 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,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.