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

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

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

- Asked in: 2082 Baishakh, 5 marks
- 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-two
- 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 and the surface S is the upper half of the sphere x² + y² + z² = 9, C being the boundary.

## Short answer

Line integral over boundary circle (x=3cos t, y=3sin t) yields 9π. Surface integral of curl F·n dS reduces to ∬ dxdy over disk of radius 3, also 9π, verifying Stokes' theorem.

## Answer

**Stokes' theorem:** ∮_C F·dr = ∬_S (curl F)·n dS. The boundary C of the upper hemisphere x² + y² + z² = 9, z ≥ 0 is the circle x² + y² = 9 in the plane z = 0, taken anticlockwise.

#### Line integral
F = 2y i + 3x j − z² k. On C: x = 3cos t, y = 3sin t, z = 0, dz = 0.

```text
F·dr = 2y dx + 3x dy
     = 2(3 sin t)(−3 sin t) dt + 3(3 cos t)(3 cos t) dt
     = (−18 sin²t + 27 cos²t) dt
∮_C F·dr = −18π + 27π = 9π
```

#### Surface integral

```text
curl F = (∂R/∂y − ∂Q/∂z) i + (∂P/∂z − ∂R/∂x) j + (∂Q/∂x − ∂P/∂y) k
       = 0 i + 0 j + (3 − 2) k = k
```

For the sphere, n = (x i + y j + z k)/3 and n·k = z/3. Project S on the xy-plane: dS = dx dy/|n·k|, so (curl F)·n dS = (z/3)(3/z) dx dy = dx dy.

```text
∬_S (curl F)·n dS = ∬ dx dy over x² + y² ≤ 9 = 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. The line integral around the circle is?
   A. 27pi
   B. -9pi
   C. 9pi
   D. 18pi
   Answer: C. -18pi + 27pi. (Key point: Line integral)

2. curl F equals?
   A. 3k
   B. -k
   C. 0
   D. k
   Answer: D. 3 - 2 = 1. (Key point: Curl)

3. On sphere radius 3, n.k equals?
   A. z/3
   B. 3z
   C. 1/3
   D. x/3
   Answer: A. n = (x,y,z)/3. (Key point: Normal)

4. Projecting S, dS equals?
   A. dx dy
   B. dx dy / |n.k|
   C. 3 dx dy
   D. z dx dy
   Answer: B. So (z/3)(3/z) dx dy = dx dy. (Key point: Surface area element)

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