Vector calculus identity

A vector calculus identity relates combinations of gradient, divergence and curl, subject to regularity and coordinate assumptions. For a twice continuously differentiable scalar

f

and vector field

A

in three-dimensional Euclidean space,

\nabla \times \nabla f = 0, \nabla \cdot (\nabla \times A) = 0.

These follow because mixed partial derivatives commute while the curl antisymmetrizes them. They explain why a conservative force has zero curl and a field represented as a curl has zero divergence.

The converses require domain hypotheses. A curl-free field around a puncture can have a nonzero loop integral, so local differentiation alone does not guarantee a global scalar potential. Singular fields must be handled with distributional derivatives when their classical formulas omit concentrated sources.

Table of contents
- Divergence of a curl Vector calculus identity

Divergence of a curl ( $\nabla \cdot (\nabla \times A) = 0$ )

For a twice continuously differentiable vector field in Cartesian coordinates, expand the curl and then its divergence. Every mixed partial derivative occurs twice with opposite signs, proving the identity. In electromagnetism this forces the right side of the Ampère-Maxwell equation to be divergence-free, yielding local charge conservation.

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