---
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-seven"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-seven
site: BCT Engineering
---

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

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

- Asked in: 2081 Baishakh, 1+4 marks
- 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-seven
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

State 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 by Gauss's divergence theorem.

## Short answer

Divergence is 9. Volume integral over unit sphere: 9 × (4π/3) = 12π.

## Answer

#### Statement
If S is a closed surface enclosing a volume V, n is the outward unit normal and F is a vector function with continuous first partial derivatives in V, then

∬_S F·n dS = ∭_V (∇·F) dV.

#### Evaluation
```text
F = 2x i + 3y j + 4z k
div F = 2 + 3 + 4 = 9
```

S is the unit sphere, so V is the unit ball with volume (4/3)π(1)³ = 4π/3.

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

**Answer: ∬_S F·n dS = 12π.**

## Check yourself

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

1. Gauss' theorem requires?
   A. Open surface
   B. Constant F
   C. Closed surface, outward normal, F with continuous first partials
   D. Zero curl
   Answer: C. These are the standard conditions. (Key point: Statement conditions)

2. div F?
   A. 3
   B. 2
   C. 24
   D. 9
   Answer: D. 2+3+4. (Key point: Divergence)

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

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