I have the following:
\begin{align} \mathbf{u} = \frac{(\mathbf{n}\times\mathbf{c})\times\mathbf{n}}{\sqrt{(\mathbf{n}\times\mathbf{c})\!\cdot\!(\mathbf{n}\times\mathbf{c})}} \end{align}
where $\mathbf{c}$ is a constant vector and $\mathbf{n}$ is defined as
\begin{align} \mathbf{n}(\theta,\phi) = \begin{bmatrix} \cos\theta\sin\phi \\ \sin\theta\sin\phi \\ \cos\phi \end{bmatrix} \end{align}
Can anyone help me find the partial derivatives $\frac{\partial \mathbf{u}}{\partial \theta}$, $\frac{\partial \mathbf{u}}{\partial \phi}$, $\frac{\partial^2 \mathbf{u}}{\ \partial \theta^2}$, and $\frac{\partial^2 \mathbf{u}}{\ \partial \phi^2}$?
Hint \begin{equation} (f g^{-1/2})' = f' g^{-1/2} - \frac{1}{2}f g^{-3/2} g' \end{equation}