ψ η δ) (cos ψ) (sin ψ) (sin η)^2)
+;      (* -8 (a|4 θ φ ψ η δ) (a θ φ ψ η δ) (cos η) (sin η))
+;      )
+;   (* (a θ φ ψ η δ)^4 r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2))
+
+;;
+;; Weyl curvature tensor
+;;
+(define $C_i_k_l_m
+  (+ (. R_i_k_l_m)
+     (+ (- (. Ric_i_m g_k_l) (. Ric_i_l g_k_m))
+        (- (. Ric_k_l g_i_m) (. Ric_k_m g_i_l)))
+     (* (/ scalar-curvature 2) (- (. g_i_l g_k_m) (. g_i_m g_k_l)))))
+
+C_#_#_#_#
+
+(define $C~___ (with-symbols {i} (. g~i~# C_i_#_#_#)))
+C~#_#_#_#
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(let {[[$es $os] (even-and-odd-permutations 5)]}
+  (- (sum' (map (lambda [$σ] (.' C~u_1_s_(σ 1) C~s_t_(σ 3)_(σ 2) C~t_u_(σ 5)_(σ 4))) es))
+     (sum' (map (lambda [$σ] (.' C~u_1_s_(σ 1) C~s_t_(σ 3)_(σ 2) C~t_u_(σ 5)_(σ 4))) os))))
+;0
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S5-conformal.egi b/sample/math/geometry/riemann-curvature-tensor-of-S5-conformal.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S5-conformal.egi
@@ -0,0 +1,110 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|θ φ ψ η δ|])
+
+(define $X [|(* r (cos θ))
+             (* r (sin θ) (cos φ))
+             (* r (sin θ) (sin φ) (cos ψ))
+             (* r (sin θ) (sin φ) (sin ψ) (cos η))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos δ))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η) (sin δ))
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[| (* -1 r (sin θ)) (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ) (cos ψ)) (* r (cos θ) (sin φ) (sin ψ) (cos η)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (cos δ)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (sin δ)) |]
+7;  [| 0 (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ) (cos ψ)) (* r (sin θ) (cos φ) (sin ψ) (cos η)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (cos δ)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (sin δ)) |]
+;  [| 0 0 (* -1 r (sin θ) (sin φ) (sin ψ)) (* r (sin θ) (sin φ) (cos ψ) (cos η)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (cos δ)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (sin δ)) |]
+;  [| 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (cos δ)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (sin δ)) |]
+;  [| 0 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η) (sin δ)) (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos δ)) |] |]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(* (a θ φ ψ η δ)^2 (V.* e_%1 e_%2)) {5 5}))
+(define $g~~ (M.inverse g_#_#))
+g_#_#
+g~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature
+;(/ (+ (* 20 (a θ φ ψ η δ)^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -8 (a|1|1 θ φ ψ η δ) (a θ φ ψ η δ) (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -8 (a|2|2 θ φ ψ η δ) (a θ φ ψ η δ) (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -8 (a|3|3 θ φ ψ η δ) (a θ φ ψ η δ) (sin ψ)^2 (sin η)^2)
+;      (* -8 (a|4|4 θ φ ψ η δ) (a θ φ ψ η δ) (sin η)^2)
+;      (* -8 (a|5|5 θ φ ψ η δ) (a θ φ ψ η δ))
+;      (* -4 (a|1 θ φ ψ η δ)^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -4 (a|2 θ φ ψ η δ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -4 (a|3 θ φ ψ η δ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -4 (a|4 θ φ ψ η δ)^2 (sin η)^2)
+;      (* -4 (a|5 θ φ ψ η δ)^2)
+;      (* -32 (a|1 θ φ ψ η δ) (a θ φ ψ η δ) (cos θ) (sin θ) (sin φ)^2 (sin ψ)^2 (sin η)^2)
+;      (* -24 (a|2 θ φ ψ η δ) (a θ φ ψ η δ) (cos φ) (sin φ) (sin ψ)^2 (sin η)^2)
+;      (* -16 (a|3 θ φ ψ η δ) (a θ φ ψ η δ) (cos ψ) (sin ψ) (sin η)^2)
+;      (* -8 (a|4 θ φ ψ η δ) (a θ φ ψ η δ) (cos η) (sin η))
+;      )
+;   (* (a θ φ ψ η δ)^4 r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2))
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(let {[[$es $os] (even-and-odd-permutations 5)]}
+  (- (sum' (map (lambda [$σ] (debug (.' R~u_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4)))) es))
+     (sum' (map (lambda [$σ] (debug (.' R~u_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4)))) os))))
+;0
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S5-weyl.egi b/sample/math/geometry/riemann-curvature-tensor-of-S5-weyl.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S5-weyl.egi
@@ -0,0 +1,113 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|θ φ ψ η ζ|])
+
+(define $X [|(* r (cos θ))
+             (* r (sin θ) (cos φ))
+             (* r (sin θ) (sin φ) (cos ψ))
+             (* r (sin θ) (sin φ) (sin ψ) (cos η))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos ζ))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η) (sin ζ))
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[| (* -1 r (sin θ)) (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ) (cos ψ)) (* r (cos θ) (sin φ) (sin ψ) (cos η)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (cos ζ)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (sin ζ)) |]
+;  [| 0 (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ) (cos ψ)) (* r (sin θ) (cos φ) (sin ψ) (cos η)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (cos ζ)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (sin ζ)) |]
+;  [| 0 0 (* -1 r (sin θ) (sin φ) (sin ψ)) (* r (sin θ) (sin φ) (cos ψ) (cos η)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (cos ζ)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (sin ζ)) |]
+;  [| 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (cos ζ)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (sin ζ)) |]
+;  [| 0 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η) (sin ζ)) (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos ζ)) |] |]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {5 5}))
+(define $g~~ (M.inverse g_#_#))
+g_#_#
+g~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;[| [| 1 0 0 0 0 |] [| 0 1 0 0 0 |] [| 0 0 1 0 0 |] [| 0 0 0 1 0 |] [| 0 0 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ___
+  (with-symbols {j k l}
+    (* (/ 1 2)
+       (+ (∂/∂ g_j_l x_k)
+          (∂/∂ g_j_k x_l)
+          (* -1 (∂/∂ g_k_l x_j))))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__
+  (with-symbols {i j k l}
+    (. g~i~j Γ_j_k_l)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+;(tensor {5 5 5 5} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2) 0 0 0 (sin θ)^2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 0 0 (* (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (sin θ)^2 0 0 0 (* -1 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (sin θ)^2 0 0 0 0 0 0 0 (* -1 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2) 0 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 (sin θ)^2 0 0 0 0 0 0 0 0 0 0 0 (* -1 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* -1 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 (* -1 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )~#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i j k} (contract + R~i_j_k_i)))
+
+Ric_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature
+
+;;
+;; Weyl curvature tensor
+;;
+(define $δ [| [| 1 0 0 0 0 |] [| 0 1 0 0 0 |] [| 0 0 1 0 0 |] [| 0 0 0 1 0 |] [| 0 0 0 0 1 |] |])
+(define $Ric~_ (with-symbols {i k h} (. g~i~h Ric_k_h)))
+
+(define $C~___
+  (with-symbols {i j k l}
+    (+ R~i_j_k_l
+       (* (/ -1 3) (+ (- (. δ~i_k Ric_j_l) (. δ~i_l Ric_j_k))
+                      (- (. Ric~i_k g_j_l) (. Ric~i_l g_j_k))))
+       (* (/ scalar-curvature 12) (- (. δ~i_k g_j_l) (. δ~i_l g_j_k))))))
+
+C~#_#_#_#
+;(tensor {5 5 5 5} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2) 0 0 0 (* 2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 2 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2) 0 0 0 (* 2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 2 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2) 0 0 0 (* -2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2) 0 0 0 0 0 0 0 (* -2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2) 0 0 0 (* -2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 (* -2 (sin θ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 (* -2 (sin θ)^2 (sin φ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (* 2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 (* -2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )~#_#_#_#
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(let {[[$es $os] (even-and-odd-permutations 5)]}
+  (- (sum' (map (lambda [$σ] (.' C~u_1_s_(σ 1) C~s_t_(σ 3)_(σ 2) C~t_u_(σ 5)_(σ 4))) es))
+     (sum' (map (lambda [$σ] (.' C~u_1_s_(σ 1) C~s_t_(σ 3)_(σ 2) C~t_u_(σ 5)_(σ 4))) os))))
+;0
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S5.egi b/sample/math/geometry/riemann-curvature-tensor-of-S5.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S5.egi
@@ -0,0 +1,109 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|θ φ ψ η δ|])
+
+(define $X [|(* r (cos θ))
+             (* r (sin θ) (cos φ))
+             (* r (sin θ) (sin φ) (cos ψ))
+             (* r (sin θ) (sin φ) (sin ψ) (cos η))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos δ))
+             (* r (sin θ) (sin φ) (sin ψ) (sin η) (sin δ))
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[| (* -1 r (sin θ)) (* r (cos θ) (cos φ)) (* r (cos θ) (sin φ) (cos ψ)) (* r (cos θ) (sin φ) (sin ψ) (cos η)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (cos δ)) (* r (cos θ) (sin φ) (sin ψ) (sin η) (sin δ)) |]
+;  [| 0 (* -1 r (sin θ) (sin φ)) (* r (sin θ) (cos φ) (cos ψ)) (* r (sin θ) (cos φ) (sin ψ) (cos η)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (cos δ)) (* r (sin θ) (cos φ) (sin ψ) (sin η) (sin δ)) |]
+;  [| 0 0 (* -1 r (sin θ) (sin φ) (sin ψ)) (* r (sin θ) (sin φ) (cos ψ) (cos η)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (cos δ)) (* r (sin θ) (sin φ) (cos ψ) (sin η) (sin δ)) |]
+;  [| 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (cos δ)) (* r (sin θ) (sin φ) (sin ψ) (cos η) (sin δ)) |]
