---
title: "Surface integral over a surface F=y²z²i+z²x²j+x²y²k"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:surface-integral-over-a-surface-f-four"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/surface-integral-over-a-surface-f-four
site: BCT Engineering
---

# Surface integral over a surface F=y²z²i+z²x²j+x²y²k

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

- Asked in: 2080 Baishakh, 5 marks; 2079 Bhadra, 5 marks (asked 2 times)
- Video: [Vector Integration | Surface Integral | Vector Calculus 2.O by GP Sir](https://www.youtube.com/watch?v=X2Z0tJG0rjU) — Dr.Gajendra Purohit
- 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-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Evaluate ∬_S F·n ds where F = y²z² i + z²x² j + x²y² k and S is the surface of the sphere x² + y² + z² = 1 above the xy-plane.

## Short answer

Divergence ∇·F = 0, so total flux through closed hemisphere-plus-base is zero. Base flux (disk z=0) is −π/24; hence flux through upper hemisphere alone is π/24.

## Answer

F = y²z² i + z²x² j + x²y² k and S is the upper hemisphere x² + y² + z² = 1, z ≥ 0 (curved surface only).

#### Use the divergence theorem on the closed surface (hemisphere + base disc)
```text
∇·F = ∂(y²z²)/∂x + ∂(z²x²)/∂y + ∂(x²y²)/∂z = 0
```
So the total flux through the closed surface (curved part + base) is zero:

```text
∬_{curved} F·n dS + ∬_{base} F·n dS = 0
```

#### Flux through the base
The base is the disc x² + y² ≤ 1 at z = 0, with n = −k:

```text
F·n = −x²y²
∬_{base} F·n dS = −∬_{x²+y²≤1} x²y² dx dy
```
In polar form x = r cos θ, y = r sin θ:

```text
∬ x²y² dx dy = ∫₀^{2π} cos²θ sin²θ dθ · ∫₀¹ r⁵ dr
             = (π/4)(1/6) = π/24
```
so the base flux is −π/24.

#### Curved surface
```text
∬_{curved} F·n dS = 0 − (−π/24) = π/24
```

**Answer: ∬_S F·n dS = π/24.**

(Direct check: on the sphere n = (x, y, z), so F·n = xyz(xy + yz + zx). The terms xy²z² and x²yz² are odd in x or y and integrate to zero over the hemisphere, leaving ∬ x²y²z dS = π/24 ✓.)

## Check yourself

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

1. What is div F for F = y^2z^2 i + z^2x^2 j + x^2y^2 k?
   A. 0
   B. 2xyz
   C. x^2+y^2+z^2
   D. 1
   Answer: A. Each component is independent of its own variable, so every partial is 0. (Key point: Divergence)

2. How is the hemisphere flux obtained?
   A. Directly with n = k
   B. Total flux through closed surface is 0, so curved flux = - base flux
   C. By doubling the base flux
   D. By setting it to zero
   Answer: B. Curved + base = 0 by Gauss, so curved = -base. (Key point: Closing the surface)

3. What normal is used on the base disc z = 0?
   A. n = k
   B. n = i
   C. n = -k (outward, downward)
   D. n = (x,y,z)
   Answer: C. Outward from the closed region at the bottom points down. (Key point: Base normal)

4. What is the flux through the curved surface?
   A. -pi/24
   B. pi/6
   C. 0
   D. pi/24
   Answer: D. Base flux is -pi/24, so curved flux is +pi/24. (Key point: Result)

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