Methods | Part IB, 2002

If B\mathbf{B} is a vector, and

Tij=αBiBj+βBkBkδijT_{i j}=\alpha B_{i} B_{j}+\beta B_{k} B_{k} \delta_{i j}

show for arbitrary scalars α\alpha and β\beta that TijT_{i j} is a symmetric second-rank tensor.

Find the eigenvalues and eigenvectors of TijT_{i j}.

Suppose now that B\mathbf{B} depends upon position x\mathbf{x} and that B=0\nabla \cdot \mathbf{B}=0. Find constants α\alpha and β\beta such that

xjTij=[(×B)×B]i.\frac{\partial}{\partial x_{j}} T_{i j}=[(\nabla \times \mathbf{B}) \times \mathbf{B}]_{i} .

Hence or otherwise show that if B\mathbf{B} vanishes everywhere on a surface SS that encloses a volume VV then

V(×B)×BdV=0\int_{V}(\nabla \times \mathbf{B}) \times \mathbf{B} d V=0

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