Partial Differential

Idea

Partial differential of $f(x,y,z)$ is a differential of $f$ in the direction $x,y$ or $z$, when the rest of variables are taken as constant.

Computation and notation

For multivariate function $f(x,y)$ its partial differentials with respect to $x$ or $y$ respectively are defined by:

(1)
\begin{align} \partial f = f_xdx \end{align}
(2)
\begin{align} \partial f = f_y dy \end{align}

Note: There is an intrinsic problem with this notation: $\partial f$ in Eq.(1) and Eq.(2) is not the same. Unfortunately, the symbols $\partial_x f$ and $\partial_y f$ are reserved for partial derivatives.

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