Compute the divergence in spherical coordinates.

To compute the divergence in spherical coordinates, we need to understand the formula for divergence in this coordinate system. Spherical coordinates are defined by three parameters: the radial distance $r$, the polar angle $\theta$ (measured from the positive z-axis), and the azimuthal angle $\phi$ (measured in the x-y plane from the positive x-axis).

The divergence of a vector field $\vec{F} = F_r \hat{r} + F_\theta \hat{\theta} + F_\phi \hat{\phi}$ in spherical coordinates is given by the following formula:

$$\mathrm{\nabla }\cdot \overrightarrow{F}=\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{F}_r)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta F_{\theta })+\frac{1}{r\sin \theta }\frac{\partial F_{\varphi }}{\partial \varphi }$$

Here, $F_r$, $F_\theta$, and $F_\phi$ are the components of the vector field $\vec{F}$ in the radial, polar, and azimuthal directions, respectively.

Let's go through the steps to compute the divergence of a vector field in spherical coordinates:

Step 1: Identify the components of the vector field
First, you need to have the vector field $\vec{F}$ expressed in terms of its spherical components $F_r$, $F_\theta$, and $F_\phi$.

Step 2: Compute the partial derivative with respect to $r$
Calculate the partial derivative of the radial component $F_r$ with respect to $r$, and then multiply by $r^2$. This gives you the first term in the divergence formula.

$$\frac{\partial }{\partial r}(r^2{F}_r)$$

Step 3: Compute the partial derivative with respect to $\theta$
Calculate the partial derivative of the polar component $F_\theta$ with respect to $\theta$, multiply by $\sin \theta$, and then divide by $r \sin \theta$. This gives you the second term in the divergence formula.

$$\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta F_{\theta })$$

Step 4: Compute the partial derivative with respect to $\phi$
Calculate the partial derivative of the azimuthal component $F_\phi$ with respect to $\phi$, and then divide by $r \sin \theta$. This gives you the third term in the divergence formula.

$$\frac{1}{r\sin \theta }\frac{\partial F_{\varphi }}{\partial \varphi }$$

Step 5: Sum the terms
Add the three terms obtained from steps 2, 3, and 4 to get the divergence of the vector field in spherical coordinates.

$$\mathrm{\nabla }\cdot \overrightarrow{F}=\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{F}_r)+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta F_{\theta })+\frac{1}{r\sin \theta }\frac{\partial F_{\varphi }}{\partial \varphi }$$

Step 6: Simplify the expression
Finally, simplify the expression obtained in step 5 to get the divergence in its simplest form.

By following these steps, you can compute the divergence of any vector field in spherical coordinates. Remember that the divergence is a scalar field that represents the magnitude of a vector field's source or sink at a given point.