Allied & ICAR Exam Bank • ALLIED_2695 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Stokes, Gauss and Green's theorems

Question Statement:

Evaluate $\oint_C (y^2 dx + x^2 dy)$ where $C$ is the boundary of the unit square $[0, 1] \times [0, 1]$ traversed counterclockwise. Using Green's theorem, the integral equals:

Answer Key:

0

Step-by-Step Solution & Explanation

By Green's theorem: $\iint_D \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) dA = \int_0^1 \int_0^1 (2x - 2y) dx dy$. Since $\int_0^1 2x dx = 1$ and $\int_0^1 2y dy = 1$, the integral evaluates to $1 - 1 = 0$.
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