UPSC Maths 2013 Paper 1 Q8a — Solution

10 marks · Section B

Question

Calculate $\nabla^{2}(r^{n})$ and find its expression in terms of $r$ and $n$ , $r$ being the distance of any point $(x,y,z)$ from the origin, $n$ being a constant and $\nabla^{2}$ being the Laplace operator.

Technique

Use the radial Laplacian formula for spherically symmetric functions.

Solution

Strategy. $r^{n}$ is a function of $r$ only (spherically symmetric). For any $f(r)$ in 3D:

\nabla^{2}f(r)=f''(r)+\frac{2}{r}f'(r).

(This is the radial Laplacian; comes from $\nabla^{2}=\dfrac{1}{r^{2}}\dfrac{\partial}{\partial r}\!\left(r^{2}\dfrac{\partial}{\partial r}\right)$ in spherical coordinates.)

Step 1 — Compute derivatives of $f(r)=r^{n}$

$f'(r)=nr^{n-1}$ , $f''(r)=n(n-1)r^{n-2}$ .

Step 2 — Combine

\nabla^{2}(r^{n})=n(n-1)r^{n-2}+\frac{2}{r}\cdot nr^{n-1}=n(n-1)r^{n-2}+2nr^{n-2}=nr^{n-2}\bigl[(n-1)+2\bigr]=n(n+1)r^{n-2}.

Answer

\boxed{\;\nabla^{2}(r^{n})=n(n+1)\,r^{n-2}.\;}

UPSC Maths 2013 Paper 1 Q8a — Solution

Question

Technique

Solution

Step 1 — Compute derivatives of f(r)=rnf(r)=r^{n}f(r)=rn

Step 2 — Combine

Answer

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Step 1 — Compute derivatives of $f(r)=r^{n}$