Paper 3, Section I, C

Vector Calculus | Part IA, 2011

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}|.

Vector fields u\mathbf{u} and v\mathbf{v} in R3\mathbb{R}^{3} are given by u=rαx\mathbf{u}=r^{\alpha} \mathbf{x} and v=k×u\mathbf{v}=\mathbf{k} \times \mathbf{u}, where α\alpha is a constant and k\mathbf{k} is a constant vector. Calculate the second-rank tensor dij=ui/xjd_{i j}=\partial u_{i} / \partial x_{j}, and deduce that ×u=0\boldsymbol{\nabla} \times \mathbf{u}=\mathbf{0} and v=0\boldsymbol{\nabla} \cdot \mathbf{v}=0. When α=3\alpha=-3, show that u=0\boldsymbol{\nabla} \cdot \mathbf{u}=0 and

×v=3(kx)xkr2r5\nabla \times \mathbf{v}=\frac{3(\mathbf{k} \cdot \mathbf{x}) \mathbf{x}-\mathbf{k} r^{2}}{r^{5}}

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