Pre-Quantum Electrodynamics

$\mathbf{\nabla }\cdot (\mathbf{\nabla }T)\equiv {\mathbf{\nabla }}^{2}T$ is called the Laplacian of the scalar field $T$. The Laplacian of a vector field ${\mathbf{\nabla }}^2\mathbf{v}$ is also defined as the vector with components given by the Laplacian of the corresponding vector elements.

This always vanishes.

$\mathbf{\nabla }(\mathbf{\nabla }\cdot \mathbf{v})$ does not appear often in physics. No special name.

This always vanishes.

$$\begin{array}{}\text{(curlcurl)}& \mathbf{\nabla }\times (\mathbf{\nabla }\times \mathbf{v})=\mathbf{\nabla }(\mathbf{\nabla }\cdot \mathbf{v})-{\mathbf{\nabla }}^2\mathbf{v}\end{array}$$