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.

Because 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 $-\operatorname {grad} \Phi +\operatorname {curl} \mathbf {A}$ where $Φ$ is a scalar field, called scalar potential, and $A$ is a vector field called a vector potential.

Let $F$ be a vector field on a bounded domain $V \subseteq R 3$ , which is twice continuously differentiable, and let $S$ be the surface that encloses the domain $V$ . Then $F$ can be decomposed into a curl-free component and a divergence-free component:

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