3 Vector Fields and One-Form Fields, 3.1

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Functional Differential Geometry

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p. 22

Eq. (3.4):

\displaystyle{ \begin{aligned} \textbf{v}(\text{f})(\textbf{m}) &= (D( \textbf{f} \circ \chi^{-1}) b_{\chi,\mathbf{v}}) \circ \chi) \\ \end{aligned} }

(define (components->vector-field components coordsys)
  (define (v f)
    (compose (* (D (compose f (point coordsys)))
        components)
         (chart coordsys)))
  (procedure->vector-field v))

An example:

(define R2->R (-> (UP Real Real) Real))

(define v
  (components->vector-field
   (up (literal-function 'b^0 R2->R)
       (literal-function 'b^1 R2->R))
   R2-rect))

(define R2-rect-chi-inverse
  (point R2-rect))

(define R2-rect-point
  (R2-rect-chi-inverse (up 'x_0 'y_0)))

((v (literal-manifold-function 'f-rect R2-rect))
 R2-rect-point)

(show-expression
 ((v (literal-manifold-function 'f-rect R2-rect))
  R2-rect-point))


(define v
  (literal-vector-field 'b R2-rect))

(show-expression
 ((v (literal-manifold-function 'f-rect R2-rect))
  R2-rect-point))

— Me@2024-08-10 07:06:38 AM

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