Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 391

Fig. A.5 An illustration of the geometric elements appearing in the divergence theorem

ð
R rr d v

ð

¼ r n d A ; (A.183)

@ R

where r represents any vector field, R is a region of three-dimensional space and R

is the entire boundary of that region (see Fig. A.5 ). There are some mathematical
restrictions on the validity of (A.183). The vector field r ( x 1 , x 2 , x 3 , t ) must be
defined and continuously differentiable in the region R . The region R is subject to
mathematical restrictions, but any region of interest satisfies these restrictions. For
the second order tensor the divergence theorem takes the form
ð
R r

∂

ð

T d v ¼ T n d A : (A.184)

@R

To show that this version of the theorem is also true if (A.183) is true, the
constant vector c is introduced and used with the tensor field T ( x 1 , x 2 , x 3 , t ) to form a
vector function field r ( x 1 , x 2 , x 3 , t ), thus

r ¼ c Tð x 1 ; x 2 ; x 3 ; t Þ: (A.185)

Substitution of (A.185) into (A.183) for r yields
ð
R rð

ð

c T Þ d v ¼ c T n d A ; (A.186)

@ R

and, since c is a constant vector, (A.186) may be rewritten as

8
<

9
=
; ¼

ð
R rð

ð

c T Þ d v T n d A 0 : (A.187)

:

@R