In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference): $$C=\frac{d}{dr}(r^2\pi)=2r\pi.$$
The same holds for a sphere and its area:
$$A=\frac{d}{dr}\left(\frac{4}{3}r^3\pi \right)=4r^2\pi$$
Than I wondered does it hold for arbitrary smooth objects with arbitrary genus.
I looked up the volume and surface area of a torus.
$$A=\frac{\partial}{\partial r}\left(2\pi ^{2}Rr^{2}\right)=\left(4\pi ^{2}Rr\right)$$
It also seems to hold.
Does it hold for surfaces of arbitrary genus in arbitrary dimensions?
The answer to the linked question says that the following holds for a bounded smooth body $S$:
$$|\partial S| = \frac{d}{dr} |S_r| |_{r=0}$$
where $S_r$ is the volume of the body.
But I am not really sure how the genus of a body (if at all) influences this.
