Vector calculus | Analytic geometry | Theorems in analysis

Helmholtz decomposition

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz. As an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field (satisfying appropriate smoothness and decay conditions) can be decomposed as the sum of the form , where is a scalar field called "scalar potential", and A is a vector field, called a vector potential. (Wikipedia).

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Related pages

Curl (mathematics) | Helmholtz decomposition | Clebsch representation | Vector Laplacian | Poloidal–toroidal decomposition | Sobolev space | Kronecker delta | Solenoidal vector field | Hodge theory | Maxwell's equations | Vector calculus | Divergence theorem | Divergence | Lipschitz domain | Green's function | Multi-index notation | Mathematics | Polar factorization theorem | Distribution (mathematics) | Riemannian manifold | Orthogonality | Harmonic function | Levi-Civita symbol | Vector potential | Compact space | Scalar potential | Einstein notation | Newtonian potential | Fourier transform | Vector field