Let $\;u:\mathbb R^n \rightarrow \mathbb R^m\;$. I want to know what the Laplacian operator of $\;u\;$ will be like. Searching on the web, I found the following formula:
$\;{\nabla}^2 u=\nabla ( \nabla \cdot u)-\nabla \times (\nabla \times u)\;$
which I have trouble applying on $\;u\;$ for several reasons. For example, I don't know how to compute the gradient of the matrix $\; \nabla \cdot u \;$.
I know $\; \nabla \cdot u=\begin{pmatrix} \frac{\partial u_1}{\partial x_1} \dots \frac{\partial u_1}{\partial x_n} \\ \dots \\ \frac{\partial u_m}{\partial x_1} \dots \frac{\partial u_m}{\partial x_n}\\ \end{pmatrix}\;$
but I have no clue how should I compute the gradient of the above.
I've only seen the Laplacian operator for these kind of functions: $\;f:\mathbb R^n \rightarrow \mathbb R\;$, so I 'm a bit lost now.
Any help would be valuable. Any suggestions at books that I could study from, are also welcome.
Thanks in advance!