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

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

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

- Asked in: 2079 Baishakh, 1+4 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/evaluate-using-stokes-theorem-f-2x
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

State Stokes' theorem. Apply it to evaluate ∬_S (∇×F)·n ds where F = (2x − y) i − yz² j − y²z k, S is the upper half surface of the sphere x² + y² + z² = a² and C is its boundary.

## Short answer

By Stokes’ theorem, the surface integral equals the line integral around the boundary circle x²+y²=a², z=0. Parametrizing the circle gives πa².

## Answer

#### Statement
If S is an open surface bounded by a simple closed curve C, and F has continuous first partial derivatives, then

∮_C F·dr = ∬_S (∇ × F)·n dS,

where C is traversed anticlockwise as seen from the tip of n.

#### Application
By Stokes' theorem, ∬_S (∇×F)·n dS equals the line integral round the boundary C: x² + y² = a², z = 0.

F = (2x − y) i − yz² j − y²z k. On C, z = 0, so F·dr = (2x − y) dx. With x = a cos t, y = a sin t:

```text
∮_C (2x − y) dx = ∫₀^2π (2a cos t − a sin t)(−a sin t) dt
                = ∫₀^2π (−2a² sin t cos t + a² sin²t) dt
                = 0 + a² π = π a²
```

**Check using the curl:** ∇×F = (−2yz + 2yz) i + 0 j + (0 + 1) k = k. On the hemisphere, (∇×F)·n dS = dx dy, so the integral equals the area of the disc x² + y² ≤ a², that is πa².

**Answer: ∬_S (∇×F)·n dS = πa².**

## Check yourself

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

1. Stokes' theorem equates:
   A. flux with volume
   B. line with volume
   C. closed line integral of F.dr with surface integral of curl F.n
   D. curl with divergence
   Answer: C. Standard. (Key point: Statement)

2. Boundary C of upper hemisphere:
   A. sphere
   B. z=a
   C. line
   D. circle x^2+y^2=a^2, z=0
   Answer: D. Equator. (Key point: Boundary)

3. curl F =
   A. k
   B. 0
   C. i
   D. 2k
   Answer: A. Computed. (Key point: Curl)

4. Result:
   A. 2 pi a^2
   B. pi a^2
   C. pi a
   D. 0
   Answer: B. Area of disc. (Key point: Value)

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