---
title: "Irrotational field and scalar potential F=(2x+z²)i+zj+(y+2xz)k"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:irrotational-field-and-scalar-potential-f-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/irrotational-field-and-scalar-potential-f-three
site: BCT Engineering
---

# Irrotational field and scalar potential F=(2x+z²)i+zj+(y+2xz)k

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

- Asked in: 2073 Shrawan, 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-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

Show that F = (2x + z²) i + z j + (y + 2xz) k is irrotational and find its scalar potential.

## Short answer

Curl F = 0, so it is irrotational. Integrating ∂φ/∂x = 2x+z² gives φ = x²+xz²+f(y,z); from ∂φ/∂y = z we get f = yz+g(z); then ∂φ/∂z = 2xz+y+g' matches R, so g'=0. Potential: φ = x²+xz²+yz+c.

## Answer

F is irrotational if curl F = 0. Here P = 2x + z², Q = z, R = y + 2xz.

```text
curl F = (∂R/∂y − ∂Q/∂z) i + (∂P/∂z − ∂R/∂x) j + (∂Q/∂x − ∂P/∂y) k
       = (1 − 1) i + (2z − 2z) j + (0 − 0) k
       = 0
```

So F is irrotational, and a scalar potential φ exists with F = ∇φ.

```text
∂φ/∂x = 2x + z²   → φ = x² + xz² + f(y,z)
∂φ/∂y = f_y = z    → f = yz + g(z)
∂φ/∂z = 2xz + y + g'(z) = y + 2xz  → g' = 0
```

**Answer: φ = x² + xz² + yz + c** (c is an arbitrary constant). Check: ∇φ = (2x + z²) i + z j + (2xz + y) k = F.

## Check yourself

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

1. curl F?
   A. i
   B. k
   C. 0
   D. 2z
   Answer: C. All components cancel. (Key point: Curl)

2. phi?
   A. x^2 + yz
   B. x^2 + xz^2 + c
   C. x^2z + yz
   D. x^2 + xz^2 + yz + c
   Answer: D. Gradient. (Key point: Potential)

3. c is?
   A. Arbitrary constant
   B. Zero
   C. 1
   D. Determined
   Answer: A. Gradient unaffected. (Key point: Constant)

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