---
title: "Evaluate using Gauss' divergence theorem F=x²i+zj+yzk"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-using-gauss-divergence-theorem-f-five"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-five
site: BCT Engineering
---

# Evaluate using Gauss' divergence theorem F=x²i+zj+yzk

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

- Asked in: 2072 Kartik, 5 marks; 2068 Baishakh, 5 marks (asked 2 times)
- Video: [Verify Gauss divergence theorem for F = x²i + zj + yzk over the cube bounded by x=0, x=1, y=0, y=1, ](https://www.youtube.com/watch?v=lq9-J1-j2mA) — JeyZee Maths
- 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-five
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Apply Gauss divergence theorem to evaluate ∬_S F·n ds where F = x² i + z j + yz k taken over the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.

## Short answer

Divergence of x² i + z j + yz k is 2x + y. Volume integral over the unit cube gives 3/2.

## Answer

F = x² i + z j + yz k, over the closed surface of the unit cube 0 ≤ x, y, z ≤ 1.

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

#### Volume integral
```text
∬_S F·n dS = ∭_V (2x + y) dV
           = ∫₀¹∫₀¹∫₀¹ (2x + y) dx dy dz
           = ∫₀¹∫₀¹ [x² + xy]₀¹ dy dz
           = ∫₀¹∫₀¹ (1 + y) dy dz
           = ∫₀¹ [y + y²/2]₀¹ dz
           = 3/2
```

**Answer: ∬_S F·n dS = 3/2.**

## Check yourself

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

1. What is div F for F = x^2 i + z j + yz k?
   A. 2x + z
   B. 2x + y + z
   C. x^2 + y
   D. 2x + y
   Answer: D. d(x^2)/dx = 2x, d(z)/dy = 0, d(yz)/dz = y. (Key point: Divergence)

2. Over what region is the triple integral taken?
   A. The unit cube 0 to 1 in each variable
   B. The cube's surface
   C. A sphere
   D. The square face only
   Answer: A. Gauss converts the surface integral to a volume integral. (Key point: Region)

3. The flux equals?
   A. 1
   B. 2
   C. 3/2
   D. 5/2
   Answer: C. 1 + 1/2 = 3/2. (Key point: Result)

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