EVALUATION OF VOLUME INTEGRALS

Above we have formally defined the equations which govern the averaging process (2,3 and 5). Instead of calculating the volume integral, the problem can be transformed into an evaluation of surface integrals. Greens Theorem is valid for tensors of any order. In its most general form it is written as  

$$\oint_V { {\nabla}_{} } \cdot {A}_{} ~dV = \oint_S { {A}_{} }\cdot {n}_{} dS$$ (6)

A can be a vector, then ${\nabla}_{} \cdot$ is the divergence operator and the theorem is termed ``Gauss' Theorem''; or A can be a tensor, with the appropriate definition for ${\nabla}_{} \cdot$.