Tensors for Physics

88 7 Fields, Spatial Differential Operators

Fig. 7.5 Solid-like rotation
field

Fig. 7.6 Simple shear field

7.2.2 Differential Change of a Vector Fields.

The difference dvμ=vμ(r+dr)−vμ(r)of the vector fieldvμbetween the positions
r+drandrcan be expanded with respect to the small, differential change dr.Up
to linear terms, one has

dvμ=

∂vμ

∂rν

drν=(∇νvμ)drν. (7.29)

The quantity∇νvμwhich is sometimes calledgradientofv, is a second rank tensor.
It can be decomposed into its symmetric and antisymmetric parts or its isotropic,
symmetric traceless and antisymmetric parts, just like any dyadic tensor, cf. (6.4).
These decompositions are

∇νvμ=

1

2

(∇νvμ+∇νvμ)+

1

2

(∇νvμ−∇νvμ), (7.30)

and

∇νvμ=

1

3

(∇λvλ)δμν+∇νvμ+

1

2

ενμλ(∇×v)λ. (7.31)