Differentiability of Real Vector Functions

Definition: Differentiability of Real Vector Functions

Let $f:\mathcal{D}\to {\mathbb{R}}^n$ be a real vector function on an open set $\mathcal{D}\subseteq {\mathbb{R}}^m$ and let $\mathbf{a}$ be an accumulation point of $\mathcal{D}$.

We say that $f$ is (totally) differentiable at $\mathbf{a}$ iff there exists some linear transformation $L_a:\mathcal{D}\to {\mathbb{R}}^n$ (which usually depends on $\mathbf{a}$) such that the following limit is zero.

$$\underset{\mathbf{x}\to \mathbf{a}}{\lim }\frac{||f(\mathbf{x})-f(\mathbf{a})-L_a(\mathbf{x}-\mathbf{a})||}{||\mathbf{x}-\mathbf{a}||}=0$$

In this case, the transformation $L_a$ is known as the (total) derivative or (total) differential of $f$ at $\mathbf{a}$.

If $f$ is differentiable at every $\mathbf{a}$ in some $S\subseteq {\mathbb{R}}^m$, then we say that $f$ is (totally) differentiable on $S$. If $S=\mathcal{D}$, we can also just say that $f$ is (totally) differentiable.

NOTATION

We usually denote $L_{\mathbf{a}}$ as $\mathrm{d}{f}_{\mathbf{a}}$. If it is clear what $\mathbf{a}$ is, then we can also drop it and write just $\mathrm{d}f$.

Theorem: Uniqueness of the Derivative

If $f$ is differentiable at $\mathbf{a}$, then its derivative $\mathrm{d}{f}_{\mathbf{a}}$ is unique.

PROOF

TODO

Theorem: Differentiability via Component Functions

A real vector function $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ is differentiable at $\mathbf{a}\in \mathcal{D}$ if and only if its component functions are differentiable there.

PROOF

TODO

Theorem: Total Derivative and the Jacobian

Let $f:\mathcal{D}\to {\mathbb{R}}^n$ be a real vector function on an open subset $\mathcal{D}\subseteq {\mathbb{R}}^m$ and let $\mathbf{a}$ be an accumulation point of $\mathcal{D}$.

If $f$ is differentiable at $\mathbf{a}$, then it is partially differentiable there with respect to Cartesian coordinates and the matrix representation of its derivative $\mathrm{d}{f}_{\mathbf{a}}$ (with respect to the standard bases of ${\mathbb{R}}^m$ and ${\mathbb{R}}^n$) is the Jacobian matrix $Df(\mathbf{a})$.

PROOF

TODO

Theorem: Continuous Differentiability

Let $f:\mathcal{D}\to {\mathbb{R}}^n$ be a real vector function on an open subset $\mathcal{D}\subseteq {\mathbb{R}}^m$ and let $\mathbf{a}$ be an accumulation point of $\mathbf{a}$.

If there exists an open neighbourhood $N$ of $\mathbf{a}$ on which the restriction $f{|}_N$ is continuously partially differentiable in Cartesian coordinates, then $f$ is totally differentiable at $\mathbf{a}$.

PROOF

TODO

Definition: Continuous Differentiability

In this case, we say that $f$ is continuously differentiable at $\mathbf{a}$.