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Homework Statement:

In spherical polar coordinates, the element of volume for a body that is symmetrical about the polar axis is,
$$\begin{equation}
dV=2\pi sin{\theta} dr d\theta
\end{equation}$$
Whilst its element of surface area is,
$$\begin{equation}
dS=2\pi rsin{\theta} \sqrt{dr^2+r^2d\theta^2}
\end{equation}$$
Although the homework statement continues, my question is actually about how the expression for dS given in the problem statement was arrived at in the first place.
Relevant Equations:

This question is at the end of a chapter covering multiple integrals, including a change of variables in multiple integrals where, for a change of variables from Cartesian coordinates to spherical polar coordinates, we have,
$$\begin{equation}
dxdydz = Det[J] dr d\phi d\theta = r^2 sin\theta dr d\phi d\theta
\end{equation}$$
Where ## Det[J]## is the determinant of the 3x3 Jacobian Matrix containing all the partial derivatives of ##x,y,z## with respect to ##r,θ,ϕ##
r,θ,ϕ
For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at. That's what I know. However, trying to take the determinant of the Jacobian in this case obviously does not get me the expression given in the problem statement and I'm not sure where to begin. Thank you for your help. I'm not actually in school but looking to go back for Physics and don't have any teacher to turn to.
For integration over the ##x y plane## the area element in polar coordinates is obviously ##r d \phi dr ## I can also easily see ,geometrically, how an area element on a sphere is ##r^2 sin\theta d\phi ## And I can verify these two cases with the Jacobian matrix. So that's where I'm at. That's what I know. However, trying to take the determinant of the Jacobian in this case obviously does not get me the expression given in the problem statement and I'm not sure where to begin. Thank you for your help. I'm not actually in school but looking to go back for Physics and don't have any teacher to turn to.