### Vector analysis

We use $\mathsf{rot} (\mathbf{A} \times \mathbf{B}) = \nabla \times (\mathbf{A} \times \mathbf{B})= \mathbf{A}( \nabla \cdot \mathbf{B} ) + (\mathbf{B}\cdot \nabla) \mathbf{A} -\mathbf{B} ( \nabla \cdot \mathbf{A} ) - (\mathbf{A}\cdot \nabla) \mathbf{B}$ (Cartesian coordinates only !).
Here $\mathbf{A}=\mathbf{m}=\text{const}$, $\mathbf{B}=\mathbf{r}/r^3$, so that
$\mathbf{C}=\mathbf{m} \mathsf{div} \left(\frac{\mathbf{r}}{r^3}\right) -(\mathbf{m} \cdot \nabla)\left(\frac{\mathbf{r}}{r^3}\right)\\ =\mathbf{m} \left(\frac{1}{r^3} \mathsf{div} \mathbf{r} + \mathbf{r} \cdot \mathsf{grad}(\frac{1}{r^3}\right) \\ -\frac{1}{r^3} (\mathbf{m} \cdot \nabla)\mathbf{r} - \mathbf{r} (\mathbf{m} \cdot \mathsf{grad}\frac{1}{r^3})\\ =\frac{3\mathbf{m}}{r^3} -\frac{3\mathbf{m}}{r^3} -\frac{\mathbf{m}}{r^3} + \frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{r^5}\\ =\frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r}) - r^2\mathbf{m}}{r^5}\\ ^$
Another method.
We use $\mathsf{rot} (f\mathbf{A})= f\mathsf{rot}{\mathbf{A}}+ \mathsf{grad} f\times \mathbf{A}$:
$\mathsf{rot} (\frac{\mathbf{m\times r}}{r^3} ) =\frac{1}{r^3} \mathsf{rot}(\mathbf{m\times r}) + (\mathsf{grad}\frac{1}{r^3}) \times (\mathbf{m\times r})\\ = \frac{2\mathbf{m}}{r^3}-\frac{3}{r^5} \mathbf{r}\times (\mathbf{m}\times\mathbf{r})\\ =\frac{3\mathbf{r}(\mathbf{m}\cdot \mathbf{r})-r^2\mathbf{m}}{r^5}$
(for $\mathsf{rot} (\mathbf{m}\times\mathbf{r})=2\mathbf{m}$ see below).