+;  [| 0 0 0 0 (* -1 r (sin θ) (sin φ) (sin ψ) (sin η) (sin δ)) (* r (sin θ) (sin φ) (sin ψ) (sin η) (cos δ)) |] |]
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {5 5}))
+(define $g~~ (M.inverse g_#_#))
+g_#_#;[| [| r^2 0 0 0 0 |] [| 0 (* r^2 (sin θ)^2) 0 0 0 |] [| 0 0 (* r^2 (sin θ)^2 (sin φ)^2) 0 0 |] [| 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 |] [| 0 0 0 0 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) |] |]_#_#
+g~#~#;[| [| (/ 1 r^2) 0 0 0 0 |] [| 0 (/ 1 (* r^2 (sin θ)^2)) 0 0 0 |] [| 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2)) 0 0 |] [| 0 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2)) 0 |] [| 0 0 0 0 (/ 1 (* r^2 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2)) |] |]~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;[| [| 1 0 0 0 0 |] [| 0 1 0 0 0 |] [| 0 0 1 0 0 |] [| 0 0 0 1 0 |] [| 0 0 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_#_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~#_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#
+
+(define $R____ (with-symbols {i} (. g_i_# R~i_#_#_#)))
+
+R_#_#_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#;[| [| 4 0 0 0 0 |] [| 0 (* 4 (sin θ)^2) 0 0 0 |] [| 0 0 (* 4 (sin θ)^2 (sin φ)^2) 0 0 |] [| 0 0 0 (* 4 (sin θ)^2 (sin φ)^2 (sin ψ)^2) 0 |] [| 0 0 0 0 (* 4 (sin θ)^2 (sin φ)^2 (sin ψ)^2 (sin η)^2) |] |]_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature;(/ 20 r^2)
+
+;;
+;; Conformal curvature tensor
+;;
+
+(define $C_i_k_l_m
+  (+ (. R_i_k_l_m)
+     (+ (- (. Ric_i_m g_k_l) (. Ric_i_l g_k_m))
+        (- (. Ric_k_l g_i_m) (. Ric_k_m g_i_l)))
+     (* (/ scalar-curvature 2) (- (. g_i_l g_k_m) (. g_i_m g_k_l)))))
+
+C_#_#_#_#
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(let {[[$es $os] (even-and-odd-permutations 5)]}
+  (- (sum (map (lambda [$σ] (. R~u_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4))) es))
+     (sum (map (lambda [$σ] (. R~u_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4))) os))))
+;0
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S7-conformal.egi b/sample/math/geometry/riemann-curvature-tensor-of-S7-conformal.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S7-conformal.egi
@@ -0,0 +1,81 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|α β γ δ ε ζ η|])
+
+(define $X [|(* r (cos α))
+             (* r (sin α) (cos β))
+             (* r (sin α) (sin β) (cos γ))
+             (* r (sin α) (sin β) (sin γ) (cos δ))
+             (* r (sin α) (sin β) (sin γ) (sin δ) (cos ε))
+             (* r (sin α) (sin β) (sin γ) (sin δ) (sin ε) (cos ζ))
+             (* r (sin α) (sin β) (sin γ) (sin δ) (sin ε) (sin ζ) (cos η))
+             (* r (sin α) (sin β) (sin γ) (sin δ) (sin ε) (sin ζ) (sin η))
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(* (a α β γ δ ε ζ η)^2 (V.* e_%1 e_%2)) {7 7}))
+(define $g~~ (M.inverse g_#_#))
+g_#_#;
+g~#~#;
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#;
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(let {[[$es $os] (even-and-odd-permutations 7)]}
+  (- (sum (map (lambda [$σ] (debug (. R~v_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4) R~u_v_(σ 7)_(σ 6)))) es))
+     (sum (map (lambda [$σ] (debug (. R~v_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4) R~u_v_(σ 7)_(σ 6)))) os))))
+;
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-S7.egi b/sample/math/geometry/riemann-curvature-tensor-of-S7.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-S7.egi
@@ -0,0 +1,92 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|α β γ δ ε ζ η|])
+
+(define $X [|(* r (cos α))
+             (* r (sin α) (cos β))
+             (* r (sin α) (sin β) (cos γ))
+             (* r (sin α) (sin β) (sin γ) (cos δ))
+             (* r (sin α) (sin β) (sin γ) (sin δ) (cos ε))
+             (* r (sin α) (sin β) (sin γ) (sin δ) (sin ε) (cos ζ))
+             (* r (sin α) (sin β) (sin γ) (sin δ) (sin ε) (sin ζ) (cos η))
+             (* r (sin α) (sin β) (sin γ) (sin δ) (sin ε) (sin ζ) (sin η))
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {7 7}))
+(define $g~~ (M.inverse g_#_#))
+g_#_#;[| [| r^2 0 0 0 0 0 0 |] [| 0 (* r^2 (sin α)^2) 0 0 0 0 0 |] [| 0 0 (* r^2 (sin α)^2 (sin β)^2) 0 0 0 0 |] [| 0 0 0 (* r^2 (sin α)^2 (sin β)^2 (sin γ)^2) 0 0 0 |] [| 0 0 0 0 (* r^2 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2) 0 0 |] [| 0 0 0 0 0 (* r^2 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2 (sin ε)^2) 0 |] [| 0 0 0 0 0 0 (* r^2 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2 (sin ε)^2 (sin ζ)^2) |] |]_#_#
+g~#~#;[| [| (/ 1 r^2) 0 0 0 0 0 0 |] [| 0 (/ 1 (* r^2 (sin α)^2)) 0 0 0 0 0 |] [| 0 0 (/ 1 (* r^2 (sin α)^2 (sin β)^2)) 0 0 0 0 |] [| 0 0 0 (/ 1 (* r^2 (sin α)^2 (sin β)^2 (sin γ)^2)) 0 0 0 |] [| 0 0 0 0 (/ 1 (* r^2 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2)) 0 0 |] [| 0 0 0 0 0 (/ 1 (* r^2 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2 (sin ε)^2)) 0 |] [| 0 0 0 0 0 0 (/ 1 (* r^2 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2 (sin ε)^2 (sin ζ)^2)) |] |]~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;[| [| 1 0 0 0 0 0 0 |] [| 0 1 0 0 0 0 0 |] [| 0 0 1 0 0 0 0 |] [| 0 0 0 1 0 0 0 |] [| 0 0 0 0 1 0 0 |] [| 0 0 0 0 0 1 0 |] [| 0 0 0 0 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_l x_k)
+        (∂/∂ g_j_k x_l)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#;
+;[|[| 6 0 0 0 0 0 0 |]
+;  [| 0 (* 6 (sin α)^2) 0 0 0 0 0 |]
+;  [| 0 0 (* 6 (sin α)^2 (sin β)^2) 0 0 0 0 |]
+;  [| 0 0 0 (* 6 (sin α)^2 (sin β)^2 (sin γ)^2) 0 0 0 |]
+;  [| 0 0 0 0 (* 6 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2) 0 0 |]
+;  [| 0 0 0 0 0 (* 6 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2 (sin ε)^2) 0 |]
+;  [| 0 0 0 0 0 0 (* 6 (sin α)^2 (sin β)^2 (sin γ)^2 (sin δ)^2 (sin ε)^2 (sin ζ)^2) |]
+;  |]_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature;(/ 42 r^2)
+
+;;
+;; Wodzicki-Chern-Simons class
+;;
+
+(let {[[$es $os] (even-and-odd-permutations 7)]}
+  (- (sum (map (lambda [$σ] (. R~v_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4) R~u_v_(σ 7)_(σ 6))) es))
+     (sum (map (lambda [$σ] (. R~v_1_s_(σ 1) R~s_t_(σ 3)_(σ 2) R~t_u_(σ 5)_(σ 4) R~u_v_(σ 7)_(σ 6))) os))))
+;0
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-Schwarzschild-metric.egi b/sample/math/geometry/riemann-curvature-tensor-of-Schwarzschild-metric.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-Schwarzschild-metric.egi
@@ -0,0 +1,87 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|t r θ φ|])
+
+;;
+;; Metric tensor
+;;
+
+(define $g__
+  [|[| (/ '(- (* c^2 r) (* 2 G M)) (* c^2 r)) 0 0 0 |]
+    [| 0 (/ -1 (/ '(- (* c^2 r) (* 2 G M)) (* c^2 r))) 0 0 |]
+    [| 0 0 (* -1 r^2) 0 |]
+    [| 0 0 0 (* -1 r^2 (sin θ)^2) |]
+    |])
+
+(define $g~~ (M.inverse g_#_#))
+g~#~#
+;[|[| (/ (* c^2 r) '(+ (* c^2 r) (* -2 G M))) 0 0 0 |]
+;  [| 0 (/ (* -1 '(+ (* c^2 r) (* -2 G M))) (* c^2 r)) 0 0 |]
+;  [| 0 0 (/ -1 r^2) 0 |]
+;  [| 0 0 0 (/ -1 (* r^2 (sin θ)^2)) |]|]~#~#
+
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;[| [| 1 0 0 0 |] [| 0 1 0 0 |] [| 0 0 1 0 |] [| 0 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_j_k_l
+  (* (/ 1 2)
+     (+ (∂/∂ g_j_k x_l)
+        (∂/∂ g_j_l x_k)
+        (* -1 (∂/∂ g_k_l x_j)))))
+
+Γ_1_#_#;[| [| 0 (/ (+ (* c^2 r) (* -1 '(+ (* c^2 r) (* -2 G M)))) (* 2 c^2 r^2)) 0 0 |] [| (/ (+ (* c^2 r) (* -1 '(+ (* c^2 r) (* -2 G M)))) (* 2 c^2 r^2)) 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#_#
+Γ_2_#_#;[| [| (/ (+ (* -1 c^2 r) '(+ (* c^2 r) (* -2 G M))) (* 2 c^2 r^2)) 0 0 0 |] [| 0 (/ (+ (* -1 c^2 '(+ (* c^2 r) (* -2 G M))) (* c^4 r)) (* 2 '(+ (* c^2 r) (* -2 G M))^2)) 0 0 |] [| 0 0 r 0 |] [| 0 0 0 (* r (sin θ)^2) |] |]_#_#
+Γ_3_#_#;[| [| 0 0 0 0 |] [| 0 0 (* -1 r) 0 |] [| 0 (* -1 r) 0 0 |] [| 0 0 0 (* r^2 (sin θ) (cos θ)) |] |]_#_#
+Γ_4_#_#;[| [| 0 0 0 0 |] [| 0 0 0 (* -1 r (sin θ)^2) |] [| 0 0 0 (* -1 r^2 (sin θ) (cos θ)) |] [| 0 (* -1 r (sin θ)^2) (* -1 r^2 (sin θ) (cos θ)) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~1_#_#;[| [| 0 (/ (+ (* c^2 r) (* -1 '(+ (* c^2 r) (* -2 G M)))) (* 2 '(+ (* c^2 r) (* -2 G M)) r)) 0 0 |] [| (/ (+ (* c^2 r) (* -1 '(+ (* c^2 r) (* -2 G M)))) (* 2 '(+ (* c^2 r) (* -2 G M)) r)) 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#_#
+Γ~2_#_#;[| [| (/ (+ (* '(+ (* c^2 r) (* -2 G M)) c^2 r) (* -1 '(+ (* c^2 r) (* -2 G M))^2)) (* 2 c^4 r^3)) 0 0 0 |] [| 0 (/ (+ '(+ (* c^2 r) (* -2 G M)) (* -1 c^2 r)) (* 2 r '(+ (* c^2 r) (* -2 G M)))) 0 0 |] [| 0 0 (/ (* -1 '(+ (* c^2 r) (* -2 G M))) c^2) 0 |] [| 0 0 0 (/ (* -1 '(+ (* c^2 r) (* -2 G M)) (sin θ)^2) c^2) |] |]_#_#
+Γ~3_#_#;[| [| 0 0 0 0 |] [| 0 0 (/ 1 r) 0 |] [| 0 (/ 1 r) 0 0 |] [| 0 0 0 (* -1 (sin θ) (cos θ)) |] |]_#_#
+Γ~4_#_#;[| [| 0 0 0 0 |] [| 0 0 0 (/ 1 r) |] [| 0 0 0 (/ (cos θ) (sin θ)) |] [| 0 (/ 1 r) (/ (cos θ) (sin θ)) 0 |] |]_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (expand-all (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+                   (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l))))))
+
+R~#_#_1_1;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_2;[| [| 0 (/ (* 2 G M) (+ (* c^2 r^3) (* -2 G M r^2))) 0 0 |] [| (/ (+ (* 2 G M c^2 r) (* -4 G^2 M^2)) (* c^4 r^4)) 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_3;[| [| 0 0 (/ (* -1 G M) (* c^2 r)) 0 |] [| 0 0 0 0 |] [| (/ (+ (* -1 G M c^2 r) (* 2 G^2 M^2)) (* c^4 r^4)) 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_1_4;[| [| 0 0 0 (/ (* -1 G M (sin θ)^2) (* c^2 r)) |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| (/ (+ (* -1 G M c^2 r) (* 2 G^2 M^2)) (* c^4 r^4)) 0 0 0 |] |]~#_#
+R~#_#_2_1;[| [| 0 (/ (* -2 G M) (+ (* c^2 r^3) (* -2 G M r^2))) 0 0 |] [| (/ (+ (* -2 G M c^2 r) (* 4 G^2 M^2)) (* c^4 r^4)) 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_2;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_3;[| [| 0 0 0 0 |] [| 0 0 (/ (* -1 G M) (* c^2 r)) 0 |] [| 0 (/ (* G M) (+ (* r^3 c^2) (* -2 r^2 G M))) 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_2_4;[| [| 0 0 0 0 |] [| 0 0 0 (/ (* -1 G M (sin θ)^2) (* c^2 r)) |] [| 0 0 0 0 |] [| 0 (/ (* G M) (+ (* r^3 c^2) (* -2 r^2 G M))) 0 0 |] |]~#_#
