Derivative of area of sphere 
Possible Duplicate:
Why is the derivative of a circle's area its perimeter (and similarly for spheres)? 

The derivative of the area $\pi r^2$ of a disk is the perimeter $2\pi r$ of the circumference. I know some explanations of this link.
1) Does there exist a explanation of this fact using Stokes's formula?
2) Does there exists a more general statement for some class of manifolds? 
 A: For part 1 at least, let
$$
F(x)=x
$$
Then the Divergence Theorem, a case of Stoke's Theorem, says that when $\mathcal{D}$ is a ball of radius $r$ in $\mathbb{R}^n$,
$$
\int_{\partial\mathcal{D}}F\cdot\vec{n}\;\mathrm{d}s=\int_{\mathcal{D}}\nabla\cdot F\;\mathrm{d}x
$$
$$
\int_{\partial\mathcal{D}}r\;\mathrm{d}s=\int_{\mathcal{D}}n\;\mathrm{d}x
$$
$$
r|\partial\mathcal{D}|=n|\mathcal{D}|
$$
Thus, $|\partial\mathcal{D}|=\frac{n}{r}|\mathcal{D}|$. Since all spheres are geometrically similar, and the volume of similar objects in $\mathbb{R}^n$ grows as their size to the $n^{\text{th}}$ power, $|\mathcal{D}|=cr^n$. Therefore,
$$
\begin{align}
\frac{\mathrm{d}}{\mathrm{d}r}|\mathcal{D}|
&=cnr^{n-1}\\
&=\frac{n}{r}|\mathcal{D}|\\
&=|\partial\mathcal{D}|
\end{align}
$$
A: Certainly nothing that has to do with Stokes theorem and with manifolds this has at first glimpse nothing to do as well. What we are dealing with here is firstly submanifolds that are embedded in some Euclidean space. Then, for the calculation, one has simply used a coordinate transformation. Please note that Stokes' theorem (one of the most beautiful theorems) expresses the duality between cohomology and homology (mostly in the de Rham sense). If you would like to establish some relationship between the volume of an n-disk and the volume of its boundary, then you will have to calculate both separately and afterwards compare the two results.
This may sound a bit harsh, but I really appreciate your questions because it is exactly what you have to think about when you work in the field of differential geometry and read more advanced books and research papers.
A: The size of the boundary times the rate at which the boundary moves equals the rate at which the size of the bounded region changes.
There appears to be no conventional name for this fact.  I've called it the boundary rule sometimes.
