Allied & ICAR Exam Bank • ALLIED_0055NAT • 2 Marks
Section 1: Engineering Mathematics
Vector Calculus
Subtopic: Stokes, Gauss and Green's theorems
Question Statement:
Using Stokes' Theorem, $\oint_C \vec{F} \cdot d\vec{r}$ where $\vec{F} = y\hat{i} - x\hat{j}$ and $C$ is the circle $x^2 + y^2 = 4$ traversed counterclockwise, divided by $\pi$ is:
Answer Key:
-8
Step-by-Step Solution & Explanation
$\nabla \times \vec{F} = \left( \frac{\partial(-x)}{\partial x} - \frac{\partial y}{\partial y} \right)\hat{k} = (-1 - 1)\hat{k} = -2\hat{k}$. By Stokes' theorem, $\iint_S (-2\hat{k}) \cdot \hat{k} dA = -2 \times \text{Area} = -2 \times \pi(2^2) = -8\pi$. Divided by $\pi$, the value is $-8$.