+R~#_#_3_1;[| [| 0 0 (/ (* G M) (* c^2 r)) 0 |] [| 0 0 0 0 |] [| (/ (+ (* G M c^2 r) (* -2 G^2 M^2)) (* c^4 r^4)) 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_2;[| [| 0 0 0 0 |] [| 0 0 (/ (* G M) (* r c^2)) 0 |] [| 0 (/ (* -1 G M) (+ (* r^3 c^2) (* -2 r^2 G M))) 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_3;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+R~#_#_3_4;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 (/ (* 2 G M (sin θ)^2) (* c^2 r)) |] [| 0 0 (/ (* -2 G M) (* c^2 r)) 0 |] |]~#_#
+R~#_#_4_1;[| [| 0 0 0 (/ (* G M (sin θ)^2) (* c^2 r)) |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| (/ (+ (* G M c^2 r) (* -2 G^2 M^2)) (* c^4 r^4)) 0 0 0 |] |]~#_#
+R~#_#_4_2;[| [| 0 0 0 0 |] [| 0 0 0 (/ (* G M (sin θ)^2) (* r c^2)) |] [| 0 0 0 0 |] [| 0 (/ (* -1 G M) (+ (* r^3 c^2) (* -2 r^2 G M))) 0 0 |] |]~#_#
+R~#_#_4_3;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 (/ (* -2 G M (sin θ)^2) (* c^2 r)) |] [| 0 0 (/ (* 2 G M) (* c^2 r)) 0 |] |]~#_#
+R~#_#_4_4;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]~#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#_#
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-T2.egi b/sample/math/geometry/riemann-curvature-tensor-of-T2.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-T2.egi
@@ -0,0 +1,126 @@
+;;;
+;;; Coordinates for Torus
+;;;
+
+(define $x [|θ φ|])
+
+(define $X [|(* '(+ (* a (cos θ)) b) (cos φ)) ; = x
+             (* '(+ (* a (cos θ)) b) (sin φ)) ; = y
+             (* a (sin θ))                    ; = z
+             |])
+
+;;
+;; Local basis
+;;
+
+(define $e ((flip ∂/∂) x~# X_#))
+e
+;[|[| (* -1 a (sin θ) (cos φ)) (* -1 a (sin θ) (sin φ)) (* a (cos θ)) |]
+;  [| (* -1 '(+ (* a (cos θ)) b) (sin φ)) (* '(+ (* a (cos θ)) b) (cos φ)) 0 |]
+;  |]~#~#
+
+;;
+;; Metric tensor
+;;
+
+(define $g__ (generate-tensor 2#(V.* e_%1 e_%2) {2 2}))
+(define $g~~ (M.inverse g_#_#))
+
+g_#_#;[| [| a^2 0 |] [| 0 '(+ (* a (cos θ)) b)^2 |] |]_#_#
+g~#~#;[| [| (/ 1 a^2) 0 |] [| 0 (/ 1 '(+ (* a (cos θ)) b)^2) |] |]~#~#
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ_i_j_k
+  (* (/ 1 2)
+     (+ (∂/∂ g_i_j x_k)
+        (∂/∂ g_i_k x_j)
+        (* -1 (∂/∂ g_j_k x_i)))))
+
+Γ_#_#_#;(tensor {2 2 2} {0 0 0 (* '(+ (* a (cos θ)) b) a (sin θ)) 0 (* -1 '(+ (* a (cos θ)) b) a (sin θ)) (* -1 '(+ (* a (cos θ)) b) a (sin θ)) 0} )_#_#_#
+Γ_1_#_#;[| [| 0 0 |] [| 0 (* '(+ (* a (cos θ)) b) a (sin θ)) |] |]_#_#
+Γ_2_#_#;[| [| 0 (* -1 '(+ (* a (cos θ)) b) a (sin θ)) |] [| (* -1 '(+ (* a (cos θ)) b) a (sin θ)) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__ (with-symbols {i} (. g~#~i Γ_i_#_#)))
+
+Γ~#_#_#;(tensor {2 2 2} {0 0 0 (/ (* '(+ (* a (cos θ)) b) (sin θ)) a) 0 (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) 0} )~#_#_#
+Γ~1_#_#;[| [| 0 0 |] [| 0 (/ (* '(+ (* a (cos θ)) b) (sin θ)) a) |] |]_#_#
+Γ~2_#_#;[| [| 0 (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) |] [| (/ (* -1 a (sin θ)) '(+ (* a (cos θ)) b)) 0 |] |]_#_#
+
+;;
+;; Covariant derivative of metric tensor
+;;
+(define $∇g___
+  (with-symbols {i j m n}
+    (- (∂/∂ g_i_j x_m)
+       (. Γ~n_m_i g_n_j)
+       (. Γ~n_m_j g_i_n))))
+
+∇g_#_#_#;=>(tensor {2 2 2} {0 0 0 0 0 0 0 0} )
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#;(tensor {2 2 2 2} {0 0 0 0 0 (/ (* '(+ (* a (cos θ)) b) (cos θ)) a) (/ (* -1 '(+ (* a (cos θ)) b) (cos θ)) a) 0 0 (/ (* -1 a (cos θ)) '(+ (* a (cos θ)) b)) (/ (* a (cos θ)) '(+ (* a (cos θ)) b)) 0 0 0 0 0} )~#_#_#_#
+R~#_#_1_1;[| [| 0 0 |] [| 0 0 |] |]~#_#
+R~#_#_1_2;[| [| 0 (/ (* '(+ (* a (cos θ)) b) (cos θ)) a) |] [| (/ (* -1 a (cos θ)) '(+ (* a (cos θ)) b)) 0 |] |]~#_#
+R~#_#_2_1;[| [| 0 (/ (* -1 '(+ (* a (cos θ)) b) (cos θ)) a) |] [| (/ (* a (cos θ)) '(+ (* a (cos θ)) b)) 0 |] |]~#_#
+R~#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]~#_#
+
+(define $R____ (with-symbols {i} (. g_i_# R~i_#_#_#)))
+
+R_#_#_#_#;(tensor {2 2 2 2} {0 0 0 0 0 (* a '(+ (* a (cos θ)) b) (cos θ)) (* -1 a '(+ (* a (cos θ)) b) (cos θ)) 0 0 (* -1 '(+ (* a (cos θ)) b) a (cos θ)) (* '(+ (* a (cos θ)) b) a (cos θ)) 0 0 0 0 0} )_#_#_#_#
+R_#_#_1_1;[| [| 0 0 |] [| 0 0 |] |]_#_#
+R_#_#_1_2;[| [| 0 (* a '(+ (* a (cos θ)) b) (cos θ)) |] [| (* -1 '(+ (* a (cos θ)) b) a (cos θ)) 0 |] |]_#_#
+R_#_#_2_1;[| [| 0 (* -1 a '(+ (* a (cos θ)) b) (cos θ)) |] [| (* '(+ (* a (cos θ)) b) a (cos θ)) 0 |] |]_#_#
+R_#_#_2_2;[| [| 0 0 |] [| 0 0 |] |]_#_#
+
+;;
+;; Ricci curvature
+;;
+
+(define $Ric__ (with-symbols {i} (contract + R~i_#_i_#)))
+
+Ric_#_#;[| [| (/ (* a (cos θ)) '(+ (* a (cos θ)) b)) 0 |] [| 0 (/ (* '(+ (* a (cos θ)) b) (cos θ)) a) |] |]_#_#
+
+;;
+;; Scalar curvature
+;;
+
+(define $scalar-curvature (with-symbols {j k} (. g~j~k Ric_j_k)))
+
+scalar-curvature;(/ (* 2 (cos θ)) (* a '(+ (* a (cos θ)) b)))
+
+;;
+;; Covariant derivative of Riemann curvature tensor
+;;
+
+(define $∇R_____
+  (with-symbols {i j k l m n}
+    (- (∂/∂ R_i_j_k_l x_m)
+       (. Γ~n_m_i R_n_j_k_l)
+       (. Γ~n_m_j R_i_n_k_l)
+       (. Γ~n_m_k R_i_j_n_l)
+       (. Γ~n_m_l R_i_j_k_n))))
+
+∇R_#_#_#_#_#
+;(tensor {2 2 2 2 2} {0 0 0 0 0 0 0 0 0 0 (+ (* -1 a '(+ (* a (cos θ)) b) (sin θ)) (* a^2 (sin θ) (cos θ))) 0 (+ (* a '(+ (* a (cos θ)) b) (sin θ)) (* -1 a^2 (sin θ) (cos θ))) 0 0 0 0 0 (+ (* '(+ (* a (cos θ)) b) a (sin θ)) (* -1 a^2 (sin θ) (cos θ))) 0 (+ (* -1 '(+ (* a (cos θ)) b) a (sin θ)) (* a^2 (sin θ) (cos θ))) 0 0 0 0 0 0 0 0 0 0 0} )_#_#_#_#_#
+
+;;
+;; Second Bianchi identity
+;;
+
+(with-symbols {i j k l m} (+ ∇R_i_j_k_l_m ∇R_i_j_l_m_k ∇R_i_j_m_k_l))
+;(tensor {2 2 2 2 2} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-empty-Schwarzschild-spacetime.egi b/sample/math/geometry/riemann-curvature-tensor-of-empty-Schwarzschild-spacetime.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-empty-Schwarzschild-spacetime.egi
@@ -0,0 +1,68 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|t r θ φ|])
+
+;;
+;; Metric tensor
+;;
+
+(define $g__
+  [|[| 1 0 0 0 |]
+    [| 0 -1 0 0 |]
+    [| 0 0 (* -1 r^2) 0 |]
+    [| 0 0 0 (* -1 r^2 (sin θ)^2) |]
+    |])
+
+(define $g~~ (M.inverse g_#_#))
+g~#~#
+;[|[| 1 0 0 0 |]
+;  [| 0 -1 0 0 |]
+;  [| 0 0 (/ 1 (* -1 r^2)) 0 |]
+;  [| 0 0 0 (/ 1 (* -1 r^2 (sin θ)^2)) |]
+;  |]~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;[| [| 1 0 0 0 |] [| 0 1 0 0 |] [| 0 0 1 0 |] [| 0 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ___
+  (with-symbols {j k l}
+    (* (/ 1 2)
+       (+ (∂/∂ g_j_l x_k)
+          (∂/∂ g_j_k x_l)
+          (* -1 (∂/∂ g_k_l x_j))))))
+
+Γ_1_#_#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#_#
+Γ_2_#_#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 r 0 |] [| 0 0 0 (* r (sin θ)^2) |] |]_#_#
+Γ_3_#_#;[| [| 0 0 0 0 |] [| 0 0 (* -1 r) 0 |] [| 0 (* -1 r) 0 0 |] [| 0 0 0 (* r^2 (sin θ) (cos θ)) |] |]_#_#
+Γ_4_#_#;[| [| 0 0 0 0 |] [| 0 0 0 (* -1 r (sin θ)^2) |] [| 0 0 0 (* -1 r^2 (sin θ) (cos θ)) |] [| 0 (* -1 r (sin θ)^2) (* -1 r^2 (sin θ) (cos θ)) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__
+  (with-symbols {i j k l}
+    (. g~i~j Γ_j_k_l)))
+
+Γ~1_#_#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]_#_#
+Γ~2_#_#;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 (* -1 r) 0 |] [| 0 0 0 (* -1 r (sin θ)^2) |] |]_#_#
+Γ~3_#_#;[| [| 0 0 0 0 |] [| 0 0 (/ -1 (* -1 r)) 0 |] [| 0 (/ -1 (* -1 r)) 0 0 |] [| 0 0 0 (* -1 (sin θ) (cos θ)) |] |]_#_#
+Γ~4_#_#;[| [| 0 0 0 0 |] [| 0 0 0 (/ -1 (* -1 r)) |] [| 0 0 0 (/ (* -1 (cos θ)) (* -1 (sin θ))) |] [| 0 (/ -1 (* -1 r)) (/ (* -1 (cos θ)) (* -1 (sin θ))) 0 |] |]_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+
+R~#_#_#_#;(tensor {4 4 4 4} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )~#_#_#_#
diff --git a/sample/math/geometry/riemann-curvature-tensor-of-spherical-space.egi b/sample/math/geometry/riemann-curvature-tensor-of-spherical-space.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/riemann-curvature-tensor-of-spherical-space.egi
@@ -0,0 +1,62 @@
+;;;
+;;; Parameters
+;;;
+
+(define $x [|r θ φ|])
+
+;;
+;; Metric tensor
+;;
+
+(define $g__
+  [|[| 1 0 0 |]
+    [| 0 r^2 0 |]
+    [| 0 0 (* r^2 (sin θ)^2) |]
+    |])
+
+(define $g~~ (M.inverse g_#_#))
+g~#~#
+;[|[| 1 0 0 |]
+;  [| 0 (/ 1 r^2) 0 |]
+;  [| 0 0 (/ 1 (* r^2 (sin θ)^2)) |]|]~#~#
+
+(with-symbols {i j k} (. g~i~j g_j_k))
+;[| [| 1 0 0 |] [| 0 1 0 |] [| 0 0 1 |] |]
+
+;;
+;; Christoffel symbols of the first kind
+;;
+
+(define $Γ___
+  (with-symbols {j k l}
+    (* (/ 1 2)
+       (+ (∂/∂ g_j_l x_k)
+          (∂/∂ g_j_k x_l)
+          (* -1 (∂/∂ g_k_l x_j))))))
+
+Γ_1_#_#;[| [| 0 0 0 |] [| 0 (* -1 r) 0 |] [| 0 0 (* -1 r (sin θ)^2) |] |]_#_#
+Γ_2_#_#;[| [| 0 r 0 |] [| r 0 0 |] [| 0 0 (* -1 r^2 (sin θ) (cos θ)) |] |]_#_#
+Γ_3_#_#;[| [| 0 0 (* r (sin θ)^2) |] [| 0 0 (* r^2 (sin θ) (cos θ)) |] [| (* r (sin θ)^2) (* r^2 (sin θ) (cos θ)) 0 |] |]_#_#
+
+;;
+;; Christoffel symbols of the second kind
+;;
+
+(define $Γ~__
+  (with-symbols {i j k l}
+    (. g~i~j Γ_j_k_l)))
+
+Γ~1_#_#;[| [| 0 0 0 |] [| 0 (* -1 r) 0 |] [| 0 0 (* -1 r (sin θ)^2) |] |]_#_#
+Γ~2_#_#;[| [| 0 (/ 1 r) 0 |] [| (/ 1 r) 0 0 |] [| 0 0 (* -1 (sin θ) (cos θ)) |] |]_#_#
+Γ~3_#_#;[| [| 0 0 (/ 1 r) |] [| 0 0 (/ (cos θ) (sin θ)) |] [| (/ 1 r) (/ (cos θ) (sin θ)) 0 |] |]_#_#
+
+;;
+;; Riemann curvature tensor
+;;
+
+(define $R~i_j_k_l
+  (with-symbols {m}
+    (+ (- (∂/∂ Γ~i_j_l x_k) (∂/∂ Γ~i_j_k x_l))
+       (- (. Γ~m_j_l Γ~i_m_k) (. Γ~m_j_k Γ~i_m_l)))))
+
+R~#_#_#_#;(tensor {3 3 3 3} {0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} )~#_#_#_#
diff --git a/sample/math/geometry/surface.egi b/sample/math/geometry/surface.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/surface.egi
@@ -0,0 +1,55 @@
+; We can bound f to a specific function.
