HelmholtzHodgeZerlegung
The Helmholtz-Hodge decomposition is a fundamental theorem in vector calculus and differential geometry that decomposes a vector field into three mutually orthogonal components. For a vector field F defined on a domain, the theorem states that F can be uniquely written as the sum of a gradient field (irrotational component), a curl field (solenoidal component), and a harmonic field. The gradient field is the curl of zero, meaning it can be expressed as the gradient of a scalar potential. The curl field is the divergence of zero, meaning it can be expressed as the curl of a vector potential. The harmonic field is both irrotational and solenoidal.
The decomposition is particularly important in the study of partial differential equations and in areas such