Prove the product rule for double inner products of second-order tensors

[josh_machine]

Some one please help me how to prove the following:

$$\dot{A}:B + A:\dot{B}=A^{\nabla J}:B+A:B^{\nabla J}$$

A and B are II order tensors and : represents the inner product.

 

[josh_machine]

How do I prove the following:

$$\dot{A}:B + A:\dot{B}=A^{\nabla J}:B+A:B^{\nabla J}$$

 

[arkajad]

Where did you get this notation from? What is your dot? What is your J? What do you mean by "inner product", for what kind of tensors? Any reference to some place where your original notation is defined?

 

[josh_machine]

The dot represents material time derivative. A and B are second order tensors, eg Stress.
I myself am not clear what $$\nabla J$$ means here. However, I guess it represents divergence.

This was as a homework question for a Continuum Mechanics course. I have not got any luck trying to understand or prove this expression. Any insight will be greatly appreciated.