---
title: "Verify Gauss' divergence theorem F=x²i+3j+yzk"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:verify-gauss-divergence-theorem-f-x2i"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-gauss-divergence-theorem-f-x2i
site: BCT Engineering
---

# Verify Gauss' divergence theorem F=x²i+3j+yzk

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

- Asked in: 2068 Chaitra, 5 marks
- Video: [Verify for F = x³i + y³j + z³k over a Cube || Gauss Divergence Theorem || Maths.Engineering](https://www.youtube.com/watch?v=osaT4KDzp50) — Maths.Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-gauss-divergence-theorem-f-x2i
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Verify Gauss divergence theorem for F = x² i + 3 j + yz k taken over the cube bounded by x = 0, x = 1, y = 0, y = 1, z = 0, z = 1.

## Short answer

Compute div F = 2x + y. Volume integral over the unit cube gives 3/2. Surface integral over the six faces also sums to 3/2, confirming the theorem.

## Answer

**Gauss' theorem:** ∬_S F·n dS = ∭_V div F dV, with F = x² i + 3 j + yz k over the unit cube 0 ≤ x, y, z ≤ 1.

#### Volume integral
```text
div F = ∂(x²)/∂x + ∂(3)/∂y + ∂(yz)/∂z = 2x + 0 + y
```

```text
∭ (2x + y) dV = ∫₀¹∫₀¹∫₀¹ (2x + y) dx dy dz
              = ∫₀¹ ∫₀¹ (1 + y) dy dz  (since ∫₀¹ 2x dx = 1)
              = 1 + 1/2 = 3/2
```

#### Surface integral (six faces, outward normals)
```text
Face x = 1 (n = i):   F·n = x² = 1       → ∬ = 1
Face x = 0 (n = −i):  F·n = −x² = 0      → 0
Face y = 1 (n = j):   F·n = 3            → 3
Face y = 0 (n = −j):  F·n = −3           → −3
Face z = 1 (n = k):   F·n = yz = y       → ∫₀¹∫₀¹ y dx dy = 1/2
Face z = 0 (n = −k):  F·n = −yz = 0      → 0
Total = 1 + 0 + 3 − 3 + 1/2 + 0 = 3/2
```

Both integrals are 3/2, so **Gauss' divergence theorem is verified.**

## Check yourself

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

1. div F?
   A. 2x + y
   B. 2x
   C. 2x+z
   D. x^2+y
   Answer: A. Partials. (Key point: Divergence)

2. Volume integral?
   A. 1
   B. 3/2
   C. 2
   D. 5/2
   Answer: B. 1+1/2. (Key point: Volume)

3. Faces y=1 and y=0 contribute?
   A. 3 each
   B. 0 and 3
   C. 3 and -3, cancel
   D. 1 and 1
   Answer: C. Normals opposite. (Key point: Faces y)

4. Surface total?
   A. 3
   B. 1/2
   C. 2
   D. 3/2
   Answer: D. 1+0+3-3+1/2+0. (Key point: Total)

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