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

# Line integral independent of path

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

- Asked in: 2079 Baishakh, 3+2 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-five
- 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 P: (0,0,0) to Q: (1,2,3).

## Short answer

curl F = 0, so F is conservative with potential φ = x³yz². The line integral is path-independent and equals φ(1,2,3)−φ(0,0,0) = 18.

## Answer

A line integral ∫_C F·dr is independent of the path when F is conservative, i.e. curl F = 0.

#### Part 1: independence (3 marks)
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 F = ∇φ for some φ and the integral does not depend on the path.

#### Part 2: value (2 marks)
```text
∂φ/∂x = 3x²yz²  → φ = x³yz² + f(y,z)
∂φ/∂y = x³z² → f_y = 0;   ∂φ/∂z = 2x³yz → f_z = 0
φ = x³yz²

∫ F·dr from (0,0,0) to (1,2,3) = 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 holds when:
   A. div F = 0
   B. F constant
   C. curl F = 0
   D. F.dr = 0
   Answer: C. Conservative field. (Key point: Criterion)

2. All curl components of F:
   A. are nonzero
   B. only i
   C. only k
   D. vanish
   Answer: D. Each pair cancels. (Key point: Curl)

3. phi =
   A. x^3 y z^2
   B. x^3 y z
   C. x^2yz^2
   D. 3x^2yz^2
   Answer: A. Integrate P. (Key point: Potential)

4. Integral:
   A. 9
   B. 18
   C. 6
   D. 36
   Answer: B. phi(1,2,3) = 1*2*9. (Key point: Value)

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