---
title: "Evaluate using Gauss' divergence theorem"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-using-gauss-divergence-theorem"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem
site: BCT Engineering
---

# Evaluate using Gauss' divergence theorem

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

- Asked in: 2074 Asoj, 5 marks; 2072 Chaitra, 5 marks; 2070 Asar, 5 marks (asked 3 times)
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Apply Gauss' divergence theorem to evaluate ∬_S [(x³ − yz) i − 2x²y j + 2k]·n ds, where S is the surface of the cube bounded by the planes x = 0, x = a, y = 0, y = a, z = 0, z = a.

## Short answer

The divergence is x². Triple integrating over the cube 0≤x,y,z≤a gives a⁵/3, which is the closed surface flux by Gauss’ theorem.

## Answer

**Gauss' divergence theorem:** ∬_S F·n dS = ∭_V ∇·F dV for a closed surface S.

F = (x³ − yz) i − 2x²y j + 2k, and S is the closed surface of the cube 0 ≤ x, y, z ≤ a.

#### Divergence
```text
∇·F = ∂(x³ − yz)/∂x + ∂(−2x²y)/∂y + ∂(2)/∂z = 3x² − 2x² + 0 = x²
```

#### Volume integral
```text
∭_V x² dV = ∫₀ᵃ ∫₀ᵃ ∫₀ᵃ x² dx dy dz = (a³/3)(a)(a) = a⁵/3
```

**Answer: ∬_S F·n dS = a⁵/3.**

## Check yourself

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

1. Gauss' theorem needs S to be:
   A. Open
   B. A closed surface
   C. A plane
   D. A line
   Answer: B. The cube surface is closed. (Key point: Closed surface)

2. ∇·F for F=(x³−yz)i−2x²y j+2k:
   A. x²
   B. 3x²
   C. 5x²
   D. 0
   Answer: A. 3x²−2x²+0. (Key point: Divergence)

3. ∭x² dV over the cube:
   A. a⁶/3
   B. a³/3
   C. a⁵
   D. a⁵/3
   Answer: D. (a³/3)·a·a. (Key point: Integral)

4. The 2k term contributes to the divergence:
   A. 2
   B. 1
   C. 0
   D. a
   Answer: C. ∂(2)/∂z = 0. (Key point: Constant term)

## Close variants set in other exams

- [Evaluate using Gauss' divergence theorem F=(2xy+z)i+y²j−(x+3y)k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-gauss-divergence-theorem-f-four) — Mathematics III, chapter 2, asked 9×
- [Evaluate using Gauss' divergence theorem F=x²yi+xy²j+2xyzk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-gauss-divergence-theorem-f) — Mathematics III, chapter 2, asked 10×
- [Evaluate using Gauss' divergence theorem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-gauss-divergence-theorem-two) — Mathematics III, chapter 2, asked 5×

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