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

# Evaluate using Stokes' theorem

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

- Asked in: 2073 Shrawan, 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-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∮_C (xy dx + xy² dy) by applying Stokes' theorem where C is the square in the xy-plane with vertices (1,0), (−1,0), (0,1), (0,−1).

## Short answer

By Stokes, ∮ F·dr = ∬ curl F·n dS. Here curl F = (y²−x)k, and S is the square |x|+|y|≤1. The x term vanishes by symmetry. Integral of y² gives 4∫₀^1 y²(1−y)dy = 1/3.

## Answer

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

Write the integrand as F·dr with F = xy i + xy² j (no k component, z = 0 on C).

```text
curl F = (∂Q/∂x − ∂P/∂y) k = (y² − x) k
```

C is the square with vertices (1,0), (0,1), (−1,0), (0,−1): the region |x| + |y| ≤ 1 in the xy-plane, traversed anticlockwise. Take S as this flat region, n = k, dS = dx dy.

```text
∬ (y² − x) dx dy
−x term: odd in x over a symmetric region → 0
y² term: 2 ∫₀¹ ∫₋(1−y)^(1−y) y² dx dy
       = 2 ∫₀¹ 2y²(1 − y) dy
       = 4 (1/3 − 1/4)
       = 1/3
```

**Answer: ∮_C (xy dx + xy² dy) = 1/3.**

## Check yourself

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

1. curl F?
   A. (x - y^2) k
   B. (y^2 - x) k
   C. 0
   D. y^2 i
   Answer: B. Only k. (Key point: Curl)

2. Region bounded by C?
   A. x^2+y^2<=1
   B. |x|<=1
   C. |x| + |y| <= 1
   D. square side 2
   Answer: C. Diamond. (Key point: Region)

3. x term gives?
   A. 1/3
   B. 2/3
   C. 4/3
   D. 0
   Answer: D. Odd. (Key point: Symmetry)

4. Value?
   A. 1/3
   B. 0
   C. 4/3
   D. 2/3
   Answer: A. 4(1/3-1/4). (Key point: Result)

## Close variants set in other exams

- [Evaluate using Stokes' theorem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-stokes-theorem) — Mathematics III, chapter 2, asked 2×

## Related in Mathematics III

- [Evaluate using Stokes' theorem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-stokes-theorem-two) — Mathematics III, chapter 2
- [Evaluate using Stokes' theorem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-stokes-theorem-five) — Mathematics III, chapter 2
- [Evaluate using Stokes' theorem](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-stokes-theorem-four) — Mathematics III, chapter 2
- [Evaluate using Stokes' theorem F=yi+x(1−2z)j−xyk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-stokes-theorem-f-yi) — Mathematics III, chapter 2, asked 2×
- [Evaluate using Gauss' divergence theorem F=x²yi+xy²j+2xyzk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-gauss-divergence-theorem-f) — Mathematics III, chapter 2, asked 10×
- [Evaluate using Stokes' theorem F=(2x−y)i−yz²j−y²zk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aevaluate-using-stokes-theorem-f-2x) — Mathematics III, chapter 2

## Related in other subjects

- [Solve x dy/dx + y log y = xy eˣ](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-x-dy-dx-y-log-y) — Mathematics I, chapter 4, asked 4×
- [Evaluate ∫∫ e^(y/x) dy dx](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-y-x-dy) — Mathematics, chapter 2
- [Evaluate ∫∫ e^(x²) dx dy](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aevaluate-e-x2-dx-dy) — Mathematics, chapter 2
- [Evaluate Stokes' theorem for G field](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Astokes-theorem-evaluate-g-field) — Electromagnetism, chapter 1