Tensors for Physics

7.3 Special Types of Vector Fields 93

Thus the yet undetermined coefficients must obey the relationc 2 − 2 c 1 =1. Clearly,

there is no unique solution. Withc 1 as open parameter, the vector potential can be

written as

Aλ=c 1 (r^2 wλ+ 2 rλrκwκ)+rλrκwκ.

The first term, multiplied byc 1 , is the gradient∇λφ(r)of the scalar functionφ(r)=
r^2 wκrκ. Since∇×∇φ(r)=0, the term proportional toc 1 does not contribute in the
calculation ofvviav=∇×A. Thus one may chosec 1 =0, orc 1 =− 1 /2. In the
latter case, one has

Aμ=−

1

2

r^2 wμ. (7.41)

When one requires that the divergence of the vector potential vanishes, i.e. that

∇μAμ=0 holds true, the coefficientsc 1 andc 2 are determined uniquely:c 1 =− 2 / 5

andc 2 = 1 /5. Then the vector potential reads

Aμ=−

2

5

r^2 wμ+

1

5

rμrκwκ. (7.42)

Application in Electrodynamics

One of the Maxwell equations of electrodynamics is∇μBμ=0. Thus the magnetic

flux densityB, in general, can be represented by a vector potential according to

Bμ=εμνλ∇νAλ. For the special case of a constantB-field,Aλ=( 1 / 2 )ελσ τBσrτ

is the pertaining vector potential.

7.3.4 Vorticity Free and Divergence Free Vector Fields,

Laplace Fields

When a vector fieldvis vorticity free,∇×v=0, there exists a potential function

Φ. When furthermore, the vector field is divergence free, or “source free”,∇·v=0,

the potential obeys theLaplace equation

∇·∇Φ≡∇ν∇νΦ=ΔΦ= 0. (7.43)

Scalar fields of this kind are calledLaplace fields.

For the examples of vector fields with potentials, listed in Sect.7.2.1, one finds

ΔΦ=0 for the cases (i), (iv) and (v), pertaining to the homogeneous field, the

uniaxial and planar biaxial squeeze-stretch fields. The Laplace operator applied on

Φyields nonzero constant values for the cases (ii) and (iii).