Paper 3, Section I, A

Vector Calculus | Part IA, 2014

(a) For xRn\mathbf{x} \in \mathbb{R}^{n} and r=xr=|\mathbf{x}|, show that

rxi=xir\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r}

(b) Use index notation and your result in (a), or otherwise, to compute

(i) (f(r)x)\nabla \cdot(f(r) \mathbf{x}), and

(ii) ×(f(r)x)\nabla \times(f(r) \mathbf{x}) for n=3n=3.

(c) Show that for each nNn \in \mathbb{N} there is, up to an arbitrary constant, just one vector field F(x)\mathbf{F}(\mathbf{x}) of the form f(r)xf(r) \mathbf{x} such that F(x)=0\nabla \cdot \mathbf{F}(\mathbf{x})=0 everywhere on Rn\{0}\mathbb{R}^{n} \backslash\{\mathbf{0}\}, and determine F\mathbf{F}.

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