4 Basis Fields, 1

Functional Differential Geometry

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: …

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

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

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

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

3.2 p. 21

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

3.3 p. 22 Eq. (3.4):

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

4.1 p. 41 Eq. (4.1):

$\displaystyle{ \textbf{v}(\text{f})(\textbf{m}) = \textbf{e}(\textbf{f})(\textbf{m})\textbf{b}(\textbf{m}) = \sum_{i} \textbf{e}_i (\textbf{f})(\textbf{m}) \textbf{b}^i(\textbf{m})}$

In other words, $\textbf{b}^i$ ‘s and $\textbf{e}_i$ ‘s represent the amount and direction of changes respectively.

4.2 p. 33

A one-form field is a generalization of this idea; it is something that measures a vector field at each point.

One-form fields are linear functions of vector fields that produce real-valued functions on the manifold.

…

— Me@2025-01-17 03:59:54 PM

Physics Town

continues

4 Basis Fields, 1

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