A prove for a conclusion about partial differential

Question

I know when second order partial differential of a function($z=f(x,y)$) is continuous, $\frac{\partial^2z}{\partial x\partial y} = \frac{\partial^2z}{\partial y\partial x}$. But I don’t know how to prove it. Can anyone help me?

Answer 1

By the fundamental theorem for line integrals, on any closed curve $\gamma$ in the we have $$ 0=\int_\gamma \frac{\partial f}{\partial x}\mathrm dx+\frac{\partial f}{\partial y}\mathrm dy $$ Now assuming the domain is simply connected, we have by Green's theorem $$ 0=\int_A \left(\frac{\partial^2 f}{\partial y\partial x}-\frac{\partial^2 f}{\partial x\partial y}\right)\mathrm dx\mathrm dy $$ where $A$ is the area enclosed by the curve. Since the second derivatives are continuous, this implies that the integrand is zero everywhere in the region. Since the region was arbitrary, we may conclude.