 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/6/v/solution

Let

B

be the rank-two tensor and

v

the vector. The components of their tensor product are

T_{ijk} = B_{ij} v_{k}

. Applying the transformation rules gives

T_{ijk}^{'} = B_{ij}^{'} v_{k}^{'} = Q_{ia} Q_{jb} Q_{k c} B_{ab} v_{c} = Q_{ia} Q_{jb} Q_{k c} T_{ab c} .

The product therefore obeys the three-index tensor component transformation law, proving that $B_{ij} v_{k}$ is a rank-three tensor. Tensor-product order adds because no pair of indices is summed out.

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