3 Vector Fields and One-Form Fields, 2.2

Functional Differential Geometry

p. 21

…

2.4

Let

$\displaystyle{ \begin{aligned} D_b(f)(x) &= ((Df)\bigg|_x b \bigg|_x \\ \end{aligned} }$ ,

where $\displaystyle{ b(x) }$ is a coordinate function.

p. 12

A coordinate function $\displaystyle{\chi}$ maps points in a coordinate patch of a manifold to a coordinate tuple:

$\displaystyle{x = \chi(\mathbf{m})}$ ,

where $\displaystyle{x}$ may have a convenient tuple structure.

2.5

$\displaystyle{ \begin{aligned} \textbf{v}(\text{f})(\textbf{m}) &\ne D_b(f)(x) \\ \end{aligned} }$

Instead, $\displaystyle{ \textbf{v}(\text{f})(\textbf{m})}$ is a further generalization of $\displaystyle{ D_b(f)(x)}$ .

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 $\displaystyle{ x = \chi (\mathbf{m}) }$ .

So, actually,

$\displaystyle{ \begin{aligned} \textbf{v}(\text{f})(\textbf{m}) &= D_b(f)(x) \\ &= ((D(f)) \bigg|_x) b \bigg|_x \\ &= (\nabla f)\bigg|_x \cdot \vec b\bigg|_x \\ \end{aligned} }$

2.6

While $\displaystyle{ x }$ is the tuple of coordinate components of a point, $\displaystyle{ \textbf{m} }$ is the abstract point itself.

3.1

… ; they measure how quickly the coordinate functions change in the direction of the vector field, scaled by the magnitude of the vector field: …

$\displaystyle{ \begin{aligned} b^i_{\chi, \mathbf{v}} &= \mathbf{v}(\chi^i) \circ \chi^{-1} \\ \end{aligned} }$

$\displaystyle{ \chi }$ inputs an abstract point and outputs its coordinates.

The first factor $\displaystyle{ \mathbf{v}(\chi^i) }$ is just the meaning of the definition of $\displaystyle{ b^i_{\chi, \mathbf{v}} }$ . The second factor is needed because the input of $\displaystyle{ b^i_{\chi, \mathbf{v}} }$ is $\displaystyle{ x \\ }$ , not $\displaystyle{ \mathbf{m} \\ }$ .

$\displaystyle{ \begin{aligned} b^i_{\chi, \mathbf{v}} (x) &= \mathbf{v}(\chi^i) (\mathbf{m}) \\ &= \mathbf{v}(\chi^i) ( \chi^{-1} (\chi (\mathbf{m}))) \\ &= \mathbf{v}(\chi^i) \circ \chi^{-1} (x) \\ \end{aligned} }$

In other words, $\displaystyle{ \mathbf{v}(\chi^i) (\mathbf{m}) \\ }$ is just the definition of $\displaystyle{ b^i_{\chi, \mathbf{v}} (x) }$ .

3.2

… $\displaystyle{ \textbf{v}(\text{f})(\textbf{m})}$ is the direction derivative of the function $\displaystyle{\text{f}}$ at the point $\displaystyle{ \textbf{m} }$ .

Note that it is not the ordinary directional derivative.

3.2.1

Instead, the ordinary directional derivative is

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

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

3.2.2

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