Finding $v$ for $v_{\xi,\eta} = 0$

Question

Studying partial differential equations, I came across following implication within a proof:

$$v_{\xi\eta} = 0 \implies v(\xi,\eta) = \Phi(\xi) + \Psi(\eta)$$

But I did not see exactly how this follows. I tried following:

$v_{\xi\eta}=0 \implies v_{\xi} = \int \underbrace{v_{\xi\eta}}_{=0} d\eta = f(\xi)$ and similarly $v_{\eta} = g(\eta)$ but then I just did not see how to continue, I think I'm just missing something that is actually very obvious. Can anyone help me out?

Answer 1

I think I've got it now: Let $\Phi$ be the antiderivative of $f$, and $\Psi$ the one of $g$. Then we get on the one hand

$$v = \Phi(\xi) + c_1(\eta)$$

and on the other hand

$$v = c_2(\xi) + \Psi(\eta)$$

Comparing these two results in $v = \Phi(\xi) + \Psi(\eta)$.