+; (define $f (lambda [$x $y] (+ (** x 2) (** y 2))))
+
+(define $v1 [| 1 0 (∂/∂ (f x y) x) |])
+(define $v2 [| 0 1 (∂/∂ (f x y) y) |])
+
+v1;[| 1 0 (f|1 x y) |]
+v2;[| 0 1 (f|2 x y) |]
+
+(define $v3 (cross-product v1 v2))
+
+v3;[| (* -1 (f|1 x y)) (* -1 (f|2 x y)) 1 |]
+
+(define $e3 (/ v3 (sqrt '(V.* v3 v3))))
+
+e3
+;[|(/ (* -1 (f|1 x y))
+;     (sqrt '(+ (f|1 x y)^2 (f|2 x y)^2 1)))
+;  (/ (* -1 (f|2 x y))
+;     (sqrt '(+ (f|1 x y)^2 (f|2 x y)^2 1)))
+;  (/ 1
+;     (sqrt '(+ (f|1 x y)^2 (f|2 x y)^2 1)))|]
+
+(define $E (V.* v1 v1))
+(define $F (V.* v1 v2))
+(define $G (V.* v2 v2))
+
+E;(+ 1 (f|1 x y)^2)
+F;(* (f|1 x y) (f|2 x y)
+G;(+ 1 (f|2 x y)^2)
+
+(define $L (V.* (∂/∂ v1 x) e3))
+(define $M (V.* (∂/∂ v1 y) e3))
+;(define $M (V.* (∂/∂ v2 x) e3))
+(define $N (V.* (∂/∂ v2 y) e3))
+
+L;(/ (f|1|1 x y) (sqrt '(+ (f|1 x y)^2 (f|2 x y)^2 1)))
+M;(/ (f|1|2 x y) (sqrt '(+ (f|1 x y)^2 (f|2 x y)^2 1)))
+N;(/ (f|2|2 x y) (sqrt '(+ (f|1 x y)^2 (f|2 x y)^2 1)))
+
+(define $K (/ (- (* L N) (** M 2))
+              '(- (* E G) (** F 2))))
+(define $H (/ (+ (* 'E N) (* 'G L) (* -2 F M))
+              (* 2 '(- (* E G) (** F 2)))))
+
+K
+;(/ (+ (* (f|1|1 x y) (f|2|2 x y)) (* -1 (f|1|2 x y)^2))
+;   '(+ (f|1 x y)^2 (f|2 x y)^2 1)^2)
+H
+;(/ (+ (* '(+ 1 (f|1 x y)^2) (f|2|2 x y))
+;      (* '(+ 1 (f|2 x y)^2) (f|1|1 x y))
+;      (* -2 (f|1 x y) (f|2 x y) (f|1|2 x y)))
+;   (* 2
+;      (sqrt '(+ (f|1 x y)^2 (f|2 x y)^2 1))
+;      '(+ 1 (f|2 x y)^2 (f|1 x y)^2)))
diff --git a/sample/math/geometry/trigonometric-identities.egi b/sample/math/geometry/trigonometric-identities.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/trigonometric-identities.egi
@@ -0,0 +1,42 @@
+(coefficients (* (+ (cos α) (* i (sin α))) (+ (cos β) (* i (sin β))))
+              i)
+;{(+ (* (cos α) (cos β)) (* -1 (sin α) (sin β))) (+ (* (cos α) (sin β)) (* (sin α) (cos β)))}
+
+;(cos (+ α β)) = (+ (* (cos α) (cos β)) (* -1 (sin α) (sin β)))
+;(sin (+ α β)) = (+ (* (cos α) (sin β)) (* (sin α) (cos β)))
+
+
+(coefficients (* (+ (cos α) (* i (sin α))) (- (cos β) (* i (sin β))))
+              i)
+;{(+ (* (cos α) (cos β)) (* (sin α) (sin β))) (+ (* -1 (cos α) (sin β)) (* (sin α) (cos β)))}
+
+;(cos (- α β)) = (+ (* (cos α) (cos β)) (* (sin α) (sin β)))
+;(sin (- α β)) = (+ (* -1 (cos α) (sin β)) (* (sin α) (cos β)))
+
+
+(coefficients (+ (* (+ (cos α) (* i (sin α))) (+ (cos β) (* i (sin β))))
+                 (* (+ (cos α) (* i (sin α))) (- (cos β) (* i (sin β)))))
+              i)
+;{(* 2 (cos α) (cos β)) (* 2 (sin α) (cos β))}
+
+;(* (cos α) (cos β)) = (* (/ 1 2) (+ (cos (+ α β)) (cos (- α β))))
+;(* (sin α) (cos β)) = (* (/ 1 2) (+ (sin (+ α β)) (sin (- α β))))
+
+
+(coefficients (- (* (+ (cos α) (* i (sin α))) (+ (cos β) (* i (sin β))))
+                 (* (+ (cos α) (* i (sin α))) (- (cos β) (* i (sin β)))))
+              i)
+;{(* -2 (sin α) (sin β)) (* 2 (cos α) (sin β))}
+
+;(* (sin α) (sin β)) = (* (/ -1 2) (- (cos (+ α β)) (cos (- α β))))
+;(* (cos α) (sin β)) = (* (/  1 2) (- (sin (+ α β)) (sin (- α β))))
+
+
+(coefficients (** (+ (cos α) (* i (sin α))) 3)
+              i)
+;{(+ (cos α)^3 (* -3 (cos α) (sin α)^2)) (+ (* 3 (cos α)^2 (sin α)) (* -1 (sin α)^3))}
+
+;(cos (* 3 α)) = (+ (cos α)^3 (* -3 (cos α) (sin α)^2))
+;              = (+ (* 4 (cos α)^3) (* -3 (cos α)))
+;(sin (* 3 α)) = (+ (* 3 (cos α)^2 (sin α)) (* -1 (sin α)^3))
+;              = (+ (* -4 (sin α)^3 (* 3 (sin α))))
diff --git a/sample/math/geometry/vector-analysis.egi b/sample/math/geometry/vector-analysis.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/vector-analysis.egi
@@ -0,0 +1,57 @@
+(define $N 3)
+
+(define $g [| [| 1 0 0 |] [| 0 1 0 |] [| 0 0 1 |] |])
+
+(define $d
+  (lambda [%X]
+    !((flip ∂/∂) [| x y z |] X)))
+
+(define $hodge
+  (lambda [%A]
+    (let {[$k (df-order A)]}
+      (with-symbols {i j}
+        (* (sqrt (abs (M.det g_#_#)))
+           (foldl . (. (subrefs A (map 1#j_%1 (between 1 k)))
+                       (subrefs (ε' N k) (map 1#i_%1 (between 1 N))))
+                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))
+
+(define $δ
+  (lambda [%A]
+    (let {[$r (df-order A)]}
+      (* (** -1 (+ (* N r) 1))
+         (hodge (d (hodge A)))))))
+
+(define $grad d)
+(define $rot d)
+(define $div δ)
+
+(define $Δ
+  (lambda [%A]
+    (match (tensor-order A) integer
+      {[,0 (δ (d A))]
+       [,N (d (δ A))]
+       [_ (+ (d (δ A)) (δ (d A)))]})))
+
+(grad (+ (** x 2) (** y 2) (** z 2)))
+;[| (* 2 x) (* 2 y) (* 2 z) |]
+
+(rot [| (** y 2) (** x 2) 0 |])
+;[| [| 0 (* 2 x) 0 |] [| (* 2 y) 0 0 |] [| 0 0 0 |] |]
+
+(div [| (** y 2) (** x 2) 0 |])
+;0
+
+(rot [| (** x 2) (** y 2) (** z 2) |])
+;[| [| (* 2 x) 0 0 |] [| 0 (* 2 y) 0 |] [| 0 0 (* 2 z) |] |]
+
+(div [| (** x 2) (** y 2) (** z 2) |])
+;(+ (* 2 z) (* 2 y) (* 2 x))
+
+(rot [| (* x 2) (* y 2) (* z 2) |])
+;[| [| 2 0 0 |] [| 0 2 0 |] [| 0 0 2 |] |]
+
+(div [| (* x 2) (* y 2) (* z 2) |])
+;6
+
+(Δ (+ (** x 2) (** y 2) (** z 2)))
+;6
diff --git a/sample/math/geometry/wedge-product.egi b/sample/math/geometry/wedge-product.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/wedge-product.egi
@@ -0,0 +1,23 @@
+(define $N 3)
+(define $params [| x y z |])
+(define $g [| [| 1 0 0 |] [| 0 1 0 |] [| 0 0 1 |] |])
+
+(define $wedge
+  (lambda [%X %Y]
+    !(. X Y)))
+
+(define $dx [| 1 0 0 |])
+(define $dy [| 0 1 0 |])
+(define $dz [| 0 0 1 |])
+
+(wedge dx dy)
+;[| [| 0 1 0 |] [| 0 0 0 |] [| 0 0 0 |] |]
+
+(df-normalize (wedge dx dy))
+;[| [| 0 (/ 1 2) 0 |] [| (/ -1 2) 0 0 |] [| 0 0 0 |] |]
+
+(wedge dz dz)
+;[| [| 0 0 0 |] [| 0 0 0 |] [| 0 0 1 |] |]
+
+(df-normalize (wedge dz dz))
+;[| [| 0 0 0 |] [| 0 0 0 |] [| 0 0 0 |] |]
diff --git a/sample/math/geometry/yang-mills-equation-of-U1-gauge-theory.egi b/sample/math/geometry/yang-mills-equation-of-U1-gauge-theory.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/geometry/yang-mills-equation-of-U1-gauge-theory.egi
@@ -0,0 +1,77 @@
+(define $N 4)
+
+(define $g [| [| -1 0 0 0 |] [| 0 1 0 0 |] [| 0 0 1 0 |] [| 0 0 0 1 |] |])
+
+(define $d
+  (lambda [%X]
+    !((flip ∂/∂) [| t x y z |] X)))
+
+(define $hodge
+  (lambda [%A]
+    (let {[$k (df-order A)]}
+      (with-symbols {i j}
+        (* (sqrt (abs (M.det g_#_#)))
+           (foldl . (. (subrefs A (map 1#j_%1 (between 1 k)))
+                       (subrefs (ε' N k) (map 1#i_%1 (between 1 N))))
+                  (map 1#g~[i_%1]~[j_%1] (between 1 k))))))))
+
+(define $δ
+  (lambda [%A]
+    (let {[$r (df-order A)]}
+      (* (** -1 (+ (* N r) 1))
+         (hodge (d (hodge A)))))))
+
+(define $Δ
+  (lambda [%A]
+    (match (dfr-order A) integer
+      {[,0 (δ (d A))]
+       [,4 (d (δ A))]
+       [_ (+ (d (δ A)) (δ (d A)))]})))
+
+(define $normalize2
+  (lambda [%A]
+    (with-symbols {t1 t2}
+      (- A_t1_t2 A_t2_t1))))
+
+; *(dt^dx) = -dy^dz
+(hodge (wedge [| 1 0 0 0 |] [| 0 1 0 0 |]))
+;[| [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 -1 |] [| 0 0 0 0 |] |]
+
+; *(dy^dz) = dt^dx
+(hodge (wedge [| 0 0 1 0 |] [| 0 0 0 1 |]))
+;[| [| 0 1 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] [| 0 0 0 0 |] |]
+
+(df-normalize (d [| (φ t x y z) (Ax t x y z) (Ay t x y z) (Az t x y z) |]))
+;[|[| 0 (+ (Ax|1 t x y z) (* -1 (φ|2 t x y z))) (+ (Ay|1 t x y z) (* -1 (φ|3 t x y z))) (+ (Az|1 t x y z) (* -1 (φ|4 t x y z))) |]
+;  [| (+ (φ|2 t x y z) (* -1 (Ax|1 t x y z))) 0 (+ (Ay|2 t x y z) (* -1 (Ax|3 t x y z))) (+ (Az|2 t x y z) (* -1 (Ax|4 t x y z))) |]
+;  [| (+ (φ|3 t x y z) (* -1 (Ay|1 t x y z))) (+ (Ax|3 t x y z) (* -1 (Ay|2 t x y z))) 0 (+ (Az|3 t x y z) (* -1 (Ay|4 t x y z))) |]
+;  [| (+ (φ|4 t x y z) (* -1 (Az|1 t x y z))) (+ (Ax|4 t x y z) (* -1 (Az|2 t x y z))) (+ (Ay|4 t x y z) (* -1 (Az|3 t x y z))) 0 |]|]
