Tensors for Physics

7.3 Special Types of Vector Fields 95

Letvμbe given byvμ=∇μΦ, withΦ=Φ(r). Then

vμ=

dΦ

dr

∇μr=

dΦ

dr

r−^1 rμ=r−^1 Φ′rμ,

cf. (7.13). The prime indicates the derivative with respect tor. Nabla applied to the
vector field yields

∇νvμ=∇ν∇μΦ(r)=rμ∇ν(r−^1 Φ′)+r−^1 Φ′δμν=r−^1 (r−^1 Φ′)′rνrμ+r−^1 Φ′δμν.
(7.45)
The special caseΦ=( 1 / 2 )r^2 , treated previously, yields∇ν∇μΦ=δμν, and con-

sequently,∇νvμ =0. On the other hand, forΦ=r−^1 , one finds

∇ν∇μr−^1 = 3 r−^5 rνrμ−r−^3 δμν= 3 r−^3 r̂μr̂ν, (7.46)

withtheunit vector̂r=r−^1 r. Inthis case,∇vis “automatically”symmetrictraceless.
In general,∇νvμ=∇ν∇μΦ(r)has an isotropic part, proportional toδμνand a

symmetric traceless part, proportional tor̂μr̂ν. Relation (7.45) implies

∇νvμ =∇ν∇μΦ(r)=r(r−^1 Φ′)′r̂μr̂ν. (7.47)

On the other hand, settingμ=νin (7.45), one finds

∇μ∇μΦ(r)=ΔΦ(r)=r(r−^1 Φ′)′+ 3 r−^1 Φ′=Φ′′+ 2 r−^1 Φ′, (7.48)

for the Laplace operator applied to a function, which depends onrviar=|r|only.

7.4 Tensor Fields

7.4.1 Graphical Representations of Symmetric Second Rank

Tensor Fields

Examples for second rank tensor fields are the pressure or the stress tensor, as well
as thealignment tensorof molecular fluids or liquid crystals, which describes the
the orientation of molecules or of non-spherical particles. Sometimes, it is desirable
to have a graphical representation of such a tensor field. In the case of a tensor
with uniaxial symmetry, this can be accomplished by displaying the direction of the
principal axis of the tensor, which is associated with the tensor’s symmetry axis, i.e.
with its largest or its smallest principal value. Such a representation then looks like
that of a vector field, but here the directions are indicated by lines without arrowheads.