UPSC 2018 Maths Optional Paper 1 Q8b — Step-by-Step Solution

13 marks · Section B

Stokes' theorem · Vector Analysis · asked 10× in 13 yrs · Read the full method →

Question

Evaluate the line integral $\displaystyle\int_C(-y^3\,dx+x^3\,dy+z^3\,dz)$ using Stokes’ theorem. Here $C$ is the intersection of the cylinder $x^2+y^2=1$ and the plane $x+y+z=1$ . The orientation on $C$ corresponds to counterclockwise motion in the $xy$ -plane.

Technique

Stokes’ theorem; compute $\nabla\times\vec F=(0,0,3(x^2+y^2))$ ; cap with the plane (giving $\hat n\,dS=(1,1,1)\,dx\,dy$ ); integrate over the projected unit disk.

Solution

Step 1 — Identify the field and its curl

$\vec F=(-y^3,\ x^3,\ z^3)$ . Then

\nabla\times\vec F= \Big(\tfrac{\partial z^3}{\partial y}-\tfrac{\partial x^3}{\partial z},\ \tfrac{\partial(-y^3)}{\partial z}-\tfrac{\partial z^3}{\partial x},\ \tfrac{\partial x^3}{\partial x}-\tfrac{\partial(-y^3)}{\partial y}\Big) =\big(0,\ 0,\ 3x^2+3y^2\big).

Step 2 — Choose the capping surface

Let $S$ be the portion of the plane $z=1-x-y$ bounded by $C$ . Writing $z=g(x,y)$ with the upward orientation (consistent with CCW in the $xy$ -plane), the vector area element is

\hat n\,dS=(-g_x,\,-g_y,\,1)\,dx\,dy=(1,\,1,\,1)\,dx\,dy.

Step 3 — Apply Stokes’ theorem

\oint_C\vec F\cdot d\vec r=\iint_S(\nabla\times\vec F)\cdot\hat n\,dS =\iint_D\big(0,0,3x^2+3y^2\big)\cdot(1,1,1)\,dx\,dy =\iint_D 3(x^2+y^2)\,dx\,dy,

where $D$ is the projection onto the $xy$ -plane: the unit disk $x^2+y^2\le1$ .

Step 4 — Evaluate over the unit disk (polar coordinates)

\iint_D 3(x^2+y^2)\,dx\,dy=\int_0^{2\pi}\!\!\int_0^1 3r^2\cdot r\,dr\,d\theta =3\cdot2\pi\cdot\int_0^1 r^3\,dr=6\pi\cdot\frac14=\frac{3\pi}{2}.

Answer

\boxed{\;\int_C(-y^3\,dx+x^3\,dy+z^3\,dz)=\frac{3\pi}{2}.\;}