Second derivatves with respect to rotations in SO(3) For some background, I'm trying to calculate some physical properties such as magnetotropic coefficients for crystals in a magnetic field. I want to avoid doing discrete derivatives where I can and I need some help with the calculus.
If $R^\alpha_{\hat u}$ is a matrix implementing an rotation of angle $\alpha$ about $\hat u$ and $F(\vec r)$ is a smooth, differentiable scalar funtion over $\vec r \in \mathbb R^3$.
By using the infintessimal rotation
$$\tilde R^\alpha_{\hat u}  = \left [ \begin{matrix}
1 & -\alpha u_z & \alpha u_y\\
\alpha u_z & 1 & -\alpha u_x \\
-\alpha u_y & \alpha u_x & 1 
\end{matrix} \right ]$$
It's easy enough to show that
$$ \frac {\partial}{\partial \alpha} F(R^\alpha_{\hat u}\, \vec r)  = \hat u \cdot (\vec r \times\vec \nabla F)$$

*

*Is there a similarly simple expression for the second derivative?

*Are there any general statements that can be made about the higher-order derivatives with respect to these rotations?
$$ \frac {\partial^n}{\partial \alpha^n} F(R^\alpha_{\hat u}\, \vec r) $$
 A: From
$$
\tilde R_\hat{u}^\alpha=I+\alpha\begin{bmatrix}0&-u_z&u_y\\u_z&0&-u_x\\-u_y&u_x&0\end{bmatrix}
$$
we see that
$$
\tilde R_\hat{u}^\alpha\,\vec{r}=\vec{r}+\alpha\,\vec{u}\times\vec{r}\,.
$$
Therefore, by the chain rule,
$$
\frac{\partial}{\partial \alpha}F(\tilde R_\hat{u}^\alpha\,\vec{r})=(\vec{u}\times\vec{r})\cdot\nabla F(\tilde R_\hat{u}^\alpha\,\vec{r})
=\nabla F(\tilde R_\hat{u}^\alpha\,\vec{r})\cdot(\vec{u}\times\vec{r})
\,.
$$
(By the triple product identity this is equivalent to your equation.)
Therefore, by the chain rule again,
$$
\frac{\partial^2}{\partial \alpha^2}F(\tilde R_\hat{u}^\alpha\,\vec{r})=\sum_{i,j=1}^3\frac{\partial^2}{\partial x_i\partial x_j} F(\tilde R_\hat{u}^\alpha\,\vec{r})\,(\vec{u}\times\vec{r})_i\,(\vec{u}\times\vec{r})_j\,.
$$
Obviously,
$$
\frac{\partial^n}{\partial \alpha^n}F(\tilde R_\hat{u}^\alpha\,\vec{r})=\sum_{i_1,...,i_n=1}^3\frac{\partial^n}{\partial x_{i_1}...\partial x_{i_n}} F(\tilde R_\hat{u}^\alpha\,\vec{r})\,(\vec{u}\times\vec{r})_{i_1}...(\vec{u}\times\vec{r})_{i_n}\,.
$$
