Chapter 10.11, Problem 12MP

The square matrix in equation (10.3) is called the Jacobian matrix $J$; the determinant of this matrix is the Jacobian $J=$ det $J$ which we used in Chapter 5, Section 4 to find volume elements in multiple integrals. (Note that as in Chapter 3, J represents a matrix; $J$ in italics is its determinant.) For the transformation to spherical coordinates in (10.1) and (10.2) show that $J=\det \text{J}=r^2\sin \theta .$ Recall that the spherical coordinate volume element is $r^2$ sin $\theta$ dr $d\theta d\varphi .$ Hint: Find ${\text{J}}^{\text{T}}\text{J}$ and note that $\det \left(J^{\text{T}}J\right)={(\det J)}^2.$