---
title: "Verify Stokes' theorem F=(2x−y)i−yz²j−y²zk"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:verify-stokes-theorem-f-2x-y-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-stokes-theorem-f-2x-y-three
site: BCT Engineering
---

# Verify Stokes' theorem F=(2x−y)i−yz²j−y²zk

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

- Asked in: 2071 Shrawan, 5 marks
- Video: [Verify Stoke's theorem for f=(2x-y)i - yz^2 j -  y^2 zk where S is upper half surface of the sphere](https://www.youtube.com/watch?v=MApB4d1QJpE) — Azhagappan Meyyappan
- 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-2x-y-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Verify Stokes' theorem for F(x, y, z) = (2x − y) i − yz² j − y²z k where S is the upper half of the sphere x² + y² + z² = 4 and C is its boundary.

## Short answer

The line integral around the boundary circle equals 4π, and the surface integral of curl F = k over the hemisphere yields 4π. Both sides match, verifying Stokes’ theorem.

## Answer

**Stokes' theorem:** ∮_C F·dr = ∬_S (curl F)·n dS. Here S is the upper half of x² + y² + z² = 4 and its boundary C is the circle x² + y² = 4, z = 0 (anticlockwise).

F = (2x − y) i − yz² j − y²z k, so P = 2x − y, Q = −yz², R = −y²z.

#### Line integral
On C: x = 2cos t, y = 2sin t, z = 0, dz = 0, and Q = 0.

```text
∮_C F·dr = ∮ (2x − y) dx
         = ∫₀^2π (4cos t − 2sin t)(−2 sin t) dt
         = ∫₀^2π (−8 sin t cos t + 4 sin²t) dt
         = 0 + 4π = 4π
```

#### Surface integral
```text
curl F = (∂R/∂y − ∂Q/∂z) i + (∂P/∂z − ∂R/∂x) j + (∂Q/∂x − ∂P/∂y) k
       = (−2yz + 2yz) i + 0 j + (0 + 1) k = k
```

For the hemisphere, n·k = z/2 and dS = dx dy/(z/2), so (curl F)·n dS = dx dy:

```text
∬_S (curl F)·n dS = ∬ dx dy over x² + y² ≤ 4 = π(2)² = 4π
```

Both sides equal 4π, so **Stokes' theorem is verified.**

## Check yourself

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

1. The boundary C is:
   A. sphere
   B. line
   C. circle x^2+y^2=4 in z=0
   D. x^2+y^2=2
   Answer: C. Edge of hemisphere. (Key point: Boundary)

2. On C, F.dr reduces to (2x-y)dx because:
   A. F is zero
   B. dy = 0
   C. x = 0
   D. z=0 so dz=0 and Q=0
   Answer: D. -yz^2 vanishes. (Key point: Line integral)

3. Line integral equals:
   A. 4 pi
   B. 0
   C. 2 pi
   D. 8 pi
   Answer: A. 4 integral sin^2 t = 4pi. (Key point: Line value)

4. curl F is:
   A. i
   B. k
   C. 2k
   D. 0
   Answer: B. Only (0+1)k remains. (Key point: Curl)

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