---
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-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-stokes-theorem-f-x2-y2-three
site: BCT Engineering
---

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

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

- Asked in: 2070 Asar, 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-three
- 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 round the rectangle in the xy-plane bounded by x = 0, x = a, y = 0, y = b.

## Short answer

The surface integral of curl F over the rectangle is −2ab². The line integral along the boundary also gives −2ab², confirming Stokes’ theorem.

## Answer

**Stokes' theorem:** ∮_C F·dr = ∬_S (curl F)·n dS. F = (x² + y²) i − 2xy j over the rectangle with corners O(0,0), A(a,0), B(a,b), D(0,b) in the xy-plane (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 O → A → B → D → O)
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, −2xy = 0: 0
Total = a³/3 − ab² − a³/3 − ab² = −2ab²
```

Both are −2ab², so **Stokes' theorem is verified.**

## Check yourself

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

1. k-component of curl F is:
   A. 4y
   B. -2y
   C. -4y
   D. 0
   Answer: C. -2y - 2y. (Key point: Curl)

2. Surface integral equals:
   A. -2a^2 b
   B. 2ab^2
   C. -ab^2
   D. -2 a b^2
   Answer: D. Integral of -4y dy = -2b^2 times a. (Key point: Surface integral)

3. The line integral is taken:
   A. anticlockwise O-A-B-D-O
   B. clockwise
   C. only on OA
   D. along diagonal
   Answer: A. Consistent with n = k. (Key point: Orientation)

4. Line integral total:
   A. 0
   B. -2 a b^2
   C. a^3/3
   D. -ab^2
   Answer: B. a^3/3 - ab^2 - a^3/3 - ab^2. (Key point: Line total)

## Close variants set in other exams

- [Verify Stokes' theorem F=(x²+y²)i−2xyj](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Averify-stokes-theorem-f-x2-y2) — Mathematics III, chapter 2, asked 5×

## Related in Mathematics III

- [Verify Stokes' theorem F=(x²−y²)i+2xyj](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Averify-stokes-theorem-f-x2-y2-two) — Mathematics III, chapter 2, asked 4×
- [Verify Stokes' theorem F=(2x−y)i−yz²j−y²zk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Averify-stokes-theorem-f-2x-y) — Mathematics III, chapter 2, asked 3×
- [Apply Green's theorem F=(x²−xy³)i+(y²−2xy)j](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Aapply-greens-theorem-f-x2-xy3) — Mathematics III, chapter 2
- [Verify Stokes' theorem F=xi+z²j+y²k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Averify-stokes-theorem-f-xi-z2j) — Mathematics III, chapter 2
- [Verify Stokes' theorem F=2yi+3xj−z²k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Averify-stokes-theorem-f-2yi-3xj) — Mathematics III, chapter 2, asked 6×
- [Verify Stokes' theorem F=(2x−y)i−yz²j−y²zk](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Averify-stokes-theorem-f-2x-y-three) — Mathematics III, chapter 2, asked 6×

## Related in other subjects

- [Verify Stokes' theorem, conical surface](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Averify-stokes-conical-surface) — Electromagnetism, chapter 1
- [Double integral xy over region bounded by x²=4ay](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adouble-integral-xy-over-region) — Mathematics, chapter 2