---
title: "Verify Stokes' theorem F=2yi+3xj−z²k"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:verify-stokes-theorem-f-2yi-3xj"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/verify-stokes-theorem-f-2yi-3xj
site: BCT Engineering
---

# Verify Stokes' theorem F=2yi+3xj−z²k

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

- Asked in: 2069 Chaitra, 5 marks; 2069 Asar, 5 marks (asked 2 times)
- Video: [Verify Stokes theorem for F = (2x-y)i + yz²j - yz²k for the upper half of the sphere x²+y²+z² = 1.](https://www.youtube.com/watch?v=giuD-SLwlOU) — 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-2yi-3xj
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Verify Stokes' theorem for F = 2y i + 3x j − z² k where S is the upper half of the sphere x² + y² + z² = 9 and C is its boundary.

## Short answer

Line integral around the circle yields 9π. Curl F = k, surface integral over the disc of radius 3 gives 9π. Both sides agree, verifying the theorem.

## Answer

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

F = 2y i + 3x j − z² k. S: upper half of x² + y² + z² = 9 (radius 3). Boundary C: the circle x² + y² = 9, z = 0.

#### Left side: line integral
On C: x = 3 cos t, y = 3 sin t, z = 0, 0 ≤ t ≤ 2π; dx = −3 sin t dt, dy = 3 cos t dt, dz = 0.

```text
F·dr = 2y dx + 3x dy − z² dz
     = 2(3 sin t)(−3 sin t) dt + 3(3 cos t)(3 cos t) dt
     = (−18 sin²t + 27 cos²t) dt

∮ F·dr = −18π + 27π = 9π
```

#### Right side: surface integral
```text
∇×F = i(0 − 0) − j(0 − 0) + k(∂(3x)/∂x − ∂(2y)/∂y) = (3 − 2) k = k
```
So (∇×F)·n dS = n_z dS = dx dy, the projection on the xy-plane, which is the disc x² + y² ≤ 9:

```text
∬_S (∇×F)·n dS = area of disc = π(3)² = 9π
```

**Both sides equal 9π, so Stokes' theorem is verified.** ∎

## Check yourself

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

1. How is C parametrised?
   A. x = 9 cos t, y = 9 sin t
   B. x = cos t, y = sin t
   C. x = 3cos t, y = 3sin t, z = 3
   D. x = 3cos t, y = 3sin t, z = 0, 0 <= t <= 2pi
   Answer: D. The boundary circle has radius 3 in the plane z = 0. (Key point: Parametrisation)

2. What is the line integral?
   A. 9pi
   B. -9pi
   C. 45pi
   D. 0
   Answer: A. -18pi + 27pi = 9pi. (Key point: Line integral)

3. What is curl F for F = 2y i + 3x j - z^2 k?
   A. 5k
   B. k
   C. -k
   D. 0
   Answer: B. The k component is d(3x)/dx - d(2y)/dy = 3 - 2 = 1. (Key point: Curl)

4. Why is the surface integral 9pi?
   A. The sphere area is 36pi
   B. curl F is 9
   C. curl F.n dS = dx dy over the disc of radius 3, area 9pi
   D. The hemisphere area is 18pi
   Answer: C. Projection onto the xy-plane is the disc of area pi(3)^2. (Key point: Surface integral)

## Close variants set in other exams

- [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×
- [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-two) — Mathematics III, chapter 2, asked 5×
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