Page 447 - Introduction to Continuum Mechanics

Divergence Theorem 431

where 5 is a surface forming the complete boundary of a bounded closed region R in space
and n is the outward unit normal of S. Equation (7.2.2) is known as the diveigence theorem (
or Gauss theorem). The theorem is valid if the components of v are continuous and have
continuous first partial derivatives in R. It is also valid under less restrictive conditions on the
derivatives.
Next, if TIJ are components of a tensor T, then the application of Eq. (7.2.2a) gives

Or in invariant notation,

Equation (7.2.3) is the divergence theorem for a tensor field. It is obvious that for tensor
fields of higher order, Eq. (7.2.3b) is also valid provided the Cartesian components of divT are
defined to be dl/^/ s / dx s.

Example 7.2.1
Let T be a stress tensor field and let S be a closed surface. Show that the resultant force of
the distributive forces on S is given by

Solution. Let f be the resultant force, then

where t is the stress vector. But t = Tn, therefore from the divergence theorem, we have

i.e.,