Directional Differentiability (Real Vector Functions)#
Definition: Directional Differentiability (Real Vector Functions)
Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function, \(\boldsymbol{\hat{r}} \in \mathbb{R}^n\) be a unit vector and let \(\boldsymbol{p} \in \mathcal{D}\).
We say that \(f\) is (directionally) differentiable at \(\boldsymbol{p}\) along \(\boldsymbol{\hat{r}}\) if the following limit exists:
\[\lim_{h\to 0}\frac{f(\boldsymbol{p} + h \cdot \boldsymbol{\hat{r}} ) - f(\boldsymbol{p})}{h}\]
Definition: Directional Derivative
In this case, this limit is known as \(f\)'s directional derivative at \(\boldsymbol{p}\) along \(\boldsymbol{\hat{r}}\).
Notation
\[\frac{\partial f}{\partial \boldsymbol{\hat{r}}}(\boldsymbol{p}) \qquad \partial_{\boldsymbol{\hat{r}}}f(\boldsymbol{p}) \qquad f_{\boldsymbol{\hat{r}}}(\boldsymbol{p}) \qquad D_{\boldsymbol{\hat{r}}} f(\boldsymbol{p})\]