Homework Help: Showing T_ij*S_ij is pseudoscalar

1. Oct 12, 2016

spacetimedude

1. The problem statement, all variables and given/known data
If Tij is a tensor of rank two and Sij is a pseudotensor of rank two, show that the numbers TijSik transform as a pseudotensor of rank two, and that TijSij is a pseudoscalar.

2. Relevant equations

3. The attempt at a solution
I've got the transformation T'ijS'ik as a pesudotensor of rank two, since taking the product of T and S in their respective transformation form results in a solution which consists of product of |S|, two transformation matrix and product of T and S with a common index, calling it matrix R.

In the second part, I used the result I got from the first part and changed the index k into j, giving a kronecker delta and resulting in |S|Rbb (any index suffice) where Rbb is a scalar.

I was wondering, is showing T'ijS'ij is a pseudoscalar≡TijSij is a pseudoscalar? Do tensors hold the if and only if property?

Thanks

2. Oct 17, 2016