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 there exist and such that

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 that goes to zero faster than as can be expressed as the gradient of a scalar plus the curl of a vector.¹ vec

Calculation

Let be the divergence of and be the curl. Then, the scalar potential of the irrotational component is given by

and the vector potential of the incompressible component is given by

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 and the curl of a vector function are specified, and if they both go to zero faster than as , then may be determined uniquely.¹ vec

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Footnotes

1. 2013. Introduction to electrodynamicsp. 582 (Appendix B) ↩ ↩² ↩³