Paper 3, Section I, C

Vector Calculus | Part IA, 2011

Cartesian coordinates x,y,zx, y, z and spherical polar coordinates r,θ,ϕr, \theta, \phi are related by

x=rsinθcosϕ,y=rsinθsinϕ,z=rcosθx=r \sin \theta \cos \phi, \quad y=r \sin \theta \sin \phi, \quad z=r \cos \theta

Find scalars hr,hθ,hϕh_{r}, h_{\theta}, h_{\phi} and unit vectors er,eθ,eϕ\mathbf{e}_{r}, \mathbf{e}_{\theta}, \mathbf{e}_{\phi} such that

dx=hrer dr+hθeθdθ+hϕeϕdϕ\mathrm{d} \mathbf{x}=h_{r} \mathbf{e}_{r} \mathrm{~d} r+h_{\theta} \mathbf{e}_{\theta} \mathrm{d} \theta+h_{\phi} \mathbf{e}_{\phi} \mathrm{d} \phi

Verify that the unit vectors are mutually orthogonal.

Hence calculate the area of the open surface defined by θ=α,0rR\theta=\alpha, 0 \leqslant r \leqslant R, 0ϕ2π0 \leqslant \phi \leqslant 2 \pi, where α\alpha and RR are constants.

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