# necessary and sufficient conditions for the existence of solutions.

Let $\phi$ and $\psi$ be two smooth functions on $\Omega$ open subset of $\mathbb{R}^2$. There exists a function $u=u(x,y)$ such that: $$\left\{\begin{array}{lll} \frac{\partial u}{\partial x}=\phi(x,y)\\ \frac{\partial u}{\partial y}=\psi(x,y) \end{array}\right.$$ if and only if $\frac{\partial \phi}{\partial y}=\frac{\partial \psi}{\partial x}$.

Where I can find the proof of this statement? Is it a obvious consecuence of some famous theorem (like implicit funtion)?

Many thank!

• – flawr May 30 '16 at 16:18
• Thanks! It is clear that if $u$ is a solution, then $\partial_y \phi$ must be equal to $\partial_x\psi$. How can I prove the converse? – FUUNK1000 May 30 '16 at 17:05
• Integrate those terms again, and do some detail work with the integration constants. – flawr May 30 '16 at 17:11
• @flawr weird..... I always cite that as clairaut's – qbert May 30 '16 at 18:20
• @qbert I probably depends on where you are / what language it is used. Among german speakers it is widely known as Satz von Schwarz. – flawr May 30 '16 at 20:21

Your statement is false in general, unless $\Omega$ is simply connected.