+
+(define $F
+  [|[| 0 (Ex t x y z) (Ey t x y z) (Ez t x y z) |]
+    [| (* -1 (Ex t x y z)) 0 (Bz t x y z) (* -1 (By t x y z)) |]
+    [| (* -1 (Ey t x y z)) (* -1 (Bz t x y z)) 0 (Bx t x y z) |]
+    [| (* -1 (Ez t x y z)) (By t x y z) (* -1 (Bx t x y z)) 0 |]
+    |])
+
+(hodge (d F))
+;[|(+ (* -2 (Bz|4 t x y z)) (* -2 (By|3 t x y z)) (* -2 (Bx|2 t x y z)))
+;  (+ (* -2 (Ey|4 t x y z)) (* 2 (Ez|3 t x y z)) (* -2 (Bx|1 t x y z)))
+;  (+ (* 2 (Ex|4 t x y z)) (* -2 (Ez|2 t x y z)) (* -2 (By|1 t x y z)))
+;  (+ (* -2 (Ex|3 t x y z)) (* 2 (Ey|2 t x y z)) (* -2 (Bz|1 t x y z)))|]
+
+;(∇ B) = 0
+;(rot x E) = ∂t B
+;(rot y E) = ∂t B
+;(rot z E) = ∂t B
+
+(δ F)
+;[|(+ (* -2 (Ez|4 t x y z)) (* -2 (Ey|3 t x y z)) (* -2 (Ex|2 t x y z)))
+;  (+ (* 2 (By|4 t x y z)) (* -2 (Bz|3 t x y z)) (* -2 (Ex|1 t x y z)))
+;  (+ (* -2 (Bx|4 t x y z)) (* 2 (Bz|2 t x y z)) (* -2 (Ey|1 t x y z)))
+;  (+ (* 2 (Bx|3 t x y z)) (* -2 (By|2 t x y z)) (* -2 (Ez|1 t x y z)))|]
+
+;(∇ E) = 0
+;(rot x B) = ∂t E
+;(rot y B) = ∂t E
+;(rot z B) = ∂t E
diff --git a/sample/math/number/10bonacci.egi b/sample/math/number/10bonacci.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/10bonacci.egi
@@ -0,0 +1,37 @@
+(define $m 10)
+
+(define $A
+  (generate-tensor
+    (match-lambda [integer integer]
+      {[[,1 _] 1]
+       [[$x ,(- x 1)] 1]
+       [[_ _] 0]})
+    {m m}))
+
+A
+;[| [| 1 1 1 1 1 1 1 1 1 1 |] [| 1 0 0 0 0 0 0 0 0 0 |] [| 0 1 0 0 0 0 0 0 0 0 |] [| 0 0 1 0 0 0 0 0 0 0 |] [| 0 0 0 1 0 0 0 0 0 0 |] [| 0 0 0 0 1 0 0 0 0 0 |] [| 0 0 0 0 0 1 0 0 0 0 |] [| 0 0 0 0 0 0 1 0 0 0 |] [| 0 0 0 0 0 0 0 1 0 0 |] [| 0 0 0 0 0 0 0 0 1 0 |] |]
+
+(define $B
+  (generate-tensor
+    (match-lambda integer
+      {[,1 1]
+       [_ 0]})
+    {m}))
+
+B
+;[| 1 0 0 0 0 0 0 0 0 0 |]
+
+(M.* A B)
+;[| 1 1 0 0 0 0 0 0 0 0 |]
+
+(define $M.*-mod
+  (lambda [%m1 %m2]
+    (modulo (b..' m1~#~j m2_j) (** 10 9))))
+
+(M.*-mod A A)
+;[| [| 2 2 2 2 2 2 2 2 2 1 |] [| 1 1 1 1 1 1 1 1 1 1 |] [| 1 0 0 0 0 0 0 0 0 0 |] [| 0 1 0 0 0 0 0 0 0 0 |] [| 0 0 1 0 0 0 0 0 0 0 |] [| 0 0 0 1 0 0 0 0 0 0 |] [| 0 0 0 0 1 0 0 0 0 0 |] [| 0 0 0 0 0 1 0 0 0 0 |] [| 0 0 0 0 0 0 1 0 0 0 |] [| 0 0 0 0 0 0 0 1 0 0 |] |]
+
+(M.* (repeated-squaring M.*-mod A 10) B)
+;[| 512 256 128 64 32 16 8 4 2 1 |]~#
+(M.* (repeated-squaring M.*-mod A (** 10 18)) B)
+;[| 781174235709863749 377867955633934335 842430993012717568 732703024915201024 898916287400615936 291801846997259776 348909715528105216 288982486365729408 408131585481965832 584591530883971372 |]
diff --git a/sample/math/number/11th-root-of-unity.egi b/sample/math/number/11th-root-of-unity.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/11th-root-of-unity.egi
@@ -0,0 +1,54 @@
+;(gen-cyclic-group (map 1#(modulo (* %1 2) 11) (between 1 10)))
+;
+
+(define $z (rtu 11))
+(define $k (rtu 5))
+
+(define $a11 (+ z^1 z^10))
+(define $a12 (+ z^2 z^9))
+(define $a13 (+ z^3 z^8))
+(define $a14 (+ z^4 z^7))
+(define $a15 (+ z^5 z^6))
+
+(define $b10 (+ a11 a12 a13 a14 a15))
+
+(define $b10' -1);-1
+
+(define $b11 (+ a11 (* k a12) (* k^2 a13) (* k^3 a14)  (* k^4 a15)))
+(define $b12 (+ a15 (* k a11) (* k^2 a12) (* k^3 a13)  (* k^4 a14)));(* k b11)
+(define $b13 (+ a14 (* k a15) (* k^2 a11) (* k^3 a12)  (* k^4 a13)));(* k^2 b11)
+(define $b14 (+ a13 (* k a14) (* k^2 a15) (* k^3 a11)  (* k^4 a12)));(* k^3 b11)
+(define $b15 (+ a12 (* k a13) (* k^2 a14) (* k^3 a15)  (* k^4 a11)));(* k^4 b11)
+
+b11
+(* b11 b12)
+
+(rt 5 (* -1 b11 b12 b13 b14 b15));
+(define $b11' (rt 3 (+ 7 (* 21 w^2))))
+
+(define $b14 (+ a11 (* w a13) (* w^2 a12)))
+(define $b15 (+ a12 (* w a11) (* w^2 a13)));(* w b14)
+(define $b16 (+ a13 (* w a12) (* w^2 a11)));(* w^2 b14)
+
+;(rt 3 (* b14 b15 b16));(rt 3 (+ 7 (* 21 w)))
+(define $b14' (rt 3 (+ 7 (* 21 w))))
+
+(define $a11' (/ (+ b10' b11' b14') 3));;/ (+ -1 (rt 3 (+ 7 (* 21 w^2))) (rt 3 (+ 7 (* 21 w)))) 3)
+
+(define $z1' (fst (q-f' 1 (* -1 a11') 1)))
+
+z1'
+;(/ (+ -1 (rt 3 (+ 7 (* 21 w^2))) (rt 3 (+ 7 (* 21 w))) (sqrt (+ -35 (* -2 (rt 3 (+ 7 (* 21 w^2)))) (* -2 (rt 3 (+ 7 (* 21 w)))) (rt 3 (+ 7 (* 21 w^2)))^2 (* 2 (rt 3 (+ 7 (* 21 w^2))) (rt 3 (+ 7 (* 21 w)))) (rt 3 (+ 7 (* 21 w)))^2))) 6)
+
+(/ (+ -1
+      (rt 3 (+ 7 (* 21 w^2)))
+      (rt 3 (+ 7 (* 21 w)))
+      (sqrt (+ -35
+               (* -2 (rt 3 (+ 7 (* 21 w^2))))
+               (* -2 (rt 3 (+ 7 (* 21 w))))
+               (rt 3 (+ 7 (* 21 w^2)))^2
+               (* 2 (rt 3 (+ 7 (* 21 w^2))) (rt 3 (+ 7 (* 21 w))))
+               (rt 3 (+ 7 (* 21 w)))
+               ^2))
+      )
+   6)
diff --git a/sample/math/number/17th-root-of-unity.egi b/sample/math/number/17th-root-of-unity.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/17th-root-of-unity.egi
@@ -0,0 +1,76 @@
+;(gen-cyclic-group (map 1#(modulo (* %1 11) 17) (between 1 16)))
+;{{11 5 16 10 4 15 9 3 14 8 2 13 7 1 12 6} {2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15} {5 10 15 3 8 13 1 6 11 16 4 9 14 2 7 12} {4 8 12 16 3 7 11 15 2 6 10 14 1 5 9 13} {10 3 13 6 16 9 2 12 5 15 8 1 11 4 14 7} {8 16 7 15 6 14 5 13 4 12 3 11 2 10 1 9} {3 6 9 12 15 1 4 7 10 13 16 2 5 8 11 14} {16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1} {6 12 1 7 13 2 8 14 3 9 15 4 10 16 5 11} {15 13 11 9 7 5 3 1 16 14 12 10 8 6 4 2} {12 7 2 14 9 4 16 11 6 1 13 8 3 15 10 5} {13 9 5 1 14 10 6 2 15 11 7 3 16 12 8 4} {7 14 4 11 1 8 15 5 12 2 9 16 6 13 3 10} {9 1 10 2 11 3 12 4 13 5 14 6 15 7 16 8} {14 11 8 5 2 16 13 10 7 4 1 15 12 9 6 3} {1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16}}
+
+(define $z (rtu 17))
+
+;(gen-cyclic-group {16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1})
+;{{16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1} {1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16}}
+(define $a1 (+ z^1 z^16))
+(define $a2 (+ z^2 z^15))
+(define $a3 (+ z^3 z^14))
+(define $a4 (+ z^4 z^13))
+(define $a5 (+ z^5 z^12))
+(define $a6 (+ z^6 z^11))
+(define $a7 (+ z^7 z^10))
+(define $a8 (+ z^8 z^9))
+
+;(gen-cyclic-group {4 8 12 16 3 7 11 15 2 6 10 14 1 5 9 13})
+;{{4 8 12 16 3 7 11 15 2 6 10 14 1 5 9 13} {16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1} {13 9 5 1 14 10 6 2 15 11 7 3 16 12 8 4} {1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16}}
+(define $b11 (+ a1 a4))
+(define $b12 (- a1 a4));(** b12 2);(+ 4 b21 (* -2 b31))
+
+(define $b21 (+ a2 a8))
+(define $b22 (- a2 a8));(** b22 2);(+ 4 b21 (* -2 b41))
+
+(define $b31 (+ a3 a5))
+(define $b32 (- a3 a5));(** b32 2);(+ 4 b41 (* -2 b21))
+
+(define $b41 (+ a6 a7))
+(define $b42 (- a6 a7));(** b42 2);(+ 4 b31 (* -2 b21))
+
+;(gen-cyclic-group {2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15})
+;{{2 4 6 8 10 12 14 16 1 3 5 7 9 11 13 15} {4 8 12 16 3 7 11 15 2 6 10 14 1 5 9 13} {8 16 7 15 6 14 5 13 4 12 3 11 2 10 1 9} {16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1} {15 13 11 9 7 5 3 1 16 14 12 10 8 6 4 2} {13 9 5 1 14 10 6 2 15 11 7 3 16 12 8 4} {9 1 10 2 11 3 12 4 13 5 14 6 15 7 16 8} {1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16}}
+(define $c11 (+ b11 b21))
+(define $c12 (- b11 b21));(+ 8 (* -1 c11'))
+
+(define $c21 (+ b31 b41))
+(define $c22 (- b31 b41));(+ 8 (* -1 c21'))
+
+(define $d10 (+ c11 c21));-1
+(define $d11 (- c11 c21))
+(define $d12 (- c21 c11))
+
+(define $d10' -1)
+
+;(define $d11' (sqrt (* -1 d11 d12)));(sqrt 17)
+(define $d11' (sqrt 17))
+
+(define $c11' (/ (+ d10' d11') 2));(/ (+ -1 (sqrt 17)) 2)
+(define $c21' (/ (- d10' d11') 2));(/ (+ -1 (* -1 (sqrt 17))) 2)
+
+(define $c12' (sqrt (+ 8 (* -1 c11'))));(/ (sqrt (+ 34 (* -2 (sqrt 17)))) 2)
+(define $c22' (sqrt (+ 8 (* -1 c21'))));(/ (sqrt (+ 34 (* 2 (sqrt 17)))) 2)
