Let $A\subset\mathbb{R^2}$ be an open connected set, and $\Omega=\{c:[0,1]\rightarrow A| c\in C^1\textrm{ and } c(0)=c(1)\}$. Consider the $1$-form $\omega = F_1dx_1+F_2dx_2$ in $A$ such that $\omega\in C^1$ and consider $f:\Omega\rightarrow\mathbb{R}$ such that $f(c)=\int_c\omega$.
I have to show that $f$ is differentiable with $$f'(c)h=\int_0^1\bigg(\frac{\partial F_2}{\partial x_1}-\frac{\partial F_1}{\partial x_2}\bigg)(c(t))\bigg(h_1(t)c_2'(t)-h_2(t)c_1'(t)\bigg) dt. $$
What I have done so far:
Well...at this point I was able to show that $f'(c)h=\int_0^1d\omega(c(t))(h(t),c'(t))dt$, so $f'(c)$ is linear (what is expected), to prove differentiability I need to show that $$\lim_{h\rightarrow0} \frac{f(c+h)-f(c)-f'(c)h}{\|h\|}=0,$$ where $\|h\|=\displaystyle\max_{t\in[0,1]}|h(t)|+\displaystyle\max_{t\in[0,1]}|h'(t)|$ is the norm used. With some calculations the limit becomes $$\lim_{h\rightarrow0} \frac{\int_0^1\omega(h(t))h'(t)dt-\int_0^1dw(c(t))\big(h(t),c'(t)\big)dt}{\|h\|},$$ and I don't know how to proceed from here. Please help someone!
Thank you very much!