← 2022 Paper 1

UPSC 2022 Maths Optional Paper 1 Q8c — Step-by-Step Solution

15 marks · Section B

Gauss divergence theorem · Vector Analysis · asked 10× in 14 yrs · Read the full method →

Question

Using Gauss’ divergence theorem, evaluate $\iint_S\vec F\cdot\hat n\,dS$ where $\vec F=x\hat\imath-y\hat\jmath+(z^2-1)\hat k$ and $S$ is the cylinder formed by $z=0,\,z=1,\,x^2+y^2=4$.

Technique

Gauss’ divergence: convert closed-surface flux integral to volume integral of divergence. Divergence here is $2z$, integration trivial.

Solution

Setup. $S$ is the closed surface of the cylinder: bottom disk ($z=0$), top disk ($z=1$), lateral curved surface ($x^2+y^2=4$ between $z=0$ and $z=1$). Volume $V$ enclosed: $\{x^2+y^2\le 4,\;0\le z\le 1\}$.

Gauss’ theorem: $\iint_S\vec F\cdot\hat n\,dS=\iiint_V\nabla\cdot\vec F\,dV$.

Step 1 — Divergence

$\nabla\cdot\vec F=\partial_x(x)+\partial_y(-y)+\partial_z(z^2-1)=1+(-1)+2z=2z$.

Step 2 — Volume integral

(Disk area $=\pi r^2=4\pi$; $\int_0^1 z\,dz=1/2$; factor 2 from the divergence.)

Answer