+
+(define $b11' (/ (+ c11' c12') 2));(/ (+ -1 (sqrt 17) (sqrt (+ 34 (* -2 (sqrt 17))))) 4)
+(define $b21' (/ (- c11' c12') 2));(/ (+ -1 (sqrt 17) (* -1 (sqrt (+ 34 (* -2 (sqrt 17)))))) 4)
+(define $b31' (/ (+ c21' c22') 2));(/ (+ -1 (* -1 (sqrt 17)) (sqrt (+ 34 (* 2 (sqrt 17))))) 4)
+(define $b41' (/ (- c21' c22') 2));(/ (+ -1 (* -1 (sqrt 17)) (* -1 (sqrt (+ 34 (* 2 (sqrt 17)))))) 4)
+
+(define $b12' (sqrt (+ 4 b21' (* -2 b31'))))
+(define $b22' (sqrt (+ 4 b21' (* -2 b41'))))
+(define $b32' (sqrt (+ 4 b41' (* -2 b21'))))
+(define $b42' (sqrt (+ 4 b31' (* -2 b21'))))
+
+(define $a1' (/ (+ b11' b12') 2))
+
+a1';(+ z z^16) = (* 2 (cos (/ (* 2 pi) 17)))
+;(/ (+ -1 (sqrt 17) (sqrt (+ 34 (* -2 (sqrt 17)))) (* 2 (sqrt (+ 17 (* 3 (sqrt 17)) (* -1 (sqrt (+ 34 (* -2 (sqrt 17))))) (* -2 (sqrt (+ 34 (* 2 (sqrt 17))))))))) 8)
+
+(/ (+ -1
+      (sqrt 17)
+      (sqrt (+ 34 (* -2 (sqrt 17))))
+      (* 2 (sqrt (+ 17
+                    (* 3 (sqrt 17))
+                    (* -1 (sqrt (+ 34 (* -2 (sqrt 17)))))
+                    (* -2 (sqrt (+ 34 (* 2 (sqrt 17)))))))))
+   8)
diff --git a/sample/math/number/5th-root-of-unity.egi b/sample/math/number/5th-root-of-unity.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/5th-root-of-unity.egi
@@ -0,0 +1,44 @@
+;(gen-cyclic-group (map 1#(modulo (* %1 2) 5) (between 1 4)))
+;{{2 4 1 3} {4 3 2 1} {3 1 4 2} {1 2 3 4}}
+
+(define $z (rtu 5))
+
+(define $a11 (+ z^1 z^4))
+(define $a12 (+ z^2 z^3))
+
+(define $b10 (+ a11 a12))
+(define $b11 (- a11 a12))
+(define $b12 (- a12 a11))
+
+(define $b10' -1);-1
+(define $b11' (sqrt (** b11 2)));(sqrt 5)
+
+(define $a11' (/ (+ b10' b11') 2));(/ (+ -1 (sqrt 5)) 2)
+(define $a12' (/ (- b10' b11') 2));(/ (+ -1 (* -1 (sqrt 5))) 2)
+
+(define $a21 (- z^1 z^4))
+(define $a22 (- z^2 z^3))
+
+(define $b20 (+ a21 a22))
+(define $b21 (- a21 a22))
+(define $b22 (- a22 a21))
+
+;(define $b20' (sqrt (* -1 b20 b20)));(sqrt (+ (* -3 (rtu 5)^2) 4 (* -3 (rtu 5)^3) (rtu 5)^4 (rtu 5)))
+(define $b20' (sqrt (+ -3 (* 4 a12'))))
+;(define $b21' (sqrt (* -1 b21 b22)));(sqrt (+ (* -1 (rtu 5)^3) (* 3 (rtu 5)^4) (* -1 (rtu 5)^2) -4 (* 3 (rtu 5))))
+(define $b21' (sqrt (+ -3 (* 4 a11'))))
+
+(define $a21' (/ (+ b20' b21') 2))
+(define $a22' (/ (- b20' b21') 2))
+
+(define $z1' (/ (+ a11' a21') 2))
+
+z1';(/ (+ -1 (sqrt 5) (sqrt (+ -5 (* -2 (sqrt 5)))) (sqrt (+ -5 (* 2 (sqrt 5))))) 4)
+
+(** (+ (sqrt (+ -5 (* -2 (sqrt 5)))) (sqrt (+ -5 (* 2 (sqrt 5))))) 2)
+;(+ -10 (* 2 (sqrt (+ -5 (* -2 (sqrt 5)))) (sqrt (+ -5 (* 2 (sqrt 5))))))
+
+(* (+ -5 (* -2 (sqrt 5))) (+ -5 (* 2 (sqrt 5))));5
+
+; z1' is equal to
+(/ (+ -1 (sqrt 5) (sqrt (+ -10 (* -2 (sqrt 5))))) 4)
diff --git a/sample/math/number/7th-root-of-unity-2.egi b/sample/math/number/7th-root-of-unity-2.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/7th-root-of-unity-2.egi
@@ -0,0 +1,68 @@
+;(gen-cyclic-group (map 1#(modulo (* %1 3) 7) (between 1 6)))
+;{{3 6 2 5 1 4} {2 4 6 1 3 5} {6 5 4 3 2 1} {4 1 5 2 6 3} {5 3 1 6 4 2} {1 2 3 4 5 6}}
+
+(define $z (rtu 7))
+
+(define $a11 (+ z   z^2 z^4))
+(define $a12 (+ z^6 z^5 z^3))
+
+(define $b10 (+ a11 a12));-1
+
+(define $b10' b10)
+
+(define $b11 (- a11 a12))
+(define $b12 (- a12 a11))
+
+(define $b11' (sqrt (** b11 2)))
+
+(define $a11' (/ (+ b10' b11') 2));(/ (+ -1 (* i (sqrt 7))) 2)
+(define $a12' (/ (- b10' b11') 2));(/ (+ -1 (* -1 i (sqrt 7))) 2)
+
+
+(define $a21 (+ z   (* w z^2) (* w^2 z^4)))
+(define $a22 (+ z^6 (* w z^5) (* w^2 z^3)))
+
+(define $b20 (+ a21 a22))
+(define $b21 (- a21 a22))
+(define $b22 (- a22 a21))
+
+(define $b20' (rt 3 (** b20 3)))
+;(define $b21' (rt 3 (** b21 3)))
+;(** b21 3)
+;(+ (* 8 (rtu 7)) (* 8 (rtu 7)^2) (* -5 (rtu 7)^3) (* 5 (rtu 7)^4) (* -8 (rtu 7)^5) (* -8 (rtu 7)^6) (* 3 (rtu 7) w) (* 3 (rtu 7)^2 w) (* -3 (rtu 7)^5 w) (* -3 (rtu 7)^6 w) (* 3 (rtu 7)^3 w^2) (* -3 (rtu 7)^4 w^2))
+(define $b21' (rt 3 (+ (* 5 a11') (* -5 a12') (* w^2 -3 a11') (* w^2 3 a12'))));Calculate manually
+
+(define $a21' (/ (+ b20' b21') 2))
+(define $a22' (/ (- b20' b21') 2))
+
+
+(define $a31 (+ z   (* w^2 z^2) (* w z^4)))
+(define $a32 (+ z^6 (* w^2 z^5) (* w z^3)))
+
+(define $b30 (+ a31 a32))
+(define $b31 (- a31 a32))
+(define $b32 (- a32 a31))
+
+(define $b30' (rt 3 (** b30 3)))
+;(define $b31' (rt 3 (** b31 3)))
+;(** b31 3)
+;(+ (* 5 (rtu 7)) (* 8 (rtu 7)^2) (* -5 (rtu 7)^3) (* 5 (rtu 7)^4) (* -8 (rtu 7)^5) (* -5 (rtu 7)^6) (* -3 (rtu 7) w) (* 3 (rtu 7)^3 w) (* -3 (rtu 7)^4 w) (* 3 (rtu 7)^6 w) (* 3 (rtu 7)^2 w^2) (* -3 (rtu 7)^5 w^2))
+(define $b31' (rt 3 (+ (* 5 a11') (* -5 a12') (* w -3 a11') (* w 3 a12'))));Calculate manually
+
+(define $a31' (/ (+ b30' b31') 2))
+(define $a32' (/ (- b30' b31') 2))
+
+(define $z1' (/ (+ a11' a21' a31') 3))
+(define $z6' (/ (+ a12' a22' a32') 3))
+
+z1'
+;(/ (+ -1 (* i (sqrt 7)) (rt 3 (+ 14 (* 21 w))) (rt 3 (+ (* 5 i (sqrt 7)) (* -3 i (sqrt 7) w^2))) (rt 3 (+ -7 (* -21 w))) (rt 3 (+ (* 5 i (sqrt 7)) (* -3 i (sqrt 7) w)))) 6)
+
+(/ (+ -1
+      (rt 3 (+ 14 (* 21 w)))
+      (rt 3 (+ -7 (* -21 w)))
+      (* i (sqrt 7))
+      (rt 3 (+ (* 5 i (sqrt 7)) (* -3 i (sqrt 7) w)))
+      (rt 3 (+ (* 5 i (sqrt 7)) (* -3 i (sqrt 7) w^2)))
+      )
+   6)
diff --git a/sample/math/number/7th-root-of-unity.egi b/sample/math/number/7th-root-of-unity.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/7th-root-of-unity.egi
@@ -0,0 +1,53 @@
+;(gen-cyclic-group (map 1#(modulo (* %1 3) 7) (between 1 6)))
+;{{3 6 2 5 1 4} {2 4 6 1 3 5} {6 5 4 3 2 1} {4 1 5 2 6 3} {5 3 1 6 4 2} {1 2 3 4 5 6}}
+
+(define $z (rtu 7))
+
+(define $a11 (+ z^1 z^6))
+(define $a12 (+ z^2 z^5))
+(define $a13 (+ z^3 z^4))
+
+(define $b10 (+ a11 a12 a13))
+
+(define $b10' b10)
+
+b10';-1
+
+(define $b11 (+ a11 (* w a12) (* w^2 a13)))
+(define $b12 (+ a13 (* w a11) (* w^2 a12)));(* w b11)
+(define $b13 (+ a12 (* w a13) (* w^2 a11)));(* w^2 b11)
+
+(define $b11' (rt 3 (* b11 b12 b13)))
+
+b11';(rt 3 (+ 14 (* 21 w)))
+
+(define $b14 (+ a11 (* w a13) (* w^2 a12)))
+(define $b15 (+ a12 (* w a11) (* w^2 a13)));(* w b14)
+(define $b16 (+ a13 (* w a12) (* w^2 a11)));(* w^2 b14)
+
+(define $b14' (rt 3 (* b14 b15 b16)))
+
+b14';(rt 3 (+ -7 (* -21 w)))
+
+(define $a11' (/ (+ b10' b11' b14') 3))
+
+a11';(/ (+ -1 (rt 3 (+ 14 (* 21 w))) (rt 3 (+ -7 (* -21 w)))) 3)
+
+
+(define $z1' (fst (q-f' 1 (* -1 a11') 1)))
+
+z1';(/ (+ -1 (rt 3 (+ 14 (* 21 w))) (rt 3 (+ -7 (* -21 w))) (sqrt (+ -35 (* -2 (rt 3 (+ 14 (* 21 w)))) (* -2 (rt 3 (+ -7 (* -21 w)))) (rt 3 (+ 14 (* 21 w)))^2 (* 2 (rt 3 (+ 14 (* 21 w))) (rt 3 (+ -7 (* -21 w)))) (rt 3 (+ -7 (* -21 w)))^2))) 6)
+
+(/ (+ -1
+      (rt 3 (+ 14 (* 21 w)))
+      (rt 3 (+ -7 (* -21 w)))
+      (sqrt (+ -35
+               (* -2 (rt 3 (+ 14 (* 21 w))))
+               (* -2 (rt 3 (+ -7 (* -21 w))))
+               (rt 3 (+ 14 (* 21 w)))^2
+               (rt 3 (+ -7 (* -21 w)))^2
+               (* 2
+                  (rt 3 (+ 14 (* 21 w)))
+                  (rt 3 (+ -7 (* -21 w))))
+               )))
+   6)
diff --git a/sample/math/number/9th-root-of-unity.egi b/sample/math/number/9th-root-of-unity.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/9th-root-of-unity.egi
@@ -0,0 +1,48 @@
+;(map 1#(modulo (** 2 %1) 9) (between 1 6));{2 4 8 7 5 1}
+
+(define $z (rtu 9))
+
+(define $a11 (+ z^1 z^8))
+(define $a12 (+ z^2 z^7))
+(define $a13 (+ z^4 z^5))
+
+(define $b10 (+ a11 a12 a13))
+
+(define $b10' 0)
+
+(define $b11 (+ a11 (* w a12) (* w^2 a13)))
+(define $b12 (+ a13 (* w a11) (* w^2 a12)));(* w b11)
+(define $b13 (+ a12 (* w a13) (* w^2 a11)));(* w^2 b11)
+
+;(define $b11' (rt 3 (** b11 3)))
+(define $b11' (* 3 (rt 3 w)));Calculate manually
+;(** b11 3)
