---
title: "Verify Stokes' theorem F=(x²−y²)i+2xyj"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:verify-stokes-theorem-f-x2-y2-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-stokes-theorem-f-x2-y2-two
site: BCT Engineering
---

# Verify Stokes' theorem F=(x²−y²)i+2xyj

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

- Asked in: 2078 Kartik, 5 marks
- Video: [Verify Stokes theorem for F = (x²+y²)i - 2xy j  | Surface Integrals | Vector Calculus](https://www.youtube.com/watch?v=m3i-CYczr1s) — Spectrum of Mathematics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-stokes-theorem-f-x2-y2-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Verify Stokes' theorem for F = (x² − y²) i + 2xy j taken over the rectangle bounded by the lines x = 0, x = a, y = 0, y = b.

## Short answer

curl F = 4y k. The surface integral over the rectangle 0≤x≤a, 0≤y≤b gives ∬ 4y dxdy = 2ab². The line integral around the boundary also yields 2ab², verifying Stokes' theorem.

## Answer

**Stokes' theorem:** ∮_C F·dr = ∬_S (curl F)·n dS. Here F = (x² − y²) i + 2xy j, and the rectangle with vertices O(0,0), A(a,0), B(a,b), D(0,b) lies in the xy-plane, so n = k.

#### Surface integral
```text
curl F · k = ∂Q/∂x − ∂P/∂y = 2y − (−2y) = 4y
∬_S 4y dx dy = ∫₀^a ∫₀^b 4y dy dx = ∫₀^a 2b² dx = 2ab²
```

#### Line integral (anticlockwise: OA → AB → BD → DO)
F·dr = (x² − y²) dx + 2xy dy.

```text
OA: y = 0, dy = 0:       ∫₀^a x² dx = a³/3
AB: x = a, dx = 0:       ∫₀^b 2a y dy = ab²
BD: y = b, dy = 0, x: a→0: ∫ₐ⁰ (x² − b²) dx = −a³/3 + ab²
DO: x = 0, dx = 0, and 2xy = 0: 0
Sum = a³/3 + ab² − a³/3 + ab² = 2ab²
```

Both sides equal 2ab², so **Stokes' theorem is verified.**

## Check yourself

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

1. Rectangle in xy-plane, so n?
   A. k
   B. i
   C. j
   D. -k
   Answer: A. Plane z = 0. (Key point: Normal)

2. curl F . k?
   A. 2y
   B. 4y
   C. 0
   D. -4y
   Answer: B. 2y - (-2y). (Key point: curl.k)

3. Value?
   A. ab^2
   B. 4ab^2
   C. 2ab^2
   D. a^2 b
   Answer: C. Integral of 4y dy from 0 to b = 2b^2, times a. (Key point: Surface integral)

4. On side DO (x = 0), integrand?
   A. a^3/3
   B. ab^2
   C. b^2
   D. 0
   Answer: D. x = 0 so dx = 0 and 2xy = 0. (Key point: Line integral sides)

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