Given the vector field $\vec A(\vec r) = \vec r$, I have to calculate the vector flux through a sphere whose center is located in the origin. I want to apply Gauß-Theorem and use spherical coordinates.
The vector field in spherical coordinates is $\vec A (\vec r) = \begin{pmatrix} r \\ 0 \\ 0 \end{pmatrix}$. Now I have to calculate $\int \vec A d\vec F$, and here i am very unsure what $\vec F$ is because I never did it in spherical coordinates.
My attempt is $\vec F = \vec e_r = \begin{pmatrix} \cos(\phi)\sin(\theta) \\ \sin(\phi)\sin(\theta) \\ \cos(\theta)\end{pmatrix}d\phi d\theta$ .
Is this correct and if no, what did I miss?
$d\vec{F}$ is the vector pointing out of the infinitesimal area element of your sphere. Therefore: $d\vec{F}=dS\cdot\vec{n}$, where $dS=r^2\sin\theta d\theta d\phi$ and $\vec{n}=\vec{r}$ (by symmetry of the problem).