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Helmholtz' Theorem

Helmholtz’ Theorem

EE 141 Lecture Notes Topic 3

Professor K. E. Oughstun School of Engineering College of Engineering & Mathematical Sciences University of Vermont

2009 Motivation Helmholtz’ Theorem

Because 1 ∇2 = −4πδ(R) (1) R  where R = r − r′ with magnitude R = |R| and where

δ(R)= δ(r − r′)= δ(x − x ′)δ(y − y ′)δ(z − z ′) is the three-dimensional , then any sufficiently well-behaved vector function F(r)= F(x, y, z) can be represented as

′ ′ ′ 1 ′ 1 ′ F(r) = F(r )δ(r − r ) d 3r = − F(r )∇2 d 3r ZV 4π ZV R  1 F(r′) = − ∇2 d 3r ′, (2) 4π ZV R the integration extending over any region V that contains the point r. Helmholtz’ Theorem

With the identity ∇ × ∇× = ∇∇ · −∇2, Eq. (2) may be written as

1 F(r′) 1 F(r′) F(r)= ∇×∇× d 3r ′ − ∇∇ · d 3r ′. (3) 4π ZV R 4π ZV R

Consider first the term appearing in this expression. Because the vector differential operator ∇ does not operate on the primed coordinates, then

1 F(r′) 1 1 ∇· d 3r ′ = F(r′) · ∇ d 3r ′. (4) 4π ZV R 4π ZV R  Helmholtz’ Theorem

The integrand appearing in this expression may be expressed as

1 1 F(r′) · ∇ = −F(r′) · ∇′ R  R  F(r′) 1 = −∇′ · + ∇′ · F(r′), (5)  R  R where the prime on ∇′ denotes differentiation with respect to the primed coordinates alone, viz. ∂ ∂ ∂ ∇′ = 1ˆ + 1ˆ + 1ˆ x ∂x ′ y ∂y ′ z ∂z ′ when 1ˆj′ = 1ˆj , j = x, y, z. Helmholtz’ Theorem

Substitution of Eq. (5) into Eq. (4) and application of the to the first term then yields

1 F(r′) 1 F(r′) ∇· d 3r ′ = − ∇′ · d 3r ′ 4π ZV R 4π ZV  R  1 ∇′ · F(r′) + d 3r ′ 4π ZV R 1 1 = − F(r′) · ˆnd 2r ′ 4π IS R 1 ∇′ · F(r′) + d 3r ′ 4π ZV R = φ(r), (6) which is the desired form of the scalar φ(r) for the vector field F(r). Here S is the surface that encloses the regular region V containing the point r. Helmholtz’ Theorem

For the term appearing in Eq. (3) one has that

1 F(r′) 1 1 ∇ × d 3r ′ = − F(r′) × ∇ d 3r ′ 4π ZV R 4π ZV R  1 1 = F(r′) × ∇′ d 3r ′. (7) 4π ZV R  Moreover, the integrand appearing in the final form of the integral in Eq. (7) may be expressed as

′ ′ ′ ′ ′ 1 ∇ × F(r ) ′ F(r ) F(r ) × ∇ = − ∇ × , (8) R  R  R  so that Helmholtz’ Theorem

1 F(r′) 1 ∇′ × F(r′) ∇ × d 3r ′ = d 3r ′ 4π ZV R 4π ZV R 1 F(r′) − ∇′ × d 3r ′ 4π ZV  R  1 ∇′ × F(r′) = d 3r ′ 4π ZV R 1 1 + F(r′) × ˆnd 2r ′ 4π IS R = a(r), (9) which is the desired form of the . Helmholtz’ Theorem

Equations (3), (6), and (9) then show that

F(r)= −∇φ(r)+ ∇ × a(r) (10) where the φ(r) is given by Eq. (6) and the vector potential a(r) by Eq. (9).

This expression may also be written as

F(r)= Fℓ(r)+ Ft (r) (11) known as the . Helmholtz’ Theorem

In the Helmholtz decomposition,

Fℓ(r) = −∇φ(r) ′ F r′ F r′ 1 ∇ · ( ) 3 ′ 1 ( ) ˆn 2 ′ = − ∇ r r′ d r + ∇ r r′ · d r 4π ZV | − | 4π IS | − | (12) is the longitudinal or irrotational part of the vector field (with ′ ∇ × Fℓ(r )= 0), and F r′ F r a r 1 ( ) 3 ′ t ( ) = ∇ × ( )= ∇×∇× r r′ d r 4π ZV | − | ′ F r′ F r′ 1 ∇ × ( ) 3 ′ 1 ( ) ˆn 2 ′ = ∇ × ′ d r + ∇ × ′ × d r 4π ZV |r − r | 4π IS |r − r | (13) is the transverse or solenoidal part of the vector field (with ′ ∇· Fℓ(r ) = 0). Helmholtz’ Theorem

If the surface S recedes to infinity and if the vector field F(r) is regular at infinity, then the surface integrals appearing in Eqs. (12)–(13) become

Fℓ(r) = −∇φ(r) 1 ∇′ · F(r′) 3 ′, = − ∇ r r′ d r (14) 4π ZV | − | Ft (r) = ∇ × a(r) 1 ∇′ × F(r′) 3 ′. = ∇ × r r′ d r (15) 4π ZV | − | Taken together, the above results constitute what is known as Helmholtz’ theorem or the Fundamental Theorem of . Helmholtz’ Theorem

Theorem Helmholtz’ Theorem. Let F(r) be any continuous vector field with continuous first partial derivatives. Then F(r) can be uniquely expressed in terms of the negative of a scalar potential φ(r) & the curl of a vector potential a(r), as embodied in Eqs. (10)–(11).

Hermann Ludwig Ferdinand von Helmholtz (1821–1894)

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