Let $S$ be the surface of a unit sphere.
Let $\vec{A}=\frac{1}{r^2}\hat{e}_{r}$
The flux through the surface $S$ is given by: $\int_{S}\vec{A}\space\cdot\space d\vec{S}$
$$d\vec{S}=r^2\sin\theta \,d\theta d\phi\hat{e}_{r}$$
$$\int_{S}\vec{A}\space\cdot\space d\vec{S}=\int_{s}(\frac{1}{r^2}\hat{e}_{r})\cdot(r^2\sin\theta \,d\theta d\phi\hat{e}_{r})=\int_{S}\sin\theta \space d\theta d\phi=\int_{0}^{2\pi}\int_{0}^{\pi}\sin\theta \ d\theta d\phi=4\pi$$
The above steps are all correct.
My question is, how is this $d\vec{S}$ calculated?
Calculating flux using Cartesian coordinates is very familiar to me, however I can't grasp how it is done in spherical coordinates or in any other general coordinate system. Especially, "the unit vector in spherical coordinates is a function of position" part is quite confusing.
I could, of course, solve the above problem by converting everything into Cartesian coordinates. But this seems like going the "long way around" when everything is already set up so nicely in spherical coordinates.
Best Answer
There is a handy and intuitive way to derive the surface element
$$ dS=r^2\sin\theta \,d\theta d\phi\ $$
Think of the surface area $dS$ as an infinitesimal square. In spherical coordinates, one of its sides is simply the arc length $r d\theta$ along the $\theta$-direction and the other side along the $\phi$-direction is the arc length $\rho d\phi$, where the arc lies on the circle with its radius given by $\rho=r\sin\theta$. Thus, together, one has
$$ dS= (r d\theta)( \rho d\phi)=r^2\sin\theta \,d\theta d\phi\ $$
Since the surface is on a sphere, the corresponding vector form is
$$d\vec{S}=r^2\sin\theta \,d\theta d\phi\hat{e}_{r}$$