 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/6/b/i/solution

S_{ij}^{E} = \frac{1}{2} (\partial_{i} E_{j} + \partial_{j} E_{i}), S_{ij}^{B} = \frac{1}{2} (\partial_{i} B_{j} + \partial_{j} B_{i}) .

The corresponding axial parts are

u^{E} = \frac{1}{2} \nabla \times E = - \frac{1}{2} \partial_{t} B, u^{B} = \frac{1}{2} \nabla \times B = \frac{1}{2 c ^{2}} \partial_{t} E .

The second equalities use the vacuum Maxwell equations. Their divergence constraints also show that both symmetric gradient tensors are traceless.

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