Equality of first-order partial derivatives Let $f(u,v)$ be a "sufficiently good" function of two variables. 
I need to find sufficient conditions on $f$ such that 
$$
\frac{\partial f(u,v)}{\partial u}=\frac{\partial f(u,v)}{\partial v},\qquad \forall \,u,v.
$$ 
For example, if $f(u,v)=f(c(u+v))$ the two first-order partial derivatives coincide. Is this true also in other cases? 
Thanks in advance.

I'm not certain what you mean by "other cases".  However, any solution of the partial differential equation $\frac{\partial f}{\partial u}= \frac{\partial f}{\partial v}$ is of the form $f(u, v)= g(u+ v)$ where g is an arbitrary differentiable function  of a single variable.
