The function $f(x,y) = x^2y + 2xy + xy^2$ has the properties such that $\frac{\partial^2 f}{\partial x^2}=\frac{\partial^2 f}{\partial y^2}$ and $\frac{\partial^2 f}{\partial x \partial y}=\frac{\partial^2 f}{\partial y \partial x}$. Is there a name for the types of functions which satisfy this? Does one equality imply the other?
Also, is there any importance to those equalities, or is it just a pretty coincidence?
Symmetry? I don't think any other name exists.