Curlgrad

Curlgrad is the shorthand for the curl of the gradient of a scalar field, written as curlgrad f or ∇ × ∇f. In three-dimensional space, if f: R^3 → R is a scalar field with continuous second partial derivatives (i.e., f is C^2), then ∇ × ∇f = 0. This identity follows from the equality of mixed partial derivatives (Clairaut’s theorem), since the components of the curl involve second derivatives that cancel pairwise.

The result implies that gradient fields are irrotational: the flow described by ∇f has no intrinsic rotation, and line integrals of ∇f along closed curves vanish. More generally, ∇f is a conservative vector field, meaning its work along any path depends only on the endpoints through f.

Regarding the converse, a general vector field F with curl F = 0 on a simply connected region is locally the gradient of some scalar potential (F = ∇g); this is the Poincaré lemma. However, curl F = 0 does not automatically imply F is a gradient on domains that are not simply connected or if F lacks sufficient smoothness. The curlgrad identity itself does not rely on topology, but the relationship between curl-free fields and gradients does.

The statement extends to the language of differential forms, with d(d f) = 0, where curl corresponds to the exterior derivative on 1-forms. In higher dimensions, the curlgrad notation generalizes analogously as appropriate.

Example: f(x,y,z) = x^2 + y^2 + z^2 gives ∇f = (2x, 2y, 2z) and ∇ × ∇f = (0, 0, 0). Applications appear in physics and engineering, such as electromagnetism and potential-flow theory, and curlgrad serves as a basic check for potential fields in numerical methods.