Helmholtz decomposition
Definition

For a vector field F ∈ C 1 (V, R n) {\displaystyle \mathbf {F} \in C^{1}(V,\mathbb {R} ^{n})} defined on a domain V ⊆ R n {\displaystyle V\subseteq \mathbb {R} ^{n}}, a Helmholtz decomposition is a pair of vector fields G ∈ C 1 (V, R n) {\displaystyle \mathbf {G} \in C^{1}(V,\mathbb {R} ^{n})} and R ∈ C 1 (V, R n) {\displaystyle \mathbf {R} \in C^{1}(V,\mathbb {R} ^{n})} such that: F (r) = G (r) + R (r), G (r) = − ∇ Φ (r), ∇ ⋅ R (r) = 0. {\displaystyle {\begin{aligned}\mathbf {F} (\mathbf {r})&=\mathbf {G} (\mathbf {r})+\mathbf {R} (\mathbf {r}),\\\mathbf {G} (\mathbf {r})&=-\nabla \Phi (\mathbf {r}),\\\nabla \cdot \mathbf {R} (\mathbf {r})&=0.\end{aligned}}} Here, Φ ∈ C 2 (V, R) {\displaystyle \Phi \in C^{2}(V,\mathbb {R})} is a scalar potential, ∇ Φ {\displaystyle \nabla \Phi } is its gradient, and ∇ ⋅ R {\displaystyle \nabla \cdot \mathbf {R} } is the divergence of the vector field R {\displaystyle \mathbf {R} }. The irrotational vector field G {\displaystyle \mathbf {G} } is called a gradient field and R {\displaystyle \mathbf {R} } is called a solenoidal field or rotation field. This decomposition does not exist for all vector fields and is not unique.
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