k143.egi

(define $params [| r θ |])

(define $d
  (lambda [%X]
    !((flip ∂/∂) params X)))

(define $wedge
  (lambda [%X %Y]
    !(. X Y)))

(define $u
  (lambda [$r $θ]
    (* r (** e (* 2 π i θ)))))

(define $ū
  (lambda [$r $θ]
    (* r (** e (* -2 π i θ)))))

(d (u r θ))
;[| (exp (* 2 π θ i)) (* 2 r (exp (* 2 π θ i)) π i) |]

(d (ū r θ))
;;[| (exp (* -2 π θ i)) (* -2 r (exp (* -2 π θ i)) π i) |]

(df-normalize (wedge (d (u r θ))
                     (d (ū r θ))))
;[| [| 0 (* -2 r π i) |] [| (* 2 r π i) 0 |] |]