+;=>(+ (* 18 w) (* 9 (rtu 9)^6) (* 9 (rtu 9)^6 w^2) (* 9 (rtu 9)^3) (* 9 (rtu 9)^3 w^2))
+;=>(* 27 w)
+
+(define $b14 (+ a11 (* w a13) (* w^2 a12)))
+(define $b15 (+ a12 (* w a11) (* w^2 a13)));(* w b14)
+(define $b16 (+ a13 (* w a12) (* w^2 a11)));(* w^2 b14)
+
+;(define $b14' (rt 3 (** b14 3)))
+(define $b14' (* 3 (rt 3 w^2)));Caluculate manually
+;(** b14 3)
+;=>(+ (* 18 w^2) (* 9 (rtu 9)^6) (* 9 (rtu 9)^6 w) (* 9 (rtu 9)^3) (* 9 (rtu 9)^3 w))
+;=>(* 27 w^2)
+
+(define $a11' (/ (+ b10' b11' b14') 3))
+a11'
+;(+ (rt 3 w) (rt 3 w^2))
+
+(define $z1' (fst (q-f' 1 (* -1 a11') 1)))
+z1'
+;(/ (+ (rt 3 w) (rt 3 w^2) (sqrt (+ (rt 3 w)^2 (* 2 (rt 3 w) (rt 3 w^2)) (rt 3 w^2)^2 -4))) 2)
+
+(/ (+ (rt 3 w)
+      (rt 3 w^2)
+      (sqrt (+ -4
+               (rt 3 w)^2
+               (rt 3 w^2)^2
+               (* 2 (rt 3 w) (rt 3 w^2))
+               )))
+   2)
diff --git a/sample/math/number/eisenstein-primes.egi b/sample/math/number/eisenstein-primes.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/eisenstein-primes.egi
@@ -0,0 +1,38 @@
+(map 2#[(+ %1 (* w %2)) (* (+ %1 (* w %2)) (+ %1 (* w^2 %2)))] (match-all (take 10 nats) (set integer) [<cons $x <cons $y _>> [x y]]))
+
+{[(+ 1 w) 1]
+ [(+ 1 (* 2 w)) 3] [(+ 2 w) 3]
+ [(+ 1 (* 3 w)) 7] [(+ 2 (* 2 w)) 4] [(+ 3 w) 7]
+ [(+ 1 (* 4 w)) 13] [(+ 2 (* 3 w)) 7] [(+ 3 (* 2 w)) 7] [(+ 4 w) 13]
+ [(+ 1 (* 5 w)) 21] [(+ 2 (* 4 w)) 12] [(+ 3 (* 3 w)) 9] [(+ 4 (* 2 w)) 12] [(+ 5 w) 21] 
+ [(+ 1 (* 6 w)) 31] [(+ 2 (* 5 w)) 19] [(+ 3 (* 4 w)) 13] [(+ 4 (* 3 w)) 13] [(+ 5 (* 2 w)) 19] [(+ 6 w) 31]
+ [(+ 1 (* 7 w)) 43] [(+ 2 (* 6 w)) 28] [(+ 3 (* 5 w)) 19] [(+ 4 (* 4 w)) 16] [(+ 5 (* 3 w)) 19] [(+ 6 (* 2 w)) 28] [(+ 7 w) 43]
+ [(+ 1 (* 8 w)) 57] [(+ 2 (* 7 w)) 39] [(+ 3 (* 6 w)) 27] [(+ 4 (* 5 w)) 21] [(+ 5 (* 4 w)) 21] [(+ 6 (* 3 w)) 27] [(+ 7 (* 2 w)) 39] [(+ 8 w) 57] 
+ [(+ 1 (* 9 w)) 73] [(+ 2 (* 8 w)) 52] [(+ 3 (* 7 w)) 37] [(+ 4 (* 6 w)) 28] [(+ 5 (* 5 w)) 25] [(+ 6 (* 4 w)) 28] [(+ 7 (* 3 w)) 37] [(+ 8 (* 2 w)) 52] [(+ 9 w) 73] 
+ [(+ 1 (* 10 w)) 91] [(+ 2 (* 9 w)) 67] [(+ 3 (* 8 w)) 49] [(+ 4 (* 7 w)) 37] [(+ 5 (* 6 w)) 31] [(+ 6 (* 5 w)) 31] [(+ 7 (* 4 w)) 37] [(+ 8 (* 3 w)) 49] [(+ 9 (* 2 w)) 67] [(+ 10 w) 91]
+ [(+ 2 (* 10 w)) 84] [(+ 3 (* 9 w)) 63] [(+ 4 (* 8 w)) 48] [(+ 5 (* 7 w)) 39] [(+ 6 (* 6 w)) 36] [(+ 7 (* 5 w)) 39] [(+ 8 (* 4 w)) 48] [(+ 9 (* 3 w)) 63] [(+ 10 (* 2 w)) 84]
+ [(+ 3 (* 10 w)) 79] [(+ 4 (* 9 w)) 61] [(+ 5 (* 8 w)) 49] [(+ 6 (* 7 w)) 43] [(+ 7 (* 6 w)) 43] [(+ 8 (* 5 w)) 49] [(+ 9 (* 4 w)) 61] [(+ 10 (* 3 w)) 79]
+ [(+ 4 (* 10 w)) 76] [(+ 5 (* 9 w)) 61] [(+ 6 (* 8 w)) 52] [(+ 7 (* 7 w)) 49] [(+ 8 (* 6 w)) 52] [(+ 9 (* 5 w)) 61] [(+ 10 (* 4 w)) 76]
+ [(+ 5 (* 10 w)) 75] [(+ 6 (* 9 w)) 63] [(+ 7 (* 8 w)) 57] [(+ 8 (* 7 w)) 57] [(+ 9 (* 6 w)) 63] [(+ 10 (* 5 w)) 75]
+ [(+ 6 (* 10 w)) 76] [(+ 7 (* 9 w)) 67] [(+ 8 (* 8 w)) 64] [(+ 9 (* 7 w)) 67] [(+ 10 (* 6 w)) 76]
+ [(+ 7 (* 10 w)) 79] [(+ 8 (* 9 w)) 73] [(+ 9 (* 8 w)) 73] [(+ 10 (* 7 w)) 79]
+ [(+ 8 (* 10 w)) 84] [(+ 9 (* 9 w)) 81] [(+ 10 (* 8 w)) 84] 
+ [(+ 9 (* 10 w)) 91] [(+ 10 (* 9 w)) 91]
+ [(+ 10 (* 10 w)) 100]
+ }
+
+(filter 2#(prime? %2) (map 2#[(+ %1 (* w %2)) (* (+ %1 (* w %2)) (+ %1 (* w^2 %2)))] (match-all (take 10 nats) (set integer) [<cons $x <cons $y _>> [x y]])))
+
+{[(+ 1 w) 1]
+ [(+ 1 (* 2 w)) 3] [(+ 2 w) 3]
+ [(+ 1 (* 3 w)) 7] [(+ 3 w) 7]
+ [(+ 1 (* 4 w)) 13] [(+ 2 (* 3 w)) 7] [(+ 3 (* 2 w)) 7] [(+ 4 w) 13]
+ [(+ 1 (* 6 w)) 31] [(+ 2 (* 5 w)) 19] [(+ 3 (* 4 w)) 13] [(+ 4 (* 3 w)) 13] [(+ 5 (* 2 w)) 19] [(+ 6 w) 31]
+ [(+ 1 (* 7 w)) 43] [(+ 3 (* 5 w)) 19] [(+ 5 (* 3 w)) 19] [(+ 7 w) 43]
+ [(+ 1 (* 9 w)) 73] [(+ 3 (* 7 w)) 37] [(+ 7 (* 3 w)) 37] [(+ 9 w) 73]
+ [(+ 2 (* 9 w)) 67] [(+ 4 (* 7 w)) 37] [(+ 5 (* 6 w)) 31] [(+ 6 (* 5 w)) 31] [(+ 7 (* 4 w)) 37] [(+ 9 (* 2 w)) 67]
+ [(+ 3 (* 10 w)) 79] [(+ 4 (* 9 w)) 61] [(+ 6 (* 7 w)) 43] [(+ 7 (* 6 w)) 43] [(+ 9 (* 4 w)) 61] [(+ 10 (* 3 w)) 79]
+ [(+ 5 (* 9 w)) 61] [(+ 9 (* 5 w)) 61]
+ [(+ 7 (* 9 w)) 67] [(+ 9 (* 7 w)) 67]
+ [(+ 7 (* 10 w)) 79] [(+ 8 (* 9 w)) 73] [(+ 9 (* 8 w)) 73] [(+ 10 (* 7 w)) 79]
+ }
diff --git a/sample/math/number/euler-totient-function.egi b/sample/math/number/euler-totient-function.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/euler-totient-function.egi
@@ -0,0 +1,108 @@
+(define $φ
+  (lambda [$n]
+    (* n
+       (product (map (lambda [$p] (- 1 (/ 1 p)))
+                     (unique (p-f n)))))))
+
+(take 100 (map2 2#[%1 %2 (p-f %2)] nats (map φ nats)))
+
+{[1 1 {}]
+ [2 1 {}]
+ [3 2 {2}]
+ [4 2 {2}]
+ [5 4 {2 2}]
+ [6 2 {2}]
+ [7 6 {2 3}]
+ [8 4 {2 2}]
+ [9 6 {2 3}]
+ [10 4 {2 2}]
+ [11 10 {2 5}]
+ [12 4 {2 2}]
+ [13 12 {2 2 3}]
+ [14 6 {2 3}]
+ [15 8 {2 2 2}]
+ [16 8 {2 2 2}]
+ [17 16 {2 2 2 2}]
+ [18 6 {2 3}]
+ [19 18 {2 3 3}]
+ [20 8 {2 2 2}]
+ [21 12 {2 2 3}]
+ [22 10 {2 5}]
+ [23 22 {2 11}]
+ [24 8 {2 2 2}]
+ [25 20 {2 2 5}]
+ [26 12 {2 2 3}]
+ [27 18 {2 3 3}]
+ [28 12 {2 2 3}]
+ [29 28 {2 2 7}]
+ [30 8 {2 2 2}]
+ [31 30 {2 3 5}]
+ [32 16 {2 2 2 2}]
+ [33 20 {2 2 5}]
+ [34 16 {2 2 2 2}]
+ [35 24 {2 2 2 3}]
+ [36 12 {2 2 3}]
+ [37 36 {2 2 3 3}]
+ [38 18 {2 3 3}]
+ [39 24 {2 2 2 3}]
+ [40 16 {2 2 2 2}]
+ [41 40 {2 2 2 5}]
+ [42 12 {2 2 3}]
+ [43 42 {2 3 7}]
+ [44 20 {2 2 5}]
+ [45 24 {2 2 2 3}]
+ [46 22 {2 11}]
+ [47 46 {2 23}]
+ [48 16 {2 2 2 2}]
+ [49 42 {2 3 7}]
+ [50 20 {2 2 5}]
+ [51 32 {2 2 2 2 2}]
+ [52 24 {2 2 2 3}]
+ [53 52 {2 2 13}]
+ [54 18 {2 3 3}]
+ [55 40 {2 2 2 5}]
+ [56 24 {2 2 2 3}]
+ [57 36 {2 2 3 3}]
+ [58 28 {2 2 7}]
+ [59 58 {2 29}]
+ [60 16 {2 2 2 2}]
+ [61 60 {2 2 3 5}]
+ [62 30 {2 3 5}]
+ [63 36 {2 2 3 3}]
+ [64 32 {2 2 2 2 2}]
+ [65 48 {2 2 2 2 3}]
+ [66 20 {2 2 5}]
+ [67 66 {2 3 11}]
+ [68 32 {2 2 2 2 2}]
+ [69 44 {2 2 11}]
+ [70 24 {2 2 2 3}]
+ [71 70 {2 5 7}]
+ [72 24 {2 2 2 3}]
+ [73 72 {2 2 2 3 3}]
+ [74 36 {2 2 3 3}]
+ [75 40 {2 2 2 5}]
+ [76 36 {2 2 3 3}]
+ [77 60 {2 2 3 5}]
+ [78 24 {2 2 2 3}]
+ [79 78 {2 3 13}]
+ [80 32 {2 2 2 2 2}]
+ [81 54 {2 3 3 3}]
+ [82 40 {2 2 2 5}]
+ [83 82 {2 41}]
+ [84 24 {2 2 2 3}]
+ [85 64 {2 2 2 2 2 2}]
+ [86 42 {2 3 7}]
+ [87 56 {2 2 2 7}]
+ [88 40 {2 2 2 5}]
+ [89 88 {2 2 2 11}]
+ [90 24 {2 2 2 3}]
+ [91 72 {2 2 2 3 3}]
+ [92 44 {2 2 11}]
+ [93 60 {2 2 3 5}]
+ [94 46 {2 23}]
+ [95 72 {2 2 2 3 3}]
+ [96 32 {2 2 2 2 2}]
+ [97 96 {2 2 2 2 2 3}]
+ [98 42 {2 3 7}]
+ [99 60 {2 2 3 5}]
+ [100 40 {2 2 2 5}]}
diff --git a/sample/math/number/fib.egi b/sample/math/number/fib.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/fib.egi
@@ -0,0 +1,1 @@
+(define $F (lambda [$n] (* (/ 1 (sqrt 5)) (- (** (/ (+ 1 (sqrt 5)) 2) n) (** (/ (- 1 (sqrt 5)) 2) n)))))
diff --git a/sample/math/number/gaussian-primes.egi b/sample/math/number/gaussian-primes.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/gaussian-primes.egi
@@ -0,0 +1,36 @@
+(map 2#[(+ %1 (* i %2)) (* (+ %1 (* i %2)) (+ %1 (* -1 i %2)))] (match-all (take 10 nats) (set integer) [<cons $x <cons $y _>> [x y]]))
+
+{[(+ 1 i) 2] 
+ [(+ 1 (* 2 i)) 5] [(+ 2 i) 5]
+ [(+ 1 (* 3 i)) 10] [(+ 2 (* 2 i)) 8] [(+ 3 i) 10]
+ [(+ 1 (* 4 i)) 17] [(+ 2 (* 3 i)) 13] [(+ 3 (* 2 i)) 13] [(+ 4 i) 17]
+ [(+ 1 (* 5 i)) 26] [(+ 2 (* 4 i)) 20] [(+ 3 (* 3 i)) 18] [(+ 4 (* 2 i)) 20] [(+ 5 i) 26]
+ [(+ 1 (* 6 i)) 37] [(+ 2 (* 5 i)) 29] [(+ 3 (* 4 i)) 25] [(+ 4 (* 3 i)) 25] [(+ 5 (* 2 i)) 29] [(+ 6 i) 37] 
+ [(+ 1 (* 7 i)) 50] [(+ 2 (* 6 i)) 40] [(+ 3 (* 5 i)) 34] [(+ 4 (* 4 i)) 32] [(+ 5 (* 3 i)) 34] [(+ 6 (* 2 i)) 40] [(+ 7 i) 50]
+ [(+ 1 (* 8 i)) 65] [(+ 2 (* 7 i)) 53] [(+ 3 (* 6 i)) 45] [(+ 4 (* 5 i)) 41] [(+ 5 (* 4 i)) 41] [(+ 6 (* 3 i)) 45] [(+ 7 (* 2 i)) 53] [(+ 8 i) 65]
+ [(+ 1 (* 9 i)) 82] [(+ 2 (* 8 i)) 68] [(+ 3 (* 7 i)) 58] [(+ 4 (* 6 i)) 52] [(+ 5 (* 5 i)) 50] [(+ 6 (* 4 i)) 52] [(+ 7 (* 3 i)) 58] [(+ 8 (* 2 i)) 68] [(+ 9 i) 82]
