Allied & ICAR Exam Bank • ALLIED_1009 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Gauss divergence theorem

Question Statement:

Using the Gauss Divergence Theorem, the outward flux of $\vec{F} = x\hat{i} + y\hat{j} + z\hat{k}$ across the surface of a unit sphere $x^2 + y^2 + z^2 = 1$ is $\alpha \pi$. The value of $\alpha$ is:

Answer Key:

4.0

Step-by-Step Solution & Explanation

$\nabla \cdot \vec{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} = 1 + 1 + 1 = 3$. By Gauss Divergence Theorem: $\iint_S \vec{F} \cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{F}) dV = 3 \times V_{sphere} = 3 \times \left(\frac{4}{3}\pi (1)^3\right) = 4\pi$. Thus $\alpha = 4.0$.
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