Differentiating Spherical Basis Vectors With Respect to Time
\[\mathbf{e_r} = sin \theta cos \phi \mathbf{i} + sin \theta sin \phi \mathbf{j} + cos \theta \mathbf{k}\]
\[\mathbf{e_\theta } = cos \theta cos \phi \mathbf{i} + cos \theta sin \phi \mathbf{j} - sin \theta \mathbf{k}\]
\[\mathbf{e_\phi} = - sin \theta sin \phi \mathbf{i} + sin \theta cos \phi \mathbf{j} \]
Differente
\[\mathbf{e_r}\]
with respect to time to obtain\[\begin{equation} \begin{aligned} \dot{\mathbf{e_r}} &= \dot{\theta} (cos \theta cos \phi \mathbf{i} + cos \theta sin \phi \mathbf{j} -sin \theta \mathbf{k}) + \dot{\phi}(-sin \theta sin \phi \mathbf{i} + sin \theta cos \phi \mathbf{j}) \\ &= \dot{\theta} {\mathbf{e_\theta}} + \dot{\phi} sin \theta \mathbf{e_{\phi}} \end{aligned} \end{equation}\]
Differente
\[\mathbf{e_\theta}\]
with respect to time to obtain\[\begin{equation} \begin{aligned} \dot{\mathbf{e_\theta}} &= \dot{\theta} (- sin \theta cos \phi \mathbf{i} - sin \theta sin \phi \mathbf{j} -cos \theta \mathbf{k}) + \dot{\phi}(-cos \theta sin \phi \mathbf{i} + cos \theta cos \phi \mathbf{j}) \\ &= - \dot{\theta} \mathbf{e_r} + \dot{\phi} cos \theta \mathbf{e_{\phi}} \end{aligned} \end{equation}\]
Differente
\[\mathbf{e_\phi}\]
with respect to time to obtain\[\begin{equation} \begin{aligned} \dot{\mathbf{e_\phi}} &= - \dot{\phi}(cos \theta \mathbf{i} + sin \theta \mathbf{j} ) \\ &= - \dot{\phi} ( sin \theta (sin \theta cos \phi \mathbf{i} + sin \theta sin \phi \mathbf{j} + cos \theta \mathbf{k}) \\ &+ cos \theta (cos \theta cos \phi \mathbf{i} + cos \theta sin \phi \mathbf{j} - sin \theta \mathbf{k})) \\ &= -\dot{\phi} sin \theta \mathbf{e_r} - \dot{\phi} cos \theta \mathbf{e_\theta} \end{aligned} \end{equation}\]