+ [(+ 1 (* 10 i)) 101] [(+ 2 (* 9 i)) 85] [(+ 3 (* 8 i)) 73] [(+ 4 (* 7 i)) 65] [(+ 5 (* 6 i)) 61] [(+ 6 (* 5 i)) 61] [(+ 7 (* 4 i)) 65] [(+ 8 (* 3 i)) 73] [(+ 9 (* 2 i)) 85] [(+ 10 i) 101]
+ [(+ 2 (* 10 i)) 104] [(+ 3 (* 9 i)) 90] [(+ 4 (* 8 i)) 80] [(+ 5 (* 7 i)) 74] [(+ 6 (* 6 i)) 72] [(+ 7 (* 5 i)) 74] [(+ 8 (* 4 i)) 80] [(+ 9 (* 3 i)) 90] [(+ 10 (* 2 i)) 104]
+ [(+ 3 (* 10 i)) 109] [(+ 4 (* 9 i)) 97] [(+ 5 (* 8 i)) 89] [(+ 6 (* 7 i)) 85] [(+ 7 (* 6 i)) 85] [(+ 8 (* 5 i)) 89] [(+ 9 (* 4 i)) 97] [(+ 10 (* 3 i)) 109]
+ [(+ 4 (* 10 i)) 116] [(+ 5 (* 9 i)) 106] [(+ 6 (* 8 i)) 100] [(+ 7 (* 7 i)) 98] [(+ 8 (* 6 i)) 100] [(+ 9 (* 5 i)) 106] [(+ 10 (* 4 i)) 116]
+ [(+ 5 (* 10 i)) 125] [(+ 6 (* 9 i)) 117] [(+ 7 (* 8 i)) 113] [(+ 8 (* 7 i)) 113] [(+ 9 (* 6 i)) 117] [(+ 10 (* 5 i)) 125]
+ [(+ 6 (* 10 i)) 136] [(+ 7 (* 9 i)) 130] [(+ 8 (* 8 i)) 128] [(+ 9 (* 7 i)) 130] [(+ 10 (* 6 i)) 136]
+ [(+ 7 (* 10 i)) 149] [(+ 8 (* 9 i)) 145] [(+ 9 (* 8 i)) 145] [(+ 10 (* 7 i)) 149]
+ [(+ 8 (* 10 i)) 164] [(+ 9 (* 9 i)) 162] [(+ 10 (* 8 i)) 164]
+ [(+ 9 (* 10 i)) 181] [(+ 10 (* 9 i)) 181]
+ [(+ 10 (* 10 i)) 200]
+ }
+
+(filter 2#(prime? %2) (map 2#[(+ %1 (* i %2)) (* (+ %1 (* i %2)) (+ %1 (* -1 i %2)))] (match-all (take 10 nats) (set integer) [<cons $x <cons $y _>> [x y]])))
+
+{[(+ 1 i) 2]
+ [(+ 1 (* 2 i)) 5] [(+ 2 i) 5]
+ [(+ 1 (* 4 i)) 17] [(+ 2 (* 3 i)) 13] [(+ 3 (* 2 i)) 13] [(+ 4 i) 17]
+ [(+ 1 (* 6 i)) 37] [(+ 2 (* 5 i)) 29] [(+ 5 (* 2 i)) 29] [(+ 6 i) 37]
+ [(+ 2 (* 7 i)) 53] [(+ 4 (* 5 i)) 41] [(+ 5 (* 4 i)) 41] [(+ 7 (* 2 i)) 53]
+ [(+ 1 (* 10 i)) 101] [(+ 3 (* 8 i)) 73] [(+ 5 (* 6 i)) 61] [(+ 6 (* 5 i)) 61] [(+ 8 (* 3 i)) 73] [(+ 10 i) 101]
+ [(+ 3 (* 10 i)) 109] [(+ 4 (* 9 i)) 97] [(+ 5 (* 8 i)) 89] [(+ 8 (* 5 i)) 89] [(+ 9 (* 4 i)) 97] [(+ 10 (* 3 i)) 109]
+ [(+ 7 (* 8 i)) 113] [(+ 8 (* 7 i)) 113]
+ [(+ 7 (* 10 i)) 149] [(+ 10 (* 7 i)) 149]
+ [(+ 9 (* 10 i)) 181] [(+ 10 (* 9 i)) 181]
+ }
diff --git a/sample/math/number/napier.egi b/sample/math/number/napier.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/napier.egi
@@ -0,0 +1,21 @@
+;;;;;
+;;;;; Calucualate Napier's constant
+;;;;;
+
+(define $calculate-napier
+  (lambda [$n]
+    (sum (take n (map (lambda [$i] (/ 1 (fact i))) nats0)))))
+
+(test (calculate-napier 1))
+(test (calculate-napier 2))
+(test (calculate-napier 3))
+(test (calculate-napier 4))
+(test (calculate-napier 5))
+(test (calculate-napier 6))
+(test (calculate-napier 7))
+(test (calculate-napier 8))
+(test (calculate-napier 9))
+(test (calculate-napier 10))
+(test (rtof (calculate-napier 10)))
+(test (rtof (calculate-napier 100)))
+(test (rtof (calculate-napier 200)))
diff --git a/sample/math/number/pi.egi b/sample/math/number/pi.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/pi.egi
@@ -0,0 +1,32 @@
+;;;;;
+;;;;; Calucualate Pi
+;;;;;
+
+;(define $calculate-pi
+;  (lambda [$n]
+;    (foldr (lambda [$x $y] (+ x (/ 1 y))) 1 (take n {3 7 15 1 292 @(repeat1 1)}))))
+
+;(define $odds (map (compose (* $ 2) (- $ 1)) nats))
+
+;(define $calculate-pi
+;  (lambda [$n]
+;    (+ 3 (foldr (lambda [$x $y] (/ x (+ 6 y))) 1 (take n (map (power $ 2) odds))))))
+
+(define $calculate-pi
+  (lambda [$n]
+    (/ 4 (foldr (lambda [$x $y] (+ (- (* 2 x) 1) (/ (power x 2) y))) 1 (take n nats)))))
+
+(test (calculate-pi 1))
+(test (calculate-pi 2))
+(test (calculate-pi 3))
+(test (calculate-pi 4))
+(test (calculate-pi 5))
+(test (calculate-pi 6))
+(test (calculate-pi 7))
+(test (calculate-pi 8))
+(test (calculate-pi 9))
+(test (calculate-pi 10))
+(test (rtof (calculate-pi 100)))
+(test (rtof (calculate-pi 1000)))
+(test (rtof (calculate-pi 2000)))
+(test pi)
diff --git a/sample/math/number/sum-of-cubes.egi b/sample/math/number/sum-of-cubes.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/sum-of-cubes.egi
@@ -0,0 +1,23 @@
+;;;;;
+;;;;;
+;;;;; Sum of Cubes
+;;;;;
+;;;;;
+
+; Infintite list of sum of cubes.
+; -- [m n (+ m^3 n^3)]
+(define $sum-of-cubes
+  (let {[$cube (lambda [$x] (* x (* x x)))]}
+    (match-all nats (list integer)
+      [<join _ (& <cons $m _> <join _ <cons $n _>>)> [m n (+ (cube m) (cube n))]])))
+
+; sample output
+(test (take 10 sum-of-cubes))
+
+; list numbers that is the sum of two non-zero cube numbers
+(test (take 2 (match-all sum-of-cubes (list [integer integer integer])
+                [<join _ <cons [$x1 $y1 $c]
+                  <join _ <cons [$x2 $y2 ,c]
+                   _>>>>
+                 [[x1 y1 c] [x2 y2 c]]]
+                )))
diff --git a/sample/math/number/sum-of-squares.egi b/sample/math/number/sum-of-squares.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/sum-of-squares.egi
@@ -0,0 +1,36 @@
+;;;;;
+;;;;;
+;;;;; Sum of Squares
+;;;;;
+;;;;;
+
+; Infintite list of sum of squres.
+; -- [m n (+ m^2 n^2)]
+(define $sum-of-squares
+  (let {[$square (lambda [$x] (* x x))]}
+    (match-all nats (list integer)
+      [<join _ (& <cons $m _> <join _ <cons $n _>>)> [m n (+ (square m) (square n))]])))
+
+; sample output
+(test (take 30 sum-of-squares))
+
+; list numbers that is the sum of two non-zero square numbers in two distinct way
+(test (let {[$n 2]}
+        (take 5 (match-all sum-of-squares (list [integer integer integer])
+                  [<join _ <cons [$x_1 $y_1 $c]
+                    (loop $i [2 n]
+                      <join _ <cons [$x_i $y_i ,c] ...>>
+                      _)>>
+                   (map (lambda [$i] [x_i y_i c]) (between 1 n))]))))
+
+; prime-factorize sum of squares
+; -- [m n {p1 p2 ...}]
+(define $sum-of-squares-pf (map (match-lambda [integer integer integer] {[[$m $n $c] [m n (p-f c)]]}) sum-of-squares))
+
+; sample output
+(test (take 30 sum-of-squares-pf))
+
+; list prime numbers that is the sum of two non-zero square numbers
+(test (take 30 (match-all sum-of-squares-pf (list [integer integer (multiset integer)])
+                 [<join _ <cons [$m $n <cons $p <nil>>] _>> [m n p]])))
+
diff --git a/sample/math/number/tribonacci.egi b/sample/math/number/tribonacci.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/tribonacci.egi
@@ -0,0 +1,39 @@
+(define $m 3)
+
+(define $A
+  (generate-tensor
+    (match-lambda [integer integer]
+      {[[,1 _] 1]
+       [[$x ,(- x 1)] 1]
+       [[_ _] 0]})
+    {m m}))
+A
+;[| [| 1 1 1 |] [| 1 0 0 |] [| 0 1 0 |] |]
+
+(define $B
+  (generate-tensor
+    (match-lambda integer
+      {[,1 1]
+       [_ 0]})
+    {m}))
+
+B
+;[| 1 0 0 |]
+
+(M.* A B)
+;[| 1 1 0 |]
+
+(M.* (M.power A 2) B)
+;[| 2 1 1 |]
+
+(M.* (M.power A 3) B)
+;[| 4 2 1 |]
+
+(M.* (M.power A 4) B)
+;[| 7 4 2 |]
+
+(M.* (M.power A 5) B)
+;[| 13 7 4 |]
+
+(M.* (M.power A 100) B)
+;[| 180396380815100901214157639 98079530178586034536500564 53324762928098149064722658 |]
diff --git a/sample/math/number/zeta.egi b/sample/math/number/zeta.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/number/zeta.egi
@@ -0,0 +1,9 @@
+(define $zeta
+  (lambda [$n]
+    (rtof (foldl + 0 (take n (map (lambda [$n] (* (/ 1 n) (/ 1 n))) nats))))))
+
+(test (zeta 100))
+(test (zeta 1000))
+(test (zeta 10000))
+
+(test (/ (* pi pi) 6))
diff --git a/sample/math/others/mobius-transformation.egi b/sample/math/others/mobius-transformation.egi
new file mode 100644
--- /dev/null
+++ b/sample/math/others/mobius-transformation.egi
@@ -0,0 +1,24 @@
+(define $f
+  (lambda [$z]
+    (/ (+ (* a z) b) (+ (* c z) d))))
+
+(define $f1
+  (lambda [$z]
+    (+ z (/ d c))))
+
+(define $f2
+  (lambda [$z]
+    (/ 1 z)))
+
+(define $f3
+  (lambda [$z]
+    (* z
+       (/ (* -1 (- (* a d) (* b c)))
+          c^2))))
+
+(define $f4
+  (lambda [$z]
+    (+ (/ a c) z)))
+
+(f4 (f3 (f2 (f1 z))))
+;(/ (+ (* a z) b) (+ (* c z) d))