Equal partial derivatives

[sunrah]

Hi, in general can we tell anything about the partial derivatives of a differentiable function if they are equal?

for example I would like them to have to equal some constant. Would this be true?

 

[HallsofIvy]

Do you mean $\partial f/\partial x= C$ and $\partial f/\partial y= C$? The same constant or different constants? From $\partial f/\partial x= C$, we get $f(x,y)= Cx+ g(y)$ where g can be any function of y. Differentiating that with respect to y, $\partial f/\partial y= g'(y)= C$ which tells us that g(y)= Cy+ C' where C' is an arbitrary constant of integration. That is, f(x,y)= Cx+ Cy+ C'.