3 Vector Fields and One-Form Fields, 3.2

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Functional Differential Geometry

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3.2.1

Instead, the ordinary directional derivative is

\displaystyle{(Df(x)) \Delta x}

or

\displaystyle{\begin{aligned} D_{\mathbf{v}}(f) &= \frac{\left(\delta f\right)_{\mathbf{v}}}{|\mathbf{v}|} &= \left(\nabla f\right) \cdot \hat{\mathbf{v}} \\ \end{aligned}}

3.2.2

The first generalization of directional derivative is replacing \displaystyle{\Delta x}, a vector independent of \displaystyle{x}, with \displaystyle{b(x)}, a vector function of \displaystyle{x}.

1. Note that

\displaystyle{(Df(x)) \Delta x \ne D_{\mathbf{v}}(f)}

Instead,

\displaystyle{\begin{aligned} (Df(x)) \Delta x &\approx \Delta f \\ (Df(x)) &= D_{\mathbf{v}}(f) \\ \end{aligned}}

2. The physical meaning of \displaystyle{b(x)} is given by:

p. 25

The function b is the coefficient function for the vector field v. It provides a scale factor for the component in each coordinate direction.

3.

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

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

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

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

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

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


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

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

4. Eq. (3.7):

\displaystyle{\text{v}(\chi)(\chi^{-1}(x)) = b_{\chi, \text{v}} (x)}

p. 25

The vector field \displaystyle{ \textbf{v} } has a coordinate representation \displaystyle{ v}:

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

with the definitions \displaystyle{ f = \mathbf{f} \circ \chi^{-1} } and \displaystyle{ x = \chi (\mathbf{m}) }.

(define (coordinatize v coordsys)
  (define ((coordinatized-v f) x)
    (let ((b (compose (v (chart coordsys))
                      (point coordsys))))
      (* ((D f) x) (b x))))
  (make-operator coordinatized-v))

(((coordinatize v R2-rect) (literal-function 'f_rect R2->R)) (up 'x_0 'y_0))

— Me@2024-09-02 01:06:17 AM

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2024.09.03 Tuesday (c) All rights reserved by ACHK