3 Vector Fields and One-Form Fields, 2.3

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Functional Differential Geometry

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p. 21

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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}

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}.

3.2.3

The second generalization of directional derivative is replacing \displaystyle{D} or \displaystyle{\nabla} with \displaystyle{\textbf{v}}, which is a vector function chosen by you.

In differential geometry, a vector is an operator that takes directional derivatives of manifold functions at its anchor point.

The directional derivative of a scalar function f with respect to a vector \displaystyle{\mathbf{v}} at a point (e.g., position) \displaystyle{\mathbf{x}} may be denoted by any of the following:

\displaystyle{\begin{aligned} \nabla _{\mathbf {v} }{f}(\mathbf {x} ) &=f'_{\mathbf {v} }(\mathbf {x} )=D_{\mathbf {v} }f(\mathbf {x} )=Df(\mathbf {x} )(\mathbf {v} ) \\ &=\partial _{\mathbf {v} }f(\mathbf {x} )=\mathbf {v} \cdot {\nabla f(\mathbf {x} )}=\mathbf {v} \cdot {\frac {\partial f(\mathbf {x} )}{\partial \mathbf {x} }}.\end{aligned}}

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Let \displaystyle{\textit{M}} be a differentiable manifold and \displaystyle{\mathbf{p}} a point of \displaystyle{\textit{M}}.

Suppose that \displaystyle{f} is a function defined in a neighborhood of \displaystyle{\mathbf{p}}, and differentiable at \displaystyle{\mathbf{p}}.

If \displaystyle{\mathbf{v}} is a tangent vector to \displaystyle{\textit{M}} at \displaystyle{\mathbf{p}}, then the directional derivative of \displaystyle{f} along \displaystyle{\mathbf{v}}, denoted variously as \displaystyle{df(\mathbf{v})} (see Exterior derivative), \displaystyle{\nabla_{\mathbf {v} }f(\mathbf {p} )} (see Covariant derivative), \displaystyle{L_{\mathbf {v} }f(\mathbf {p} )} (see Lie derivative), or \displaystyle{ {\mathbf {v} }_{\mathbf {p} }(f)} (see Tangent space § Definition via derivations), can be defined as follows.

Let \displaystyle{\gamma: [-1, 1] \to M} be a differentiable curve with \displaystyle{\gamma(0) = \mathbf{p}} and \displaystyle{\gamma'(0) = \mathbf{v}}. Then the directional derivative is defined by

\displaystyle{\nabla _{\mathbf {v} }f(\mathbf {p} )=\left.{\frac {d}{d\tau }}f\circ \gamma (\tau )\right|_{\tau =0}.}

This definition can be proven independent of the choice of \displaystyle{\gamma}, provided \displaystyle{\gamma} is selected in the prescribed manner so that \displaystyle{\gamma(0) = \mathbf{p}} and \displaystyle{\gamma'(0) = \mathbf{v}}.

— Wikipedia on Directional derivative

Tangent vectors as directional derivatives

Another way to think about tangent vectors is as directional derivatives. Given a vector \displaystyle{v} in \displaystyle{ \mathbb {R} ^{n}}, one defines the corresponding directional derivative at a point \displaystyle{ x\in \mathbb {R} ^{n}} by

\displaystyle{ \forall f\in {C^{\infty }}(\mathbb {R} ^{n}):\qquad (D_{v}f)(x):=\left.{\frac {\mathrm {d} }{\mathrm {d} {t}}}[f(x+tv)]\right|_{t=0}=\sum _{i=1}^{n}v^{i}{\frac {\partial f}{\partial x^{i}}}(x).}

This map is naturally a derivation at \displaystyle{ x }. Furthermore, every derivation at a point in {\displaystyle \mathbb {R} ^{n}} is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point.

— Wikipedia on Tangent space

4. In a more user-friendly language:

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

\displaystyle{ \begin{aligned} b^i_{\chi, \mathbf{v}} (x) &= \mathbf{v}(\chi^i) \circ \chi^{-1} (x) \\ &= \mathbf{v}(\chi^i) (\mathbf{m}) \\ &= D_b(\chi^i \circ \chi^{-1})(x) \\ &= \left(\nabla (\chi^i \circ \chi^{-1}) \right)\bigg|_x \cdot \vec b\bigg|_x \\ &= \sum_j \frac{\partial}{\partial x^j}(\chi^i \circ \chi^{-1}) \bigg|_x b^j \bigg|_x \\ \end{aligned} }

where \displaystyle{ x = \chi (\mathbf{m}) } and

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

\displaystyle{ \begin{aligned} b^i_{\chi, \mathbf{v}} (x) &= \sum_j \frac{\partial}{\partial x^j}(\chi^i \circ \chi^{-1}) \bigg|_x b^j \bigg|_x \\ &= \sum_j \frac{\partial x^i}{\partial x^j} \bigg|_x b^j \bigg|_x \\ &= \sum_j \delta^{ij} b^j \bigg|_x \\ &= b^i(x) \\ \end{aligned} }

This is a self-consistency check.

— Me@2024-02-03 04:45:17 PM

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2024.07.13 Saturday (c) All rights reserved by ACHK