Total derivative

Description

The total derivative is the natural generalisation of a $\mathbb{R}\to \mathbb{R}$ function derivative to a ${\mathbb{R}}^n\to {\mathbb{R}}^m$ function derivative. It preserves all interesting properties of a differentiable function. For example it is continuous.

Definition

Let $U\subseteq {\mathbb{R}}^n$ open and $f;U\to {\mathbb{R}}^m$. $f$ is totally differentiable in $a$ if there exists a linear map $L:{\mathbb{R}}^n\to {\mathbb{R}}^m$ and an error term $q:U\to {\mathbb{R}}^m$ such that $f(a+h)=f(a)+L(h)+q(a+h)\text{ and }{\lim }_{h\to 0}\frac{q(a+h)}{\Vert h\Vert }=0$ The last condition means that the error is less-than-linear. The differential $L$ is usually denoted by $(Df)(a)$ and given by a matrix which is calculated as follows $(Df)(a)=\left(\begin{array}{ccc}{\partial }_{x_1}{f}_1(a)& ...& {\partial }_{x_n}{f}_1(a)\\ ⋮& \ddots & ⋮\\ {\partial }_{x_1}{f}_m(a)& ...& {\partial }_{x_n}{f}_m(a)\end{array}\right)$ Where the entries are given by Partial derivative along the coordinate axes.

This is kind similar to what we do with Tangentialraum and derivatives. There too, we have a linear map from one tangent space to another. We understand the derivative as a Tensor that takes in a vector and returns another vector. If we take $k$ total derivatives the result will become a tensor that takes in $k$ vector and returns a $k$-vector tensor.

Characterisation by Schwarz

Let $U\subseteq \mathbb{R}$ open and $f:U\to \mathbb{R}$. If the $k$-th partial derivatives of $f$ exist and if those are continuous then $f$ is a $k$-times totally differentiable map and (for $k\ge 2$) we have ${\partial }_{x_i}{\partial }_{x_j}f(a)={\partial }_{x_j}{\partial }_{x_i}f(a)$

Properties

Tip

Examples

Example