Curl

Curl is a differential operator that measures the local circulation of a vector field. It produces a vector whose direction gives the rotation axis and whose magnitude gives circulation per unit area.

Cross product: A binary operation on three-dimensional vectors that produces a vector perpendicular to both inputs. Curl’s Cartesian formula combines spatial derivatives through cross products.
Vector field: A function assigning a vector to each point in a region of space. Curl acts on vector fields and describes their local variation.
Navier–Stokes equations: Nonlinear equations governing the motion of viscous fluids. Taking curl of these equations yields evolution equations for fluid vorticity.
Exterior derivative: An operator that maps differential forms to higher-degree forms and generalizes gradient, curl, and divergence. It provides a coordinate-independent framework that includes curl.
Two-dimensional curl: The scalar measure of local rotation for a planar vector field, given by the out-of-plane component of its three-dimensional curl. In a plane, curl reduces from a vector to a single signed component.
Levi-Civita symbol: A tensor whose components encode orientation and antisymmetric permutations of indices. It compactly expresses curl’s component formula and orientation dependence.
Partial derivative: A derivative measuring how a multivariable function changes with respect to one variable while others are held fixed. Curl is assembled from spatial partial derivatives of field components.
Euler equations: Equations describing the motion of an ideal, inviscid fluid. Their rotational form tracks how curl of velocity changes in ideal flow.
Differential forms: Antisymmetric tensor fields that can be integrated over curves, surfaces, and higher-dimensional regions. They express circulation and Stokes' theorem without privileging coordinates.
Covariant derivative: A derivative that accounts for how vector or tensor components vary across curved spaces or changing bases. Ordinary curl formulas require adjustment when coordinates or geometry vary.

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