Directional Derivatives of Real Scalar Fields

Definition: Directional Derivative of a Real Scalar Field

Let $f:\mathcal{D}\to \mathbb{R}$ be a real scalar field on an open subset $\mathcal{D}\subseteq {\mathbb{R}}^n$.

The directional derivative of $f$ at ${\mathbf{x}}_0\in \mathcal{D}$ along the unit vector $\hat{\mathbf{r}}$ is the limit

$$\underset{h\to 0}{\lim }\frac{f({\mathbf{x}}_0+h\cdot \hat{\mathbf{r}})-f({\mathbf{x}}_0)}{h}$$

(if it exists).

NOTATION

$$\frac{\partial f}{\partial \hat{\mathbf{r}}}({\mathbf{x}}_0){\partial }_{\hat{\mathbf{r}}}f({\mathbf{x}}_0)f_{\hat{\mathbf{r}}}({\mathbf{x}}_0)D_{\hat{\mathbf{r}}}f({\mathbf{x}}_0)$$