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

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

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

- Asked in: 2080 Bhadra, 1+4 marks
- Video: [Gauss Divergence Theorem Lecture No 3](https://www.youtube.com/watch?v=qPp7WQdR2e0) — Navnath Gadade
- 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-eight
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

State Gauss' divergence theorem. Evaluate ∬_S F·n ds where F = x i + y j + z k and S is the surface of the sphere x² + y² + z² = 1 by using Gauss' divergence theorem.

## Short answer

Divergence is 3. Volume of unit sphere is 4π/3, so flux = 3 × 4π/3 = 4π.

## Answer

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

∬_S F·n dS = ∭_V (∇·F) dV.

#### Evaluation
```text
F = x i + y j + z k
div F = 1 + 1 + 1 = 3
Volume of unit sphere = 4π/3

∬_S F·n dS = ∭_V 3 dV = 3 × 4π/3 = 4π
```

Check: on the unit sphere n = x i + y j + z k, so F·n = x² + y² + z² = 1 and the integral is the area of the sphere, 4π(1)² = 4π.

**Answer: 4π.**

## Check yourself

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

1. div (x i + y j + z k)?
   A. 1
   B. 3
   C. 0
   D. x+y+z
   Answer: B. 1+1+1. (Key point: Divergence)

2. Volume of unit sphere?
   A. 4pi
   B. pi
   C. 4pi/3
   D. 2pi/3
   Answer: C. Standard. (Key point: Volume)

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

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