---
title: "Surface integral over a surface F=xi+yj+zk"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:surface-integral-over-a-surface-f-eleven"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/surface-integral-over-a-surface-f-eleven
site: BCT Engineering
---

# Surface integral over a surface F=xi+yj+zk

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

- Asked in: 2071 Shrawan, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/surface-integral-over-a-surface-f-eleven
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Find the surface integral ∬_S F·n ds where F = x i + y j + z k and S is the upper half of the sphere x² + y² + z² = 1.

## Short answer

F·n = x²+y²+z² = 1 on the unit sphere, so the surface integral reduces to the area of the upper hemisphere, which is 2π.

## Answer

S is the upper half of the unit sphere x² + y² + z² = 1, z ≥ 0. The outward unit normal at a point of the unit sphere is the position vector itself: n = x i + y j + z k.

```text
F = x i + y j + z k
F·n = x² + y² + z² = 1
∬_S F·n dS = ∬_S dS = area of the hemisphere = ½ (4π(1)²) = 2π
```

**Check with the projection method:** dS = dx dy/|n·k| = dx dy/z, F·n = 1, so the integral is ∬ dx dy/√(1 − x² − y²) over the unit disc = ∫₀^2π ∫₀¹ r dr/√(1 − r²) dθ = 2π [−√(1 − r²)]₀¹ = 2π.

**Answer: ∬_S F·n dS = 2π** (without the base disc).

## Check yourself

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

1. On the unit sphere outward normal is:
   A. k
   B. i+j+k
   C. the position vector x i + y j + z k
   D. -(x i+y j+z k)
   Answer: C. Radial. (Key point: Normal)

2. F.n equals:
   A. 0
   B. 2
   C. r
   D. x^2+y^2+z^2 = 1
   Answer: D. F equals n. (Key point: F.n)

3. So integral equals:
   A. hemisphere area
   B. volume
   C. 3pi
   D. base disc area
   Answer: A. Integral of dS. (Key point: Integral)

4. Value:
   A. 4 pi
   B. 2 pi
   C. pi
   D. 2 pi/3
   Answer: B. Half of 4 pi. (Key point: Value)

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