Consider function $f(x, y)$ smooth enough satisfying the following equation: $$\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y}\ .$$ It is obvious, that any function of the form $f(x + y)$ suits the above condition. How can I prove, that these are all functions or find a contradiction?
Asked
Active
Viewed 62 times
0
-
2Duplicate of Problems with partial derivatives.. Also related to Geometric interpretation of ${\partial f\over \partial x}= {\partial f \over \partial y}$. – dxiv Jul 16 '17 at 20:16
-
To be more precise: “Any function $f$ of the form $f(x,y)=g(x+y)$, where $g$ is a (nice enough) function of one variable...” – Hans Lundmark Jul 17 '17 at 11:49