---
title: "Irrotational field and scalar potential F=(2xz³+6y)i+(6x−2yz)j…"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:irrotational-field-and-scalar-potential-f-four"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/irrotational-field-and-scalar-potential-f-four
site: BCT Engineering
---

# Irrotational field and scalar potential F=(2xz³+6y)i+(6x−2yz)j…

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

- Asked in: 2069 Asar, 5 marks
- 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-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Prove that F = (2xz³ + 6y) i + (6x − 2yz) j + (3x²z² − y²) k is a conservative vector field and find its scalar potential function.

## Short answer

Curl F = 0, so it is conservative. Integrate ∂φ/∂x = 2xz³+6y → φ = x²z³+6xy+f(y,z); ∂φ/∂y = 6x−2yz → f = −y²z+g(z); ∂φ/∂z = 3x²z²−y² matches, so g'=0. Potential: φ = x²z³+6xy−y²z+c.

## Answer

A vector field is conservative if curl F = 0 (the region is simply connected), and then F = ∇φ.

P = 2xz³ + 6y, Q = 6x − 2yz, R = 3x²z² − y².

```text
i: ∂R/∂y − ∂Q/∂z = −2y − (−2y) = 0
j: ∂P/∂z − ∂R/∂x = 6xz² − 6xz² = 0
k: ∂Q/∂x − ∂P/∂y = 6 − 6 = 0
```

curl F = 0, so F is conservative.

#### Scalar potential
```text
∂φ/∂x = 2xz³ + 6y   → φ = x²z³ + 6xy + f(y,z)
∂φ/∂y = 6x + f_y = 6x − 2yz  → f_y = −2yz → f = −y²z + g(z)
∂φ/∂z = 3x²z² − y² + g'(z) = 3x²z² − y²  → g' = 0
```

**Answer: φ = x²z³ + 6xy − y²z + c.**
Check: ∇φ = (2xz³ + 6y) i + (6x − 2yz) j + (3x²z² − y²) k = F.

## Check yourself

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

1. Conservative means?
   A. div F=0
   B. curl F = 0 (simply connected)
   C. F constant
   D. F=0
   Answer: B. Definition. (Key point: Conservative)

2. j-component?
   A. 12xz^2
   B. 0 only at 0
   C. 6xz^2 - 6xz^2 = 0
   D. -6xz^2
   Answer: C. dP/dz - dR/dx. (Key point: Component)

3. phi?
   A. x^2z^3 + 6xy + y^2z
   B. x^2z^3 - 6xy
   C. 2xz^3
   D. x^2z^3 + 6xy - y^2z + c
   Answer: D. Gradient. (Key point: Potential)

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