Partial Derivatives of Real Scalar Fields

Definition: Partial Derivative of a Real Scalar Field

Let $f:\mathcal{D}\to \mathbb{R}$ be a real scalar field on an open subset $\mathcal{D}\subseteq {\mathbb{R}}^n$, let $\varphi :\mathcal{D}\to {\mathbb{R}}^n$ be a coordinate system on $\mathcal{D}$ and let $a^1={\varphi }^1(\mathbf{a}),\dots ,a^n={\varphi }^n(\mathbf{a})$ be the coordinates of some $\mathbf{a}\in \mathcal{D}$ in $\varphi$.

The partial derivative of $f$ at $\mathbf{a}$ with respect to the $i$-th coordinate ${\varphi }^i$ is the limit

$$\underset{h\to 0}{\lim }\frac{f(a^1,\dots ,a^i+h,\dots ,a^n)-f(a^1,\dots ,a^i,\dots ,a^n)}{h},$$

NOTATION

Most commonly, the partial derivative of $f$ at $\mathbf{a}$ with respect to the $i$-th coordinate ${\varphi }^i$ is denoted as:

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

If the coordinate system is evident from context, we can also write ${\partial }_{i}f(\mathbf{a})$.

Note: Partial Derivative as a Function

When there is no specific $\mathbf{a}\in \mathcal{D}$ mentioned, the term “partial derivative” refers to the the function $\frac{\partial f}{\partial {\varphi }^i}:\mathcal{D}\to \mathbb{R}$ which to each $\mathbf{x}\in \mathcal{D}$ assigns the partial derivative of $f$ at $\mathbf{a}$ with respect to the coordinate ${\varphi }^i$.

Note: Orders of Partial Derivatives

An $n$-th order partial derivative of $f$ is a partial derivative of $f$‘s $(n-1)$-th partial derivative function.