PX275 - A1a - partial differentiation

for a function, $f (x, y)$ , partial differentials can be defined as ${(\frac{\partial f}{\partial x})}_{y}$ ; ${(\frac{\partial f}{\partial y})}_{x}$

\begin{aligned} \frac{\partial}{\partial x} (\frac{\partial f}{\partial x}) & = \frac{\partial^{2} f}{\partial x^{2}} \\ \frac{\partial}{\partial y} (\frac{\partial f}{\partial y}) & = \frac{\partial^{2} f}{\partial y^{2}} \end{aligned} $ $ $ $ \frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}

\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}

the above property is always true for "well-behaved functions" routinely encountered in physics
note: $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ are operators acting on a function
if $f (x) = \sin (x)$ , $\frac{\partial}{\partial x} \sin x$ is not the same as $\sin x \frac{\partial}{\partial x}$
for $\frac{\partial^{2} f}{\partial x \partial y} = \frac{\partial^{2} f}{\partial y \partial x}$ to be true, (implies that the order of the operators can be swapped), $\frac{\partial}{\partial x} (\frac{\partial f}{\partial y}) = \frac{\partial}{\partial y} (\frac{\partial f}{\partial x})$
in general, with operators, the order of operation is important