Differentiability of Real Scalar Fields

Theorem: Differentiability of Real Scalar Fields

Let $f:\mathcal{D}\to \mathbb{R}$ be a real scalar field on an open subset $\mathcal{D}\subseteq {\mathbb{R}}^n$ and let $\mathbf{a}={\left[\begin{array}{ccc}{a}_1& \cdots & a_n\end{array}\right]}^{\mathsf{T}}\in \mathcal{D}$.

Then $f$ is differentiable at $\mathbf{a}$ if and only if it is partially differentiable there and the following limit is zero.

$$\underset{\mathbf{x}\to \mathbf{a}}{\lim }\frac{f(\mathbf{x})-[f(\mathbf{a})+\nabla f(\mathbf{a})\cdot (\mathbf{x}-\mathbf{a})]}{||\mathbf{x}-\mathbf{a}||}=0,$$

where $\nabla f(\mathbf{a})\cdot (\mathbf{x}-\mathbf{a})$ is the dot product between the gradient of $f$ at $\mathbf{a}$ and $(\mathbf{x}-\mathbf{a})$.

PROOF

TODO

Theorem: Differentiability implies Directional Differentiability

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

If $f$ is differentiable at $\mathbf{a}$, then its directional derivative ${\partial }_{\hat{\mathbf{r}}}f(\mathbf{a})$ at $\mathbf{a}$ exists along every direction $\hat{\mathbf{r}}$.

PROOF

TODO