Partial Derivatives in Different Coordinate Systems

Tip: Partial Derivatives in Cartesian Coordinates

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

The partial derivative of $f$ at $\mathbf{a}$ with respect to the $i$-th Cartesian coordinate $x^i$ coincides with the directional derivative of $f$ there along the $i$-th standard basis vector ${\mathbf{e}}_i$:

$$\frac{\partial f}{\partial x^i}(\mathbf{a})=\underset{h\to 0}{\lim }\frac{f(\mathbf{a}+h\cdot {\mathbf{e}}_i)-f(\mathbf{a})}{h}$$

NOTATION

Cartesian coordinates in partial derivatives can be denoted using either subscripts and superscripts, i.e.

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

are equivalent. Writing the coordinates using subscripts is cleaner when partial derivatives of higher orders are involved.