---
title: "Evaluate using Gauss' divergence theorem F=xi−yj+(z²−1)k"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-using-gauss-divergence-theorem-f-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-two
site: BCT Engineering
---

# Evaluate using Gauss' divergence theorem F=xi−yj+(z²−1)k

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

- Asked in: 2078 Kartik, 5 marks; 2070 Chaitra, 5 marks; 2069 Chaitra, 5 marks (asked 3 times)
- Video: [Multivariable Calculus: Divergence Theorem - Examples (16.9)](https://www.youtube.com/watch?v=qFkOfh0ENNY) — Dr. Clark Teaches Math
- 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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∬_S F·n ds by Gauss' divergence theorem, where F = x i − y j + (z² − 1) k and S is the cylinder formed by the surfaces x² + y² = 4, z = 0, z = 1.

## Short answer

Divergence ∇·F = 2z. Using cylindrical coordinates (r,θ,z) for the cylinder 0≤r≤2, 0≤z≤1, the volume integral is 4π, so the flux is 4π.

## Answer

**Gauss' divergence theorem:** ∬_S F·n dS = ∭_V ∇·F dV.

F = x i − y j + (z² − 1) k. S is the closed cylinder x² + y² = 4 with the base z = 0 and the top z = 1.

#### Divergence
```text
∇·F = ∂x/∂x + ∂(−y)/∂y + ∂(z² − 1)/∂z = 1 − 1 + 2z = 2z
```

#### Volume integral
Use cylindrical coordinates: x = r cos θ, y = r sin θ, dV = r dr dθ dz; r from 0 to 2, θ from 0 to 2π, z from 0 to 1.

```text
∭ 2z dV = ∫₀¹ 2z dz · ∫₀^{2π} dθ · ∫₀² r dr
        = [z²]₀¹ · 2π · [r²/2]₀²
        = 1 · 2π · 2 = 4π
```

**Answer: ∬_S F·n dS = 4π.**

#### Check by surfaces
- Top (z = 1, n = k): F·n = z² − 1 = 0 ⇒ contributes 0.
- Base (z = 0, n = −k): F·n = −(z² − 1) = 1 ⇒ contributes the area 4π.
- Curved surface: n = (x i + y j)/2, F·n = (x² − y²)/2, whose integral over the cylinder is 0 by symmetry.

Total = 4π ✓.

## Check yourself

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

1. ∇·F for F=x i−y j+(z²−1)k:
   A. 1
   B. 2z
   C. 2z−1
   D. 0
   Answer: B. 1−1+2z. (Key point: Divergence)

2. Which coordinates suit the cylinder?
   A. Cylindrical
   B. Spherical
   C. Cartesian only
   D. Polar in 3D
   Answer: A. dV = r dr dθ dz. (Key point: Coordinates)

3. The integral:
   A. 2π
   B. 8π
   C. 0
   D. 4π
   Answer: D. 1·2π·2 = 4π. (Key point: Result)

4. Which surface contributes all the flux?
   A. The top
   B. The curved side
   C. The base (z=0)
   D. None
   Answer: C. Top gives 0, curved side 0 by symmetry, base gives 4π. (Key point: Surface check)

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