Question Bank • QB_EM_094 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Physical interpretations – line, surface and volume integrals

Question Statement:

Using Gauss Divergence Theorem, evaluate the outward surface integral $\iint_S \vec{F} \cdot \hat{n}\, dS$, where $\vec{F} = 2x\hat{i} + 3y\hat{j} + 4z\hat{k}$ and $S$ is the surface of a sphere of radius $R = 1$ centered at origin. (Take $\pi = 3.1416$)

Answer Key:

37.7

Step-by-Step Solution & Explanation

By Gauss Divergence Theorem: $\iint_S \vec{F} \cdot \hat{n}\, dS = \iiint_V (\nabla \cdot \vec{F})\, dV$ $\nabla \cdot \vec{F} = \frac{\partial(2x)}{\partial x} + \frac{\partial(3y)}{\partial y} + \frac{\partial(4z)}{\partial z} = 2 + 3 + 4 = 9$ Volume of unit sphere $V = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (1)^3 = \frac{4\pi}{3}$: $\iint_S \vec{F} \cdot \hat{n}\, dS = 9 \times \frac{4\pi}{3} = 12\pi = 12 \times 3.14159 \approx 37.70$
← Previous (QB_EM_VC_011) Next (QB_EM_ADV_050) →

Practice In Authentic CBT Exam Interface

Test your skills under authentic timed conditions with live scoring and AIR percentile ranking.

Launch CBT Mock Test