---
title: "Line integral independent of path"
subject: "Mathematics III"
chapter: 2
key: "semester_3/mathematics_iii:line-integral-independent-of-path-three"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/line-integral-independent-of-path-three
site: BCT Engineering
---

# Line integral independent of path

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

- Asked in: 2076 Chaitra, 5 marks
- Video: [Path independence for line integrals | Multivariable Calculus | Khan Academy](https://www.youtube.com/watch?v=K_fgnCJOI8I) — Khan Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/line-integral-independent-of-path-three
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

If F = 3x²yz² i + x³z² j + 2x³yz k, show that ∫_C F·dr is independent of the path of integration. Hence evaluate the integral on any path C from (0,0,0) to (1,2,3).

## Short answer

Curl F = 0, so the integral is path-independent. A potential function is φ = x³yz², and the line integral from (0,0,0) to (1,2,3) equals φ(1,2,3) − φ(0,0,0) = 18.

## Answer

A line integral ∫ F·dr is independent of path if F is conservative, that is curl F = 0 (then F = ∇φ).

#### Test
F = 3x²yz² i + x³z² j + 2x³yz k.

```text
i: ∂R/∂y − ∂Q/∂z = 2x³z − 2x³z = 0
j: ∂P/∂z − ∂R/∂x = 6x²yz − 6x²yz = 0
k: ∂Q/∂x − ∂P/∂y = 3x²z² − 3x²z² = 0
```

curl F = 0, so the integral is independent of the path.

#### Potential function
```text
∂φ/∂x = 3x²yz²  → φ = x³yz² + f(y,z)
∂φ/∂y = x³z² + f_y = x³z²  → f_y = 0
∂φ/∂z = 2x³yz + f_z = 2x³yz → f_z = 0
φ = x³yz²
```

#### Value
```text
∫ F·dr = φ(1,2,3) − φ(0,0,0) = (1)(2)(9) − 0 = 18
```

**Answer: the integral is path independent and equals 18.**

## Check yourself

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

1. Path independence is shown by?
   A. curl F = 0
   B. div F = 0
   C. F = 0
   D. |F| constant
   Answer: A. Conservative field. (Key point: Test)

2. phi?
   A. x^3 y z
   B. x^3 y z^2
   C. 3x^2yz^2
   D. x^3 z^2
   Answer: B. Gradient matches. (Key point: Potential)

3. Integral from origin to (1,2,3)?
   A. 6
   B. 9
   C. 18
   D. 36
   Answer: C. 1 x 2 x 9. (Key point: Value)

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