Paper 3, Section I, A

Vector Calculus | Part IA, 2015

(i) For r=xr=|\mathbf{x}| with xR3\{0}\mathbf{x} \in \mathbb{R}^{3} \backslash\{\mathbf{0}\}, show that

rxi=xir(i=1,2,3).\frac{\partial r}{\partial x_{i}}=\frac{x_{i}}{r} \quad(i=1,2,3) .

(ii) Consider the vector fields F(x)=r2x,G(x)=(ax)x\mathbf{F}(\mathbf{x})=r^{2} \mathbf{x}, \mathbf{G}(\mathbf{x})=(\mathbf{a} \cdot \mathbf{x}) \mathbf{x} and H(x)=a×x^\mathbf{H}(\mathbf{x})=\mathbf{a} \times \hat{\mathbf{x}}, where a\mathbf{a} is a constant vector in R3\mathbb{R}^{3} and x^\hat{\mathbf{x}} is the unit vector in the direction of x\mathbf{x}. Using suffix notation, or otherwise, find the divergence and the curl of each of F,G\mathbf{F}, \mathbf{G} and H\mathbf{H}.

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