Theorems about properties of two-dimensional even/odd functions?

Question

As the title mentions, I would like to know the properties of two-dimensional even/odd functions. For instance, suppose that the function $f(x,y)$ is even: $$ f(-x,-y) = f(x,y) $$ Then what can we say about the properties of $F(x,y) = \int f(x,y) dy$ and $\partial_x F(x,y)$?

Answer 1

If you integrate an odd function over a spherically symmetric region around the origin, the result will be zero, just as in the one dimensional case. If you integrate an even function over a spherically symmetric region, the result will be twice the value of the integral over just one hemisphere of the region. This works in any dimensional space, including 2d.

Also the gradient of an odd function is even, and vice versa. The same goes for the components of the gradient, the partial derivatives.

Answer 2

It depends on which antiderivative $F(x,y)$ you use. If you take $F(x,y) = \int_0^y f(x,t)\; dt$, then $F$ is odd. But an arbitrary antiderivative $\int f(x,y)\; dy = F(x,y) + c(x)$ for an arbitrary function $c$, and in general this will be neither odd nor even.