---
title: "Irrotational field and scalar potential F=2xyz²i+(x²z²+zcosyz)j…"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:irrotational-field-and-scalar-potential-f-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/irrotational-field-and-scalar-potential-f-two
site: BCT Engineering
---

# Irrotational field and scalar potential F=2xyz²i+(x²z²+zcosyz)j…

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

- Asked in: 2082 Baishakh, 2+3 marks
- Video: [Show that the vector field f=2xyz^2i+(x^2z^2+zcosyz)j+(2x^2yz+ycosyz)k is irrational.](https://www.youtube.com/watch?v=De1WmBO2Gl4) — Doctor 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/irrotational-field-and-scalar-potential-f-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Prove that the force field F = 2xyz²i + (x²z² + z cos yz)j + (2x²yz + y cos yz)k is irrotational. Also find the work done by the force F along the path C from (0,0,1) to (1, π/4, 2).

## Short answer

Compute curl: ∂R/∂y = ∂Q/∂z, etc., giving curl F = 0, so F is irrotational. Scalar potential: φ = x^2 y z^2 + sin(yz). Work = φ(1,π/4,2) − φ(0,0,1) = π + 1.

## Answer

A field is irrotational if curl F = 0. Here P = 2xyz², Q = x²z² + z cos yz, R = 2x²yz + y cos yz.

#### Irrotational

```text
∂R/∂y = 2x²z + cos yz − yz sin yz
∂Q/∂z = 2x²z + cos yz − yz sin yz      → i component = 0
∂P/∂z = 4xyz
∂R/∂x = 4xyz                            → j component = 0
∂Q/∂x = 2xz²
∂P/∂y = 2xz²                            → k component = 0
```

So curl F = 0 and F is irrotational (conservative).

#### Work done
Find φ with F = ∇φ.

```text
∂φ/∂x = 2xyz²            → φ = x²yz² + f(y,z)
∂φ/∂y = x²z² + ∂f/∂y = x²z² + z cos yz   → ∂f/∂y = z cos yz → f = sin yz + g(z)
∂φ/∂z = 2x²yz + y cos yz + g'(z) = R     → g' = 0
φ = x²yz² + sin(yz)
```

Work is independent of the path:

```text
W = φ(1, π/4, 2) − φ(0, 0, 1)
  = [1·(π/4)·4 + sin(π/2)] − [0 + sin 0]
  = π + 1
```

**Answer: W = π + 1 ≈ 4.14.**

## Check yourself

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

1. How do we show F is irrotational?
   A. Show div F = 0
   B. Show F is constant
   C. Show all three components of curl F are zero
   D. Show F is zero at origin
   Answer: C. Irrotational means curl F = 0. (Key point: Irrotational check)

2. What is phi?
   A. x^2yz^2
   B. x^2yz^2 + cos(yz)
   C. 2xyz^2 + sin(yz)
   D. x^2 y z^2 + sin(yz)
   Answer: D. Its gradient reproduces P, Q and R. (Key point: Potential)

3. How is work from (0,0,1) to (1,pi/4,2) found?
   A. phi(end) - phi(start), independent of path
   B. Integrate along a straight line only
   C. Multiply F by distance
   D. Use div F
   Answer: A. For conservative F, work equals potential difference. (Key point: Work method)

4. What is the work?
   A. pi
   B. pi + 1
   C. 1
   D. pi - 1
   Answer: B. phi(1,pi/4,2) = pi + sin(pi/2) = pi + 1, phi(0,0,1)=0. (Key point: Value)

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