Are there any general statements of the curl or the divergence of a 3-dimensional vectorial function, e.g. for the magnetic field:
$$|\nabla\times\boldsymbol B\left(t,\vec{x}\right)| = ?$$ $$|\nabla\cdot\boldsymbol B\left(t,\vec{x}\right)| = ?$$
Thank you in advance. :)
On
The accepted answer is incorrect. Instead, it should be:
$$ \begin{align} |\nabla \times \vec{v}|^{2} &=(\nabla \times \vec{v}) \cdot(\nabla \times \vec{v}) \\ &=\epsilon_{i j k}\left(\partial_{j} v_{k}\right) \varepsilon_{i p q}\left(\partial_{p} v_{q}\right) \\ &=\left(\delta_{j p} \delta_{k q}-\delta_{j q} \delta_{k p}\right)\left(\partial_{j} v_{k}\right)\left(\partial_{p} v_{q}\right) \\ &=\left(\partial_{p} v_{q}\right)\left(\partial_{p} v_{q}\right)-\left(\partial_{q} v_{p}\right)\left(\partial_{p} v_{q}\right) \\ &=\sum_{p=1}^3\sum_{q=1}^3\left(\partial_{p} v_{q}\right)^{2}-\left(\partial_{q} v_{p}\right)\left(\partial_{p} v_{q}\right) \\ &=\operatorname{Tr}\left(JJ^\mathrm{T}\right)-\left(\partial_{q} v_{p}\right)\left(\partial_{p} v_{q}\right) \\ &=\operatorname{Tr}\left(JJ^\mathrm{T}\right)-\operatorname{Tr}\left(J^{2}\right) \end{align} $$
In cartesian coordinates, we have that $$(\nabla \times \vec{v})_i=\epsilon_{ijk}\partial_jv_k$$ So $$\begin{align} |\nabla \times \vec{v}|^2&=(\nabla \times \vec{v})\cdot(\nabla \times \vec{v})\\&=\epsilon_{ijk}(\partial_jv_k)\epsilon_{ipq}(\partial_pv_q)\\ &=(\delta_{jp}\delta_{kq}-\delta_{jq}\delta_{kp})(\partial_jv_k)(\partial_pv_q)\\ &=(\partial_pv_q)(\partial_pv_q)-(\partial_qv_p)(\partial_pv_q)\\ &=(\nabla\cdot v)^2-(\partial_qv_p)(\partial_pv_q)\\ &=|\nabla \cdot v|^2-J_{pq}J_{qp}\\ &=|\nabla \cdot v|^2-\operatorname{Tr}(J^2) \end{align}$$ Where $J$ is the jacobian matrix.