Directional derivative

Description

The directional derivative generalises the derivative of an $\mathbb{R}$-preimage function to a ${\mathbb{R}}^n$-preimage function. This, however is not a perfect generalisation as many nice properties of differentiable functions are lost under this definition.

Definition

Let $U\subseteq {\mathbb{R}}^n$ be an open subset. $f:U\to \mathbb{R}$. We define the directional derivative in a point $a\in U$ in direction $v\in {\mathbb{R}}^n$. Let $\varphi :I\to \mathbb{R},t\mapsto a+tv$ with $I$ suitably chosen. If the concatenation $f\circ \varphi$ is differentiable in $0$ we define the directional derivative at $a$ in direction $v$ as follows: ${\partial }_{v}f(a)=(f\circ \varphi {)}^{\prime }(0)={\lim }_{t\to 0}\frac{f(a+tv)-f(a)}{t}$

If $v$ is the standard basis vector than the directional derivative is called a partial derivative. Note that the value of the derivative depends on the length of $v$.

Properties

Tip

Examples

Unstetige Funktion mit überall definierter RIchtungsableitung

Die Funktion $f(x,y):=\{\begin{array}{ll}0& y\le 0\text{ oder }y\ge x^2\\ 1& 0<y<x^2\end{array}$ ist in $0$ unstetig hat aber überall wohl definierte RIchtungsableitung. Da sie in $0$ nicht stetig ist, ist auch ihre Totale Ableitung nicht definiert.