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

# Evaluate using Stokes' theorem

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

- Asked in: 2075 Chaitra, 1+4 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-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

State Stokes' theorem. Evaluate ∮_C (xy dx + xy² dy) by Stokes' theorem taking C to be a square in the xy-plane with vertices (1,0), (−1,0), (0,1) and (0,−1).

## Short answer

curl F = (y² − x) k. Integrating over the diamond |x|+|y|≤1 in the xy-plane, the x term vanishes by symmetry, and ∬ y² dxdy = 1/3, so the line integral equals 1/3.

## Answer

#### Statement
If S is an open surface bounded by a closed curve C, and F has continuous first partial derivatives, then

∮_C F·dr = ∬_S (curl F)·n dS,

where C is traversed anticlockwise when seen from the tip of n (right-hand rule).

#### Evaluation
F = xy i + xy² j, so

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

The square with vertices (1,0), (0,1), (−1,0), (0,−1) is a diamond |x| + |y| ≤ 1 in the xy-plane, so n = k. For a given y (0 ≤ y ≤ 1) x runs from −(1 − y) to (1 − y); the lower half is symmetrical.

```text
∬ (y² − x) dx dy
The x term: odd in x over a symmetric interval → 0.
∬ y² dx dy = 2 ∫₀¹ y² · 2(1 − y) dy
           = 4 [y³/3 − y⁴/4]₀¹
           = 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. Orientation of C is?
   A. Clockwise
   B. Any
   C. Anticlockwise seen from tip of n
   D. Along x-axis
   Answer: C. Right-hand rule. (Key point: Orientation)

2. The square with those vertices is?
   A. |x|,|y| <= 1
   B. x^2+y^2<=1
   C. x+y<=1
   D. |x| + |y| <= 1
   Answer: D. Diamond. (Key point: Region)

3. Why does x term vanish?
   A. Odd in x over symmetric interval
   B. x=0
   C. Area is zero
   D. It cancels with y^2
   Answer: A. Symmetry. (Key point: Odd term)

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

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