Multivariable Calculus Integral Proof This problem is being very difficult for me to solve, I need help.
Consider $F:\mathbb{R}^2\rightarrow\mathbb{R}$ of class $C^1$, suppose that the level curves of $F$ are closed and that $\nabla F$ is never $0$ for $x\neq0$.
Consider the region $D$ between the curves $F=a$ and $F=b$. For each $r$ in $[a,b]$, let $c_r$ be the curve $F=r$. Let $f:D\rightarrow\mathbb{R}$ continuous. 
I have to show that $$\int_Df=\int_a^b\bigg(\int_{c_r}\frac{f}{|\nabla F|}\bigg)dr$$
Usually when I ask something here I show my attempts or my observations, but in this case I couldn't even start doing this!
Thank you very much for helping!
 A: The formula that needs to be proven is basically the change of variables $(x,y)\rightarrow(F,\tau)$, where $F$ is the level of $f$ and $\tau$ is the natural parameter (curve length) on the level set. It is actually easier to go in the inverse direction: start from the expression on the right where we integrate the function $\frac{f}{|\nabla F|}$. Then, passing from $(F,\tau)$ to $(x,y)$, the integral will transform into
$$ \int_D \frac{f}{|\nabla F|} \left|\frac{D(F,\tau)}{D(x,y)}\right|dxdy,$$
where 
$$ \left|\frac{D(F,\tau)}{D(x,y)}\right|=|\mathrm{det}\left(\begin{array}{cc}
\frac{\partial F}{\partial x} & \frac{\partial F}{\partial y}\\
\frac{\partial \tau}{\partial x} & \frac{\partial \tau}{\partial y}
\end{array}\right)|$$
is the jacobian of the variable change.
Hence, all we have to show is that
$$ |\nabla F|=\biggl|\left(\frac{\partial F}{\partial x}\frac{\partial \tau}{\partial y}-\frac{\partial F}{\partial y}\frac{\partial \tau}{\partial x}\right)\biggr|.\tag{1}$$
But, since the unit vector $\vec{n}$ orthogonal to $\nabla{F}$ (and hence tangent to the level curve) has the form 
$$ \vec{n}=\left(-\frac{1}{|\nabla F|}\frac{\partial F}{\partial y},
\frac{1}{|\nabla F|}\frac{\partial F}{\partial x}\right),$$
then $d\vec{l}=\vec{n}d\tau=(dx,dy)\Rightarrow d\tau=n_xdx+n_ydy$, so that
$$ \frac{\partial\tau}{\partial x}=n_x=-\frac{1}{|\nabla F|}\frac{\partial F}{\partial y},\qquad
\frac{\partial\tau}{\partial y}=n_y=\frac{1}{|\nabla F|}\frac{\partial F}{\partial x}.$$
Substituting these expressions into (1), we see that it is indeed satisfied.
