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

# Evaluate using Gauss' divergence theorem

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

- Asked in: 2078 Bhadra, 1+4 marks; 2073 Chaitra, 5 marks (asked 2 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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

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

## Short answer

Gauss theorem equates flux to volume integral of divergence. For F = (x³−yz)i − 2x²y j + 2k, ∇·F = x². Integrating over the cube 0 ≤ x,y,z ≤ a gives a⁵/3.

## Answer

#### Gauss' divergence theorem
Let S be a closed surface enclosing a volume V, with n the outward unit normal. If F has continuous first partial derivatives in V, then

```text
∬_S F·n dS = ∭_V (∇·F) dV
```

The outward flux through a closed surface equals the integral of the divergence over the volume enclosed.

#### Application
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. What does Gauss' divergence theorem state?
   A. Outward flux through a closed surface equals the volume integral of div F
   B. Flux equals the line integral of F
   C. Flux equals the area integral of curl F
   D. Flux equals the gradient of F
   Answer: A. The surface integral of F.n equals the integral of div F over the enclosed volume. (Key point: Theorem statement)

2. What is div F for F = (x^3-yz) i - 2x^2y j + 2k?
   A. 3x^2 - 2x^2 + 2
   B. x^2
   C. 5x^2
   D. x^3
   Answer: B. 3x^2 - 2x^2 + 0 = x^2. (Key point: Divergence)

3. What is the integral of x^2 over the cube 0 to a?
   A. a^3/3
   B. a^6/3
   C. a^5/3
   D. a^5
   Answer: C. (a^3/3) times a times a. (Key point: Volume integral)

## Close variants set in other exams

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