 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-2/6/a/solution

Under an orthogonal Cartesian change of axes

Q

, the vector components transform as

v_{i}^{'} = Q_{ia} v_{a}

, while

v_{k}^{'} v_{k}^{'} = v_{a} v_{a}

and

δ_{ij} = Q_{ia} Q_{ja}

. Substitution gives

P_{ij}^{'} = δ_{ij} - \frac{Q _{ia} v _{a} Q _{jb} v _{b}}{v _{c} v _{c}} = Q_{ia} Q_{jb} (δ_{ab} - \frac{v _{a} v _{b}}{v _{c} v _{c}}) = Q_{ia} Q_{jb} P_{ab} .

This is the rank-two Cartesian tensor transformation law. The nonzero assumption ensures that the denominator is positive. In matrix form

P = I - v v^{T} / (v^{T} v)

v

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