*Find the corresponding formulae for \(\partial e_{i} / \partial q_{j}\) for spherical polar coordinates, and hence verify the results obtained in Problem \(21 .\)

Question: Compute the partial derivatives of the spherical polar basis vectors \(\frac{\partial e_i}{\partial q_j}\), and verify the results obtained in Problem 21. Answer: We computed the following partial derivatives: - With respect to \(r\): $$\frac{\partial e_r}{\partial r} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \quad \frac{\partial e_\theta}{\partial r} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}, \quad \frac{\partial e_\phi}{\partial r} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ - With respect to \(\theta\): $$\frac{\partial e_r}{\partial \theta} = \begin{pmatrix} \cos \theta \cos \phi \\ \cos \theta \sin \phi \\ -\sin \theta \end{pmatrix}, \quad \frac{\partial e_\theta}{\partial \theta} = \begin{pmatrix} -\sin \theta \cos \phi \\ -\sin \theta \sin \phi \\ -\cos \theta \end{pmatrix}, \quad \frac{\partial e_\phi}{\partial \theta} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ - With respect to \(\phi\): $$\frac{\partial e_r}{\partial \phi} = \begin{pmatrix} -\sin \theta \sin \phi \\ \sin \theta \cos \phi \\ 0 \end{pmatrix}, \quad \frac{\partial e_\theta}{\partial \phi} = \begin{pmatrix} -\cos \theta \sin \phi \\ \cos \theta \cos \phi \\ 0 \end{pmatrix}, \quad \frac{\partial e_\phi}{\partial \phi} = \begin{pmatrix} -\cos \phi \\ -\sin \phi \\ 0 \end{pmatrix}$$ These results are in agreement with those found in Problem 21, and thus, we have successfully computed the partial derivatives of the spherical polar basis vectors and verified the results.