Second derivatives – vector calculus formulas

Second derivatives

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.

$$\nabla \cdot(\nabla \times F)=0$$

$$\nabla^2 f = \nabla \cdot \nabla f$$
$$\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) -\nabla^2A$$