Paper 3, Section I, 4C4 \mathbf{C}

Vector Calculus | Part IA, 2018

In plane polar coordinates (r,θ)(r, \theta), the orthonormal basis vectors er\mathbf{e}_{r} and eθ\mathbf{e}_{\theta} satisfy

err=eθr=0,erθ=eθ,eθθ=er, and =err+eθ1rθ\frac{\partial \mathbf{e}_{r}}{\partial r}=\frac{\partial \mathbf{e}_{\theta}}{\partial r}=\mathbf{0}, \quad \frac{\partial \mathbf{e}_{r}}{\partial \theta}=\mathbf{e}_{\theta}, \quad \frac{\partial \mathbf{e}_{\theta}}{\partial \theta}=-\mathbf{e}_{r}, \quad \text { and } \quad \boldsymbol{\nabla}=\mathbf{e}_{r} \frac{\partial}{\partial r}+\mathbf{e}_{\theta} \frac{1}{r} \frac{\partial}{\partial \theta}

Hence derive the expression ϕ=1rr(rϕr)+1r22ϕθ2\nabla \cdot \nabla \phi=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} \phi}{\partial \theta^{2}} for the Laplacian operator 2\nabla^{2}.

Calculate the Laplacian of ϕ(r,θ)=αrβcos(γθ)\phi(r, \theta)=\alpha r^{\beta} \cos (\gamma \theta), where α,β\alpha, \beta and γ\gamma are constants. Hence find all solutions to the equation

2ϕ=0 in 0ra, with ϕ/r=cos(2θ) on r=a\nabla^{2} \phi=0 \quad \text { in } \quad 0 \leqslant r \leqslant a, \quad \text { with } \quad \partial \phi / \partial r=\cos (2 \theta) \text { on } r=a

Explain briefly how you know that there are no other solutions.

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