30 | September | 2023 | Physics Town

Chain Rule of Differentiation, 2

Functional Differential Geometry

p. 21

In multiple dimensions the derivative of a function is the multiplier for the best linear approximation of the function at each argument point:

$\displaystyle{f(x + \Delta x) \approx f(x) + (Df(x)) \Delta x}$

In other words:

$\displaystyle{ \begin{aligned} f(x + \Delta x) &= f(x) + a_1 (\Delta x) + a_2 (\Delta x)^2 + \cdots \\ (Df(x)) &= a_1 \\ \end{aligned} }$

The derivative $Df(x)$ is independent of $\Delta x$ .

… the product $\displaystyle{(Df(x))\Delta x}$ is invariant under change of coordinates …

$\displaystyle{(Df(x)) \Delta x}$ is the directional derivative of $\displaystyle{f}$ at $\displaystyle{x}$ with respect to $\displaystyle{\Delta x}$ .