I have a vector field $\vec{A}$ that is given in spherical coordinates.
$$\vec{A}=\frac{1}{r^2}\hat{e}_{r}$$
I need to calculate the flux integral over a unit sphere in origo (radius 1). I cannot use Gauss theorem since there exist a singularity in the volume. I have instead attempted to calculate it in the standard way with the integral below.
$$\int_{\phi}^{ } \int_{\theta}^{ }\vec{A}(\vec{r}(\theta, \phi))\cdot(\frac{\partial{\vec{r}}}{\partial{\theta}}\times \frac{\partial{\vec{r}}}{\partial{\phi}})\space d\theta d\phi$$
So in this calculation i need a parameterization of the surface. I came up with this but have been informed that it is not a correct parametrization.
$$\vec{r}(\theta, \phi)=1\hat{e}_{r}+\theta\hat{e}_{\theta}+\phi\hat{e}_{\phi}$$
$$\theta:0\rightarrow\pi$$
$$\phi:0\rightarrow2\pi$$
Why is that? And how should you describe a sphere in spherical coordinates? Or should I instead transform the vector field into cartesian coordinates? How is that done?
This is the rest of the calculation and my final answer.
$$\frac{\partial{\vec{r}}}{\partial{\theta}}=1\hat{e}_{\theta}$$
$$\frac{\partial{\vec{r}}}{\partial{\phi}}=1\hat{e}_{\phi}$$
$$\frac{\partial{\vec{r}}}{\partial{\theta}}\times \frac{\partial{\vec{r}}}{\partial{\phi}}=1\hat{e}_{r}$$
$$\vec{A}(\vec{r}(\theta, \phi))=\frac{1} {(1)^2}\hat{e}_{r}=1\hat{e}_{r}$$
$$\vec{A}(\vec{r}(\theta, \phi))\cdot(\frac{\partial{\vec{r}}}{\partial{\theta}}\times \frac{\partial{\vec{r}}}{\partial{\phi}})=1\hat{e}_{r}\cdot1\hat{e}_{r}=1$$
$$\int_{0}^{2\pi} \int_{0}^{\pi}1\space d\theta d\phi=2\pi^2$$
Best Answer
The parameterization is not correct. The position vector has neither a $\theta$ component nor a $\phi$ component. Note that both of those compoents are normal to the position vector. Therefore, the sperical coordinate vector parameterization of a surface would be in general
$\vec r=\hat r(\theta,\phi)r(\theta,\phi)$.
For a spherical surface of unit radius, $r(\theta,\phi)=1$ and
$$\vec r=\hat r(\theta,\phi)$$
where the unit vector $\hat r(\theta,\phi)$ can be expressed on Cartesian coordinates as
$$\hat r(\theta,\phi)=\hat x\sin \theta \cos \phi+\hat y\sin \theta \sin \phi+\hat z\cos \theta$$
Now, we can show that the unit normal to the sphere is $\hat r(\theta,\phi)$ since $\frac{\partial \hat r}{\partial \theta}\times \frac{\partial \hat r}{\partial \phi}=\hat \theta \times \hat \phi=\hat r$.
Can you finish now?