Paper 3, Section II, B

Vector Calculus | Part IA, 2009

State the value of xi/xj\partial x_{i} / \partial x_{j} and find r/xj\partial r / \partial x_{j}, where r=xr=|\mathbf{x}|.

A vector field u\mathbf{u} is given by

u=kr+(kx)xr3\mathbf{u}=\frac{\mathbf{k}}{r}+\frac{(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}}{r^{3}}

where k\mathbf{k} is a constant vector. Calculate the second-rank tensor dij=ui/xjd_{i j}=\partial u_{i} / \partial x_{j} using suffix notation, and show that dijd_{i j} splits naturally into symmetric and antisymmetric parts. Deduce that u=0\boldsymbol{\nabla} \cdot \mathbf{u}=0 and that

×u=2k×xr3\boldsymbol{\nabla} \times \mathbf{u}=\frac{2 \mathbf{k} \times \mathbf{x}}{r^{3}}

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