3 Vector Fields and One-Form Fields, 2

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Functional Differential Geometry

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p. 21

1.1

\displaystyle{(Df(x)) \Delta x} is the directional derivative of \displaystyle{f} at \displaystyle{x} with respect to \displaystyle{\Delta x}, which is a vector in terms of a coordinate component tuple.

1.2

Note that \displaystyle{\Delta x} is not actually coordinates, but a change of coordinates. Instead, \displaystyle{x} is the coordinate tuple.

Moreover, \displaystyle{x} and \displaystyle{\Delta x} are independent of each other.

1.3

Instead of being a function of position \displaystyle{x} only, \displaystyle{(Df(x)) \Delta x} depends also on \displaystyle{\Delta x}. So \displaystyle{(Df(x)) \Delta x} is NOT a vector field.

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2.1

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

Instead, \displaystyle{(Df(x)) b(x)} is a generalization of \displaystyle{(Df(x)) \Delta x}.

2.2

Note that \displaystyle{b(x)} is a function chosen by you.

2.3

Since \displaystyle{(Df(x)) b(x)} is a function of position \displaystyle{x} only, it is a vector field.

2.4

Let

\displaystyle{ \begin{aligned} D_b(f)(x) &= ((Df(x)) b(x) \\ \end{aligned} }

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)}.

2.6

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

— Me@2023-11-19 11:40:49 AM

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2023.11.21 Tuesday (c) All rights reserved by ACHK