---
title: "Evaluate using Gauss' divergence theorem F=2xi+3yj+4zk"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-using-gauss-divergence-theorem-f-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-three
site: BCT Engineering
---

# Evaluate using Gauss' divergence theorem F=2xi+3yj+4zk

Mathematics III · Chapter 2: Line, Surface and Volume Integrals

- Asked in: 2082 Shrawan, 5 marks; 2069 Asar, 5 marks (asked 2 times)
- Video: [Multivariable Calculus: Divergence Theorem - Examples (16.9)](https://www.youtube.com/watch?v=qFkOfh0ENNY) — Dr. Clark Teaches Math
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Using Gauss' divergence theorem, evaluate ∬_S F·n ds for F = 2x i + 3y j + 4z k, where S is the surface of the sphere x² + y² + z² = 1.

## Short answer

Divergence ∇·F = 9 is constant. Volume of unit sphere is 4π/3, so flux = 9 × 4π/3 = 12π.

## Answer

F = 2x i + 3y j + 4z k, S: the unit sphere x² + y² + z² = 1.

#### Apply Gauss' divergence theorem
```text
∬_S F·n dS = ∭_V (∇·F) dV
∇·F = ∂(2x)/∂x + ∂(3y)/∂y + ∂(4z)/∂z = 2 + 3 + 4 = 9
```

The divergence is constant, so the integral is 9 times the volume of the unit ball, V = (4/3)π(1)³ = 4π/3:

```text
∬_S F·n dS = 9 · (4π/3) = 12π
```

**Answer: 12π.**

(Check: on the unit sphere n = (x, y, z), so F·n = 2x² + 3y² + 4z². By symmetry the average of each of x², y², z² over the sphere is 1/3, giving (2 + 3 + 4)/3 · 4π = 12π ✓.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is div F for F = 2x i + 3y j + 4z k?
   A. 9
   B. 24
   C. 3
   D. 2+3+4 times x
   Answer: A. Add 2 + 3 + 4 = 9. (Key point: Divergence)

2. Why does the integral reduce to a simple formula?
   A. The divergence is zero
   B. The divergence is constant, so the result is 9 times the volume
   C. F is perpendicular to the sphere
   D. The curl is zero
   Answer: B. A constant integrand just multiplies the volume. (Key point: Why simple)

3. Which volume is used?
   A. Surface area of sphere, 4pi
   B. Volume of unit cube, 1
   C. The unit ball, 4pi/3
   D. Hemisphere volume, 2pi/3
   Answer: C. The theorem integrates over the solid enclosed by S: (4/3)pi(1)^3. (Key point: Volume used)

4. What is the flux?
   A. 9pi
   B. 4pi
   C. 36pi
   D. 12pi
   Answer: D. 9 x 4pi/3 = 12pi. (Key point: Result)

## Close variants set in other exams

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