Let A {\displaystyle {\boldsymbol {A}}} and B {\displaystyle {\boldsymbol {B}}} be two second-order tensor fields. Let the average of any second-order tensor field ( S {\displaystyle {\boldsymbol {S}}} ) over the region Ω {\displaystyle \Omega } (of volume V {\displaystyle V} ) be defined as
Show that
Expanding out the right hand side, we have
Now ⟨ A ⟩ {\displaystyle \langle {\boldsymbol {A}}\rangle } and ⟨ B ⟩ {\displaystyle \langle {\boldsymbol {B}}\rangle } are constants with respect to the integration. Hence,
Therefore,
![{\displaystyle {\begin{aligned}\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle &={\cfrac {1}{V}}\int _{\Omega }[{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]~{\text{dV}}\\&={\cfrac {1}{V}}\int _{\Omega }[{\boldsymbol {A}}\cdot {\boldsymbol {B}}-\langle {\boldsymbol {A}}\rangle \cdot {\boldsymbol {B}}-{\boldsymbol {A}}\cdot \langle {\boldsymbol {B}}\rangle +\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle ]~{\text{dV}}\\&={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {A}}\cdot {\boldsymbol {B}}~{\text{dV}}-{\cfrac {1}{V}}\int _{\Omega }\langle {\boldsymbol {A}}\rangle \cdot {\boldsymbol {B}}~{\text{dV}}-{\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {A}}\cdot \langle {\boldsymbol {B}}\rangle ~{\text{dV}}+{\cfrac {1}{V}}\int _{\Omega }\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle ~{\text{dV}}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/662beaac1c29aac56d9794baafc5a7b5aeeb889c)
and 
are constants with respect to the integration.
Hence,
![{\displaystyle {\begin{aligned}\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle &={\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {A}}\cdot {\boldsymbol {B}}~{\text{dV}}-\langle {\boldsymbol {A}}\rangle \cdot \left({\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {B}}~{\text{dV}}\right)-\left({\cfrac {1}{V}}\int _{\Omega }{\boldsymbol {A}}~{\text{dV}}\right)\cdot \langle {\boldsymbol {B}}\rangle +\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle \left({\cfrac {1}{V}}\int _{\Omega }~{\text{dV}}\right)\\&=\langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle +\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle \\&=\langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle ~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71933ef803f016ff9ed4377913d98e6a428e0925)
![{\displaystyle {\langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle =\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle \qquad \square }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f88ff840b1e9434bd774f61dde41f9d67d87c13)
![{\displaystyle \langle {\boldsymbol {A}}\cdot {\boldsymbol {B}}\rangle -\langle {\boldsymbol {A}}\rangle \cdot \langle {\boldsymbol {B}}\rangle =\langle [{\boldsymbol {A}}-\langle {\boldsymbol {A}}\rangle ]\cdot [{\boldsymbol {B}}-\langle {\boldsymbol {B}}\rangle ]\rangle ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9435962c03a4a157f1f7efa8c250bf1284c8720)