polar-laplacian-3d.egi

(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))))