 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-2/6/a/solution

A second-order Cartesian tensor assigns components in each orthonormal coordinate system according to

T_{ij}^{'} = R_{ik} R_{j ℓ} T_{k ℓ}, T^{'} = RT R^{T}, R^{T} R = I .

Here

R

is the matrix taking old vector components to new ones, and repeated indices are summed. This is the tensor component transformation law; an arbitrary array of nine numbers is not a tensor unless its components in different frames obey this rule. No symmetry

T_{ij} = T_{ji}

is implied by being a second-order tensor.

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