Schwarz's theorem

Interchanging partial derivatives -----------------------------------------

Partial differentiation of functions of several variables emerged in the wake of differentiation itself, at the end of the 17^th century. It means differentiating with respect to one variable while holding all the others fixed. This made it possible to solve many problems in physics. But what exactly does it involve?

Let f be a function of two variables (x, y) $\mapsto \backslash mapsto$ f (x, y); with y held fixed, consider the function *f _y: x $\mapsto \backslash mapsto$ f (x, y*).

Since this is now a function of just one variable, we can ask whether it is differentiable at a point x.

If this derivative exists, it is denoted by ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }x}}(x,y)\mathrm{.}\backslash dfrac\{\backslash partial\; f\}\{\backslash partial\; x\}\; (x,\; y).$ Similarly, if it exists, we can define ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }y}}(x,y)\mathrm{.}\backslash dfrac\{\backslash partial\; f\}\{\backslash partial\; y\}\; (x,\; y).$