Curl of the gradient:
The curl of the gradient of any scalar field φ is always the zero vector field
$$ \nabla \times (\nabla \varphi)=0 $$
Divergence of the curl:
The divergence of the curl of any vector field F is always zero.
$$ \nabla \cdot(\nabla \times F)=0 $$
Divergence of the gradient:
The Laplacian of a scalar field is the divergence of its gradient
$$ \nabla^2 f = \nabla \cdot \nabla f $$
the result is a scalar quantity.
Curl of the curl:
$$ \nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) -\nabla^2A $$
Here, ∇2 the vector Laplacian operating on the vector field A