---
title: "Evaluate using Gauss' divergence theorem F=xi+yj+z²k"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:evaluate-using-gauss-divergence-theorem-f-six"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/evaluate-using-gauss-divergence-theorem-f-six
site: BCT Engineering
---

# Evaluate using Gauss' divergence theorem F=xi+yj+z²k

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

- Asked in: 2081 Bhadra, 5 marks
- 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-f-six
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Using Gauss' divergence theorem, evaluate ∬_S F·n ds, where F = x i + y j + z² k and S is the surface bounded by the cylinder x² + y² = 9, z = 0 and z = 2.

## Short answer

Divergence is 2 + 2z. Using cylindrical coordinates, ∭ (2+2z) r dz dr dθ over 0≤r≤3, 0≤θ≤2π, 0≤z≤2 equals 72π.

## Answer

**Gauss' divergence theorem:** ∬_S F·n dS = ∭_V div F dV, for the closed surface S enclosing V. Here S is the closed cylinder x² + y² = 9 with bottom z = 0 and top z = 2.

```text
F = x i + y j + z² k
div F = ∂x/∂x + ∂y/∂y + ∂(z²)/∂z = 1 + 1 + 2z = 2 + 2z
```

Use cylindrical coordinates (x = r cos θ, y = r sin θ, dV = r dr dθ dz), with 0 ≤ r ≤ 3, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 2.

```text
∭ (2 + 2z) dV = ∫₀^2π ∫₀³ ∫₀² (2 + 2z) r dz dr dθ
inner: ∫₀² (2 + 2z) dz = [2z + z²]₀² = 4 + 4 = 8
then:  ∫₀³ 8r dr = 8 · 9/2 = 36
then:  ∫₀^2π 36 dθ = 72π
```

**Check by pieces:** curved surface gives ∬(x² + y²)/3 dS = 3 · (2π·3·2) = 36π; top (z = 2, F·k = 4) gives 4·9π = 36π; bottom (z = 0) gives 0. Total 72π.

**Answer: ∬_S F·n dS = 72π.**

## Check yourself

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

1. S includes?
   A. Curved part only
   B. Top only
   C. Curved cylinder plus top and bottom discs
   D. Bottom only
   Answer: C. Gauss needs closed surface. (Key point: Closed surface)

2. div F for F = x i + y j + z^2 k?
   A. 2 + z
   B. 1 + 2z
   C. 2z
   D. 2 + 2z
   Answer: D. 1 + 1 + 2z. (Key point: Divergence)

3. The flux is?
   A. 36pi
   B. 72pi
   C. 24pi
   D. 144pi
   Answer: B. 8 x 9/2 x 2pi = 72pi. (Key point: Result)

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