---
title: "Work done by a force field F=yzi+zxj+xyk"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:work-done-by-a-force-field-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/work-done-by-a-force-field-three
site: BCT Engineering
---

# Work done by a force field F=yzi+zxj+xyk

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

- Asked in: 2073 Chaitra, 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/work-done-by-a-force-field-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Find the work done by the force F = yz i + zx j + xy k in displacement of a particle along the straight segment C from the point (1,1,1) to the point (3,3,2).

## Short answer

F = ∇( xyz) is conservative. Work = φ(3,3,2) − φ(1,1,1) = 18 −1 = 17 units.

## Answer

F = yz i + zx j + xy k. Notice that F = ∇(xyz): ∂(xyz)/∂x = yz, ∂(xyz)/∂y = xz, ∂(xyz)/∂z = xy. So F is conservative, curl F = 0, and the work depends only on the end points.

```text
φ = xyz
W = ∫_C F·dr = φ(3,3,2) − φ(1,1,1) = 18 − 1 = 17
```

**Direct check along the line:** r = (1 + 2t, 1 + 2t, 1 + t), 0 ≤ t ≤ 1, dr = (2, 2, 1) dt.

```text
F·dr = [yz·2 + zx·2 + xy·1] dt
     = [2(1+2t)(1+t) + 2(1+t)(1+2t) + (1+2t)²] dt
     = [4(1 + 3t + 2t²) + (1 + 4t + 4t²)] dt
     = (5 + 16t + 12t²) dt
W = ∫₀¹ (5 + 16t + 12t²) dt = 5 + 8 + 4 = 17
```

**Answer: Work done = 17 units.**

## Check yourself

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

1. Why is F = yz i + zx j + xy k conservative?
   A. Its divergence is 0
   B. It is constant
   C. F = grad(xyz), so curl F = 0
   D. It has no k component
   Answer: C. Gradient fields are conservative. (Key point: Conservative)

2. The potential is:
   A. phi = xy+yz
   B. phi = x+y+z
   C. phi = x^2yz
   D. phi = xyz
   Answer: D. Partial derivatives match. (Key point: Potential)

3. Work equals:
   A. phi(3,3,2) - phi(1,1,1) = 17
   B. 18
   C. 19
   D. 9
   Answer: A. 18 - 1 = 17. (Key point: Work)

4. The work depends only on:
   A. path length
   B. the end points
   C. the straight line only
   D. time taken
   Answer: B. Conservative field. (Key point: Path)

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