Tensors for Physics

7.3 Special Types of Vector Fields 91

The arguments just presented here, can be reverted. When∇×v=0 holds true for
a vector fieldv, then it can be derived from a scalar potentialΦ, such thatv=∇Φ.
The condition of a vanishing curl or rotation of the vector field is equivalent to the
integrability condition∇μvν=∇νvμor

∂vν

∂rμ

=

∂vμ

∂rν

. (7.35)

When the integrability condition holds true for a vector field, it does posses an
“integral”, viz.: a scalar potential function. Since nabla applied to a constant yields
zero, the potential is only determined by its gradient up to an additive constant.

7.3.2 Poisson Equation, Laplace Operator

Let the divergence∇·vof the vector field be equal to a given “density” function
ρ=ρ(r)

∇νvν=ρ(r). (7.36)

The functionρ(r)is the “source” for the vector field. Whenv=∇Φholds true,
(7.36) implies, that the potentialΦobeys thePoisson equation

∇ν∇νΦ:=ΔΦ=ρ(r). (7.37)

The symbolΔstands for theLaplace operator. This second spatial derivative is
defined by

Δ:= ∇ν∇ν=

∂^2

∂rν∂rν

. (7.38)

By definition, the Laplace operator is a scalar, i.e. invariant under a rotation of the
coordinate system.

7.3.3 Divergence Free Vector Fields, Vector Potential

A vector fieldvis called divergence-free or source-free when∇·v=0, or equiva-
lently,

∇μvμ= 0

holds true. Such a field can be derived from avector potentialAaccording to

vμ=(∇×A)μ=εμνλ∇νAλ. (7.39)