Page 70 - Introduction to Continuum Mechanics

PartC Curl of a Vector Field 55

To find the Cartesian components of the vector div T, let b = div T, then (note Ve/=0 for
Cartesian coordinates), from Eq. (2C4.3),

In other words,

If a=a(r) and a=a(r), show that div(aa)=adiva+(Va) -a.
Solution. Let b=aa. Then bi—aa^ and

Example 2C4.2
Given a(r) and T(r), show that

div(aT) = T(Va)+adivT
Solution. We have, from Eq. (2C4.5),

But

and

Thus, the desired result follows.

2C5 Curt of a Vector Field
Let v(r) be a vector field. The curl of v is defined to be the vector field given by twice the
dual vector of the antisymmetric part of (Vv). That is