In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed. It is named after Hermann von Helmholtz.
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 may be calculated for vector fields that satisfy certain regularity or decay conditions. A decomposition exists for all vector fields, regardless of behavior at infinity, but it is not unique. Existence follows from surjectivity of the Laplace operator. Surjectivity of elliptic operators was proved by Lars Hörmander in his work on partial differential operators.
History The Helmholtz decomposition in three dimensions was first described in 1849 by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. Their derivation required the vector fields to decay sufficiently fast at infinity. Later, this condition could be relaxed, and the Helmholtz decomposition could be extended to higher dimensions. For Riemannian manifolds, the Helmholtz-Hodge decomposition using differential geometry and tensor calculus was derived. The decomposition has become an important tool for many problems in theoretical physics, but has also found applications in animation, computer vision as well as robotics.
Three-dimensional space In physics, the term Helmholtz theorem is sometimes used to describe a special case of Helmholt's decomposition theorem for 3 dimensions. Then, a vector potential A {\displaystyle A} can be defined, such that the rotation field is given by R = ∇ × A {\displaystyle \mathbf {R} =\nabla \times \mathbf {A} } , using the curl of a vector field. Let F {\displaystyle \mathbf {F} } be a vector field on a bounded domain V ⊆ R 3 {\displaystyle V\subseteq \mathbb {R} ^{3}} , which is twice continuously differentiable inside V {\displaystyle V} , and let S {\displaystyle S} be the surface that encloses the domain V {\displaystyle V} with outward surface normal n ^ ′ {\displaystyle \mathbf {\hat {n}} '} . Then F {\displaystyle \mathbf {F} } can be decomposed into a curl-free component and a divergence-free component as follows:
F = − ∇ Φ + ∇ × A , {\displaystyle \mathbf {F} =-\nabla \Phi +\nabla \times \mathbf {A} ,}
where
Φ ( r ) = 1 4 π ∫ V ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ − 1 4 π ∮ S n ^ ′ ⋅ F ( r ′ ) | r − r ′ | d S ′ A ( r ) = 1 4 π ∫ V ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ − 1 4 π ∮ S n ^ ′ × F ( r ′ ) | r − r ′ | d S ′ {\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&={\frac {1}{4\pi }}\int _{V}{\frac {\nabla '\cdot \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\cdot {\frac {\mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} S'\\[8pt]\mathbf {A} (\mathbf {r} )&={\frac {1}{4\pi }}\int _{V}{\frac {\nabla '\times \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\times {\frac {\mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} S'\end{aligned}}}
and ∇ ′ {\displaystyle \nabla '} is the nabla operator with respect to r ′ {\displaystyle \mathbf {r'} } , not r {\displaystyle \mathbf {r} } . If V = R 3 {\displaystyle V=\mathbb {R} ^{3}} and is therefore unbounded, and F {\displaystyle \mathbf {F} } vanishes faster than 1 / r {\displaystyle 1/r} as r → ∞ {\displaystyle r\to \infty } , then one has
Φ ( r ) = 1 4 π ∫ R 3 ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ A ( r ) = 1 4 π ∫ R 3 ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ {\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla '\cdot \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'\\[8pt]\mathbf {A} (\mathbf {r} )&={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla '\times \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'\end{aligned}}}
This holds in particular if F {\displaystyle \mathbf {F} } is twice continuously differentiable in R 3 {\displaystyle \mathbb {R} ^{3}} and of bounded support.
Derivation
Solution space If ( Φ 1 , A 1 ) {\displaystyle (\Phi _{1},{\mathbf {A} _{1}})} is a Helmholtz decomposition of F {\displaystyle \mathbf {F} } , then
( Φ 2 , A 2 ) {\displaystyle (\Phi _{2},{\mathbf {A} _{2}})} is another decomposition if, and only if,
Φ 1 − Φ 2 = λ {\displaystyle \Phi _{1}-\Phi _{2}=\lambda \quad } and A 1 − A 2 = A λ + ∇ φ , {\displaystyle \quad \mathbf {A} _{1}-\mathbf {A} _{2}={\mathbf {A} }_{\lambda }+\nabla \varphi ,}
where
λ {\displaystyle \lambda } is a harmonic scalar field,
A λ {\displaystyle {\mathbf {A} }_{\lambda }} is a vector field which fulfills ∇ × A λ = ∇ λ , {\displaystyle \nabla \times {\mathbf {A} }_{\lambda }=\nabla \lambda ,}
φ {\displaystyle \varphi } is a scalar field. Proof: Set λ = Φ 2 − Φ 1 {\displaystyle \lambda =\Phi _{2}-\Phi _{1}} and B = A 2 − A 1 {\displaystyle {\mathbf {B} =A_{2}-A_{1}}} . According to the definition of the Helmholtz decomposition, the condition is equivalent to
− ∇ λ + ∇ × B = 0 {\displaystyle -\nabla \lambda +\nabla \times \mathbf {B} =0} . Taking the divergence of each member of this equation yields
∇ 2 λ = 0 {\displaystyle \nabla ^{2}\lambda =0} , hence λ {\displaystyle \lambda } is harmonic. Conversely, given any harmonic function λ {\displaystyle \lambda } ,
∇ λ {\displaystyle \nabla \lambda } is solenoidal since
∇ ⋅ ( ∇ λ ) = ∇ 2 λ = 0. {\displaystyle \nabla \cdot (\nabla \lambda )=\nabla ^{2}\lambda =0.}
Thus, according to the above section, there exists a vector field A λ {\displaystyle {\mathbf {A} }_{\lambda }} such that
∇ λ = ∇ × A λ {\displaystyle \nabla \lambda =\nabla \times {\mathbf {A} }_{\lambda }} . If A ′ λ {\displaystyle {\mathbf {A} '}_{\lambda }} is another such vector field, then C = A λ − A ′ λ {\displaystyle \mathbf {C} ={\mathbf {A} }_{\lambda }-{\mathbf {A} '}_{\lambda }} fulfills ∇ × C = 0 {\displaystyle \nabla \times {\mathbf {C} }=0} , hence C = ∇ φ {\displaystyle C=\nabla \varphi }
for some scalar field φ {\displaystyle \varphi } .
