I have the following question:
Let $f,g\in C^1(\Bbb{R}^2)$ s.t. $\frac{\partial f}{\partial y}=\frac{\partial g}{\partial x}$. Show that $$F(x,y)=\int_0^x f(s,0)~ds+\int_0^y g(x,t)~dt$$is an antiderivative of the $1$-form $\omega=f(x,y)~dx+g(x,y)~dy$.
So for me it is clear that I want to show that $dF=\omega$. Therefore I first wanted to compute $\frac{\partial }{\partial x}\int_0^x f(s,0)~ds +\frac{\partial}{\partial x}\int_0^y g(x,t)~dt$. Now the only point where I have questions is $$\frac{\partial }{\partial x}\int_0^x f(s,0)~ds$$ then I wanted to use the main criteria of analysis and get $$\frac{\partial }{\partial x}\int_0^x f(s,0)~ds=f(x,0)-f(0,0)$$ but somehow they left $f(0,0)$ away and I don't know why one can do this, why do we know that $f(0,0)=0$?
But the rest is clear to me. Thanks for your help.