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

# Evaluate using Stokes' theorem

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

- Asked in: 2068 Baishakh, 5 marks
- Video: [Stoke's Theorem Problem 1 |  Vector Claculus | Engineering Mathematics](https://www.youtube.com/watch?v=DRWfHYiXnBs) — Alex 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/evaluate-using-stokes-theorem-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate by Stokes' theorem ∮_C (eˣ dx + 2y dy − dz) where C is the curve x² + y² = 4, z = 2.

## Short answer

By Stokes' theorem, curl of F = eˣ i + 2y j − k is zero. Hence the line integral over the closed curve is 0.

## Answer

**Stokes' theorem:** ∮_C F·dr = ∬_S (curl F)·n dS.

Here ∮_C (eˣ dx + 2y dy − dz) = ∮_C F·dr with F = eˣ i + 2y j − k.

```text
curl F = | i    j    k  |
         | ∂x   ∂y   ∂z |
         | eˣ   2y   −1 |
       = (0 − 0) i − (0 − 0) j + (0 − 0) k = 0
```

Since curl F = 0 on any surface S spanning C (for example the disc x² + y² ≤ 4 in the plane z = 2),

```text
∮_C F·dr = ∬_S 0 · n dS = 0
```

**Direct check:** on C, z = 2 so dz = 0. With x = 2cos t, y = 2sin t:
∮ eˣ dx = [eˣ] over one full loop = 0, ∮ 2y dy = [y²] = 0, ∮ (−dz) = 0.

**Answer: ∮_C (eˣ dx + 2y dy − dz) = 0.**

## Check yourself

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

1. Stokes' theorem converts the line integral round C into:
   A. The surface integral of (curl F).n over a surface bounded by C
   B. The volume integral of div F
   C. The surface integral of F.n over C
   D. The double integral of curl F over the xy-plane only
   Answer: A. Oint F.dr = double integral (curl F).n dS. (Key point: Stokes statement)

2. What is curl F for F = e^x i + 2y j - k?
   A. e^x k
   B. 0
   C. 2 j
   D. -i + 2y k
   Answer: B. Each component involves only its own variable, so every partial derivative needed is 0. (Key point: Curl)

3. What is the value of the integral?
   A. 8pi
   B. 4pi
   C. 0
   D. 2pi e^2
   Answer: C. Since curl F = 0, the surface integral is 0; a direct check also gives 0. (Key point: Result)

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