Curl of the gradient:
The curl of the gradient of any scalar field φ is always the zero vector field
Divergence of the curl:
The divergence of the curl of any vector field F is always zero.
Divergence of the gradient:
The Laplacian of a scalar field is the divergence of its gradient
the result is a scalar quantity.
Curl of the curl:
Here, ∇2 the vector Laplacian operating on the vector field A