Helmholtz decomposition

Helmholtz decomposition expresses a sufficiently regular vector field as the sum of an irrotational field and a divergence-free field. Its uniqueness and form depend on the domain and boundary conditions.

Gradient: The vector of spatial derivatives of a scalar field, pointing in the direction of greatest increase. The irrotational component can be represented as the gradient of a scalar potential.
Irrotational field: A vector field whose curl vanishes throughout its domain. This is the defining property of the potential component in the decomposition.
Electrostatics: The study of electric charges and fields that do not vary with time. Electrostatic fields are curl-free and can be described by scalar potentials.
Helmholtz theorem: A theorem stating conditions under which a vector field is determined by its divergence, curl, and boundary behavior. It is closely related, but emphasizes reconstruction and uniqueness rather than the decomposition itself.
Divergence: A scalar measure of a vector field’s local net outward flow per unit volume. The decomposition separates out a component whose divergence vanishes.
Solenoidal vector field: A vector field whose divergence vanishes throughout its domain. This is the defining property of the divergence-free component.
Magnetostatics: The study of steady electric currents and their time-independent magnetic fields. Magnetic fields have zero divergence, illustrating the solenoidal component.
Fourier decomposition: A representation of a function as a sum or integral of oscillatory modes. It decomposes fields by frequency, whereas Helmholtz decomposition separates longitudinal and transverse structure.
Curl: A vector measure of a vector field’s local rotation and circulation. The divergence-free component is determined through its curl, together with boundary data.
Scalar potential: A scalar function whose gradient gives a conservative or irrotational vector field. It represents the component with zero curl.

Linked from 3 pages

Magnetic vector potentialRelated: Its decomposition clarifies how potentials represent solenoidal fields.