UPSC 2018 Maths Optional Paper 1 Q6d — Step-by-Step Solution

12 marks · Section B

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

Question

If $S$ is the surface of the sphere $x^2+y^2+z^2=a^2$ , then evaluate

\iint_S\big[(x+z)\,dy\,dz+(y+z)\,dz\,dx+(x+y)\,dx\,dy\big]

using Gauss’ divergence theorem.

Recognise the flux form; $\nabla\cdot\vec F=2$ (constant); flux $=2\times$ volume of ball $=2\cdot\frac43\pi a^3$ .

The integral is the flux $\displaystyle\iint_S\vec F\cdot\hat n\,dS$ of

\vec F=(x+z)\,\hat i+(y+z)\,\hat j+(x+y)\,\hat k,

since $dy\,dz,\ dz\,dx,\ dx\,dy$ are the components of $\hat n\,dS$ .

\nabla\cdot\vec F=\frac{\partial}{\partial x}(x+z)+\frac{\partial}{\partial y}(y+z)+\frac{\partial}{\partial z}(x+y)=1+1+0=2.

Note the third component $x+y$ has no $z$ -dependence, so its $z$ -derivative is $0$ — the divergence is $2$ , not $3$ .

By the divergence theorem, with $V$ the solid ball of radius $a$ (volume $\tfrac43\pi a^3$ ):

\iint_S\vec F\cdot\hat n\,dS=\iiint_V(\nabla\cdot\vec F)\,dV=\iiint_V 2\,dV=2\cdot\frac43\pi a^3.

\boxed{\;\iint_S\big[(x+z)\,dy\,dz+(y+z)\,dz\,dx+(x+y)\,dx\,dy\big]=\frac{8}{3}\pi a^3.\;}