Partial Derivatives of Real Vector Functions

Definition: Partial Derivative of a Real Vector Function

Let $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ be a real vector function with component functions $f_1,\cdots ,f_n$ and let $\varphi$ be a coordinate system for ${\mathbb{R}}^m$.

The partial derivative of $f$ at $\mathbf{a}\in \mathcal{D}$ with respect to the $i$-th coordinate ${\varphi }^i$ is the vector whose entries are the partial derivatives of $f_1,\cdots ,f_n$ with respect to ${\varphi }^i$:

$$\left[\begin{array}{c}\frac{\partial f_1}{\partial {\varphi }^i}(\mathbf{a})\\ ⋮\\ \frac{\partial f_n}{\partial {\varphi }^i}(\mathbf{a})\end{array}\right]$$

NOTATION

$$\frac{\partial f}{\partial {\varphi }^i}(\mathbf{a}){\frac{\partial f}{\partial {\varphi }^i}|}_{\mathbf{a}}\frac{\partial }{\partial {\varphi }^i}f(\mathbf{a})f_{{\varphi }^i}(\mathbf{a})$$

Definition: (Continuous) Partial Differentiability

A real vector function $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ is called $k$-times (continuously) partially differentiable if all of its $k$-th order partial derivatives exist (and are continuous) on $\mathcal{D}$.

If $f$ is $k$-times continuously partially differentiable, then we also say that $f$ is of class $C^k$ if $f$.

NOTATION

$$f\in C^k$$

Definition: Piecewise (Continuous) Partial Differentiability

A real vector function $f:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ is $k$-times piecewise (continuously) partially differentiable iff $\mathcal{D}$ can be represented as a disjoint union $\mathcal{D}={\mathcal{D}}_1\cup \cdots \cup {\mathcal{D}}_k$ of finitely many subsets ${\mathcal{D}}_1,\dots ,{\mathcal{D}}_k$ of ${\mathbb{R}}^m$ such that the restrictions $f_{{\mathcal{D}}_1},\dots ,f_{{\mathcal{D}}_k}$ are $k$-times (continuously) partially differentiable.