Fields with prescribed divergence and curl The term "Helmholtz theorem" can also refer to the following. Let C be a solenoidal vector field and d a scalar field on R3 which are sufficiently smooth and which vanish faster than 1/r2 at infinity. Then there exists a vector field F such that
∇ ⋅ F = d and ∇ × F = C ; {\displaystyle \nabla \cdot \mathbf {F} =d\quad {\text{ and }}\quad \nabla \times \mathbf {F} =\mathbf {C} ;}
if additionally the vector field F vanishes as r → ∞, then F is unique. In other words, a vector field can be constructed with both a specified divergence and a specified curl, and if it also vanishes at infinity, it is uniquely specified by its divergence and curl. This theorem is of great importance in electrostatics, since Maxwell's equations for the electric and magnetic fields in the static case are of exactly this type. The proof is by a construction generalizing the one given above: we set
F = ∇ ( G ( d ) ) − ∇ × ( G ( C ) ) , {\displaystyle \mathbf {F} =\nabla ({\mathcal {G}}(d))-\nabla \times ({\mathcal {G}}(\mathbf {C} )),}
where G {\displaystyle {\mathcal {G}}} represents the Newtonian potential operator. (When acting on a vector field, such as ∇ × F, it is defined to act on each component.)
Weak formulation The Helmholtz decomposition can be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose Ω is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field u ∈ (L2(Ω))3 has an orthogonal decomposition:
u = ∇ φ + ∇ × A {\displaystyle \mathbf {u} =\nabla \varphi +\nabla \times \mathbf {A} }
where φ is in the Sobolev space H1(Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and A ∈ H(curl, Ω), the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl. For a slightly smoother vector field u ∈ H(curl, Ω), a similar decomposition holds:
u = ∇ φ + v {\displaystyle \mathbf {u} =\nabla \varphi +\mathbf {v} }
where φ ∈ H1(Ω), v ∈ (H1(Ω))d.
Derivation from the Fourier transform Note that in the theorem stated here, we have imposed the condition that if F {\displaystyle \mathbf {F} } is not defined on a bounded domain, then F {\displaystyle \mathbf {F} } shall decay faster than 1 / r {\displaystyle 1/r} . Thus, the Fourier transform of F {\displaystyle \mathbf {F} } , denoted as G {\displaystyle \mathbf {G} } , is guaranteed to exist. We apply the convention
F ( r ) = ∭ G ( k ) e i k ⋅ r d V k {\displaystyle \mathbf {F} (\mathbf {r} )=\iiint \mathbf {G} (\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}}
The Fourier transform of a scalar field is a scalar field, and the Fourier transform of a vector field is a vector field of same dimension. Now consider the following scalar and vector fields:
G Φ ( k ) = i k ⋅ G ( k ) ‖ k ‖ 2 G A ( k ) = i k × G ( k ) ‖ k ‖ 2 Φ ( r ) = ∭ G Φ ( k ) e i k ⋅ r d V k A ( r ) = ∭ G A ( k ) e i k ⋅ r d V k {\displaystyle {\begin{aligned}G_{\Phi }(\mathbf {k} )&=i{\frac {\mathbf {k} \cdot \mathbf {G} (\mathbf {k} )}{\|\mathbf {k} \|^{2}}}\\\mathbf {G} _{\mathbf {A} }(\mathbf {k} )&=i{\frac {\mathbf {k} \times \mathbf {G} (\mathbf {k} )}{\|\mathbf {k} \|^{2}}}\\[8pt]\Phi (\mathbf {r} )&=\iiint G_{\Phi }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}\\\mathbf {A} (\mathbf {r} )&=\iiint \mathbf {G} _{\mathbf {A} }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}\end{aligned}}}
Hence
G ( k ) = − i k G Φ ( k ) + i k × G A ( k ) F ( r ) = − ∭ i k G Φ ( k ) e i k ⋅ r d V k + ∭ i k × G A ( k ) e i k ⋅ r d V k = − ∇ Φ ( r ) + ∇ × A ( r ) {\displaystyle {\begin{aligned}\mathbf {G} (\mathbf {k} )&=-i\mathbf {k} G_{\Phi }(\mathbf {k} )+i\mathbf {k} \times \mathbf {G} _{\mathbf {A} }(\mathbf {k} )\\[6pt]\mathbf {F} (\mathbf {r} )&=-\iiint i\mathbf {k} G_{\Phi }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}+\iiint i\mathbf {k} \times \mathbf {G} _{\mathbf {A} }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}\\&=-\nabla \Phi (\mathbf {r} )+\nabla \times \mathbf {A} (\mathbf {r} )\end{aligned}}}
Longitudinal and transverse fields A terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component. This terminology comes from the following construction: Compute the three-dimensional Fourier transform F ^ {\displaystyle {\hat {\mathbf {F} }}} of the vector field F {\displaystyle \mathbf {F} } . Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k. So far, we have
F ^ ( k ) = F ^ t ( k ) + F ^ l ( k ) {\displaystyle {\hat {\mathbf {F} }}(\mathbf {k} )={\hat {\mathbf {F} }}_{t}(\mathbf {k} )+{\hat {\mathbf {F} }}_{l}(\mathbf {k} )}
k ⋅