Vector field

Helmholtz theorem

The Helmholtz theorem is sometimes referred to as the fundamental theorem of vector calculus states the existence of the Helmholtz decomposition for a sufficiently smooth rapidly decaying vector field, that is for any vector field ⃗𝐅 :ℝ3 →ℝ3 there exist 𝑈 :ℝ3 →ℝ and ⃗𝐖 :ℝ3 →ℝ3 such that

⃗𝐅=−⃗∇𝑈+⃗∇×⃗𝐖

In other words, a vector field may be decomposed into a

• Conservative vector field with a scalar potential
• Incompressible vector field with a vector potential

Helmholtz Theorem Any differentiable vector function ⃗𝐅(⃗𝐫) that goes to zero faster than 1/𝑟 as 𝑟 →∞ can be expressed as the gradient of a scalar plus the curl of a vector.¹ vec

Calculation

Let 𝐷 =⃗∇ ⋅⃗𝐅 be the divergence of ⃗𝐅 and ⃗𝐂 =⃗∇ ⋅⃗𝐅 be the curl. Then, the scalar potential of the irrotational component is given by

𝑈(⃗𝐫)=14𝜋∭ℝ3𝐷(⃗𝐫′)𝔯𝑑𝜏′=14𝜋∭ℝ3𝐷(⃗𝐫′)‖⃗𝐫−⃗𝐫′‖𝑑𝜏′

and the vector potential of the incompressible component is given by

⃗𝐖(⃗𝐫)=14𝜋∭ℝ3⃗𝐂(⃗𝐫′)𝔯𝑑𝜏′=14𝜋∭ℝ3⃗𝐂(⃗𝐫′)‖⃗𝐫−⃗𝐫′‖𝑑𝜏′

For a derivation, see Griffiths¹.

Corollary: Uniqueness

Part of the reason that the Helmholtz theorem is so important is it verifies the uniqueness of solutions to Maxwell’s equations, which is given by the corollary

Corollary If the divergence 𝐷(⃗𝐫) and the curl ⃗𝐂(⃗𝐫) of a vector function ⃗𝐅(⃗𝐫) are specified, and if they both go to zero faster than 1/𝑟2 as 𝑟 →∞, then ⃗𝐅 may be determined uniquely.¹ vec

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Footnotes

1. 2013. Introduction to electrodynamics, §B, p. 582. ↩ ↩² ↩³