Second derivatives – vector calculus formulas

Second derivatives

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

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