Question Bank • QB_SUB_EM_0117 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Physical interpretations – line, surface and volume integrals

Question Statement:

Using the Divergence Theorem, evaluate $\iint_S \vec{r} \cdot \hat{n}\, dS$ over the surface of a sphere of radius $R = 3$, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$:

Answer Key:

339.29

Step-by-Step Solution & Explanation

$\nabla \cdot \vec{r} = 1 + 1 + 1 = 3$. By Divergence theorem: $\iint_S \vec{r} \cdot \hat{n}\, dS = \iiint_V 3\, dV = 3 V = 3 \left(\frac{4}{3} \pi R^3\right) = 4 \pi R^3$ For $R = 3$: $4 \pi (3)^3 = 339.29$
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