---
title: "Irrotational field and scalar potential F=(x²−yz)i+(y²−zx)j…"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:irrotational-field-and-scalar-potential-f"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/irrotational-field-and-scalar-potential-f
site: BCT Engineering
---

# Irrotational field and scalar potential F=(x²−yz)i+(y²−zx)j…

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

- Asked in: 2081 Bhadra, 2+3 marks; 2068 Baishakh, 5 marks (asked 2 times)
- Video: [Prove Vector Field Is Irrotational and Find Scalar Potential φ| Most Important| Engineering Maths-II](https://www.youtube.com/watch?v=8CynEK8GWJk) — Mathematics Tutor
- 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
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Show that F = (x² − yz) i + (y² − zx) j + (z² − xy) k is irrotational. Also find its scalar potential function.

## Short answer

The curl ∇×F = 0, so the field is irrotational. Integrate ∂φ/∂x = x²−yz, etc., to obtain scalar potential φ = (x³ + y³ + z³)/3 − xyz + c.

## Answer

A vector field F is **irrotational** if ∇×F = 0. Then F = ∇φ for some scalar potential φ.

F = (x² − yz) i + (y² − zx) j + (z² − xy) k

#### Curl
```text
∇×F = | i          j          k        |
      | ∂/∂x      ∂/∂y      ∂/∂z      |
      | x²−yz     y²−zx     z²−xy     |

i : ∂(z²−xy)/∂y − ∂(y²−zx)/∂z = −x − (−x) = 0
j : ∂(x²−yz)/∂z − ∂(z²−xy)/∂x = −y − (−y) = 0
k : ∂(y²−zx)/∂x − ∂(x²−yz)/∂y = −z − (−z) = 0
```

So ∇×F = 0 and **F is irrotational**.

#### Scalar potential φ with F = ∇φ
```text
∂φ/∂x = x² − yz   ... (1)
∂φ/∂y = y² − zx   ... (2)
∂φ/∂z = z² − xy   ... (3)
```

Integrate (1) with respect to x: φ = x³/3 − xyz + f(y, z).

Differentiate with respect to y and compare with (2): −xz + ∂f/∂y = y² − zx ⇒ ∂f/∂y = y² ⇒ f = y³/3 + g(z).

So φ = x³/3 + y³/3 − xyz + g(z). Differentiate with respect to z and compare with (3): −xy + g′(z) = z² − xy ⇒ g′(z) = z² ⇒ g = z³/3 + c.

**Scalar potential: φ = (x³ + y³ + z³)/3 − xyz + c**, where c is an arbitrary constant.

(Check: ∂φ/∂x = x² − yz ✓, ∂φ/∂y = y² − xz ✓, ∂φ/∂z = z² − xy ✓.)

## Check yourself

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

1. When is F irrotational?
   A. When curl F = 0
   B. When div F = 0
   C. When F = 0
   D. When |F| is constant
   Answer: A. Irrotational means zero curl. (Key point: Irrotational definition)

2. What is the i-component of curl F for this F?
   A. -2x
   B. d(z^2-xy)/dy - d(y^2-zx)/dz = -x + x = 0
   C. 2x
   D. 0 only if x = 0
   Answer: B. Both partials equal -x, so the difference is 0. (Key point: Curl component)

3. How is the scalar potential found?
   A. Differentiate F three times
   B. Take the divergence of F
   C. Integrate dphi/dx = x^2 - yz w.r.t. x, then fix the unknown function using the other two components
   D. Take the magnitude of F
   Answer: C. Partial integration plus matching with the other components. (Key point: Potential method)

4. What is phi?
   A. x^3+y^3+z^3 + xyz
   B. (x^3+y^3+z^3)/3 + xyz
   C. x^2 y^2 z^2
   D. (x^3+y^3+z^3)/3 - xyz + c
   Answer: D. Gradient of this gives x^2 - yz, y^2 - xz, z^2 - xy. (Key point: Potential)

## Close variants set in other exams

- [Irrotational field and scalar potential F=(2x+z²)i+zj+(y+2xz)k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Airrotational-field-and-scalar-potential-f-three) — Mathematics III, chapter 2, asked 3×

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