Tensors for Physics

90 7 Fields, Spatial Differential Operators

For the three-dimensional radial field, one finds

∇·v= 3 , ∇×v= 0 , ∇νvμ = 0.

The 2D version of a radial vector field is

vμ=rμ⊥=rμ−eμeνrν,

with the unit vectore, perpendicular to the plane, in which the radial vector lies. Here

one obtains

∇·v= 2 , ∇×v= 0 , ∇νvμ =−eνeμ.

(vii) Simple Shear Flow

vμ=eμuνrν,

whereeanduare two orthogonal unit vectors,e·u=0. In this case, one finds

∇νvμ=uνeμ, and

∇·v= 0 ,(∇×v)λ=ελνμuνeμ, ∇νvμ =uνeμ.

The calculations here and in the following exercise show: the symmetric traceless
part∇v is zero for the flow fields (i), the 3D radial field of (iii) and for (vi). The
tensor∇vis uniaxial for the examples (ii), the 2D radial field of (iii), and for (iv),
it is planar biaxial for (v) and (vii).
7.1 Exercise: Compute the Spatial Derivatives of Special Vector Fields
Compute the divergence∇·v, the curl or rotation∇×vand the symmetric traceless

part∇νvμof the gradient tensor∇νvμfor the vector fields (iv)–(vi) of Sect.7.2.1.

7.3 Special Types of Vector Fields

7.3.1 Vorticity Free Vector Fields, Scalar Potential

Letvbe a vector field, which can be represented as the gradient of a scalar potential:

v=∇Φ(r), then the rotation∇×vof the vector is zero, thus

vλ=∇λΦ⇒(∇×v)μ=εμνλ∇νvλ=εμνλ∇ν∇λΦ= 0. (7.34)

Of course, it is assumed, that the scalar functionΦcan be differentiated twice and

that∇ν∇λΦ=∇λ∇νΦ.