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

# Line integral independent of path

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

- Asked in: 2074 Asoj, 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-four
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-2

## Question

If F is the gradient of some scalar point function φ, i.e. F = ∇φ, prove that the line integral is independent of the path joining any two points in the region, and conversely.

## Short answer

If F = ∇φ, then ∫_ C F·dr = ∫ dφ = φ(B)−φ(A), independent of path. Conversely, if the line integral is path independent, define φ(x,y,z) = ∫_ A^{(x,y,z)} F·dr; then ∂φ/∂x = P, ∂φ/∂y = Q, ∂φ/∂z = R, so F = ∇φ.

## Answer

**Part 1: F = ∇φ implies path independence.**

Let A and B be two points in the region, joined by any smooth curve C given by r = r(t), t from t₁ to t₂. Since F = ∇φ = (∂φ/∂x) i + (∂φ/∂y) j + (∂φ/∂z) k,

```text
∫_C F·dr = ∫_C (∂φ/∂x dx + ∂φ/∂y dy + ∂φ/∂z dz)
         = ∫_C dφ
         = φ(B) − φ(A)
```

The result depends only on the end points A and B, not on the curve joining them. Hence the line integral is independent of path (and for a closed curve, ∮ F·dr = 0).

**Part 2: Converse.** Suppose ∫_A^B F·dr is independent of the path in the region, and write F = P i + Q j + R k. Fix a point A and define

```text
φ(x,y,z) = ∫_A^(x,y,z) F·dr  (any path)
```

This is well defined by path independence. To find ∂φ/∂x, go from (x,y,z) to (x + Δx, y, z) along a straight segment parallel to the x-axis, where dy = dz = 0:

```text
φ(x+Δx,y,z) − φ(x,y,z) = ∫_x^(x+Δx) P(s,y,z) ds
```

Divide by Δx and let Δx → 0. Since P is continuous, the right side tends to P(x, y, z), so ∂φ/∂x = P. In the same way ∂φ/∂y = Q and ∂φ/∂z = R. Therefore

F = P i + Q j + R k = ∇φ,

so F is the gradient of a scalar function. Also curl F = curl(∇φ) = 0.

Thus ∫ F·dr is path independent if and only if F = ∇φ.

## Check yourself

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

1. If F = grad phi, integral equals?
   A. phi(A)+phi(B)
   B. 0 always
   C. phi(B) - phi(A)
   D. phi(B) times path length
   Answer: C. d(phi). (Key point: Forward)

2. Closed curve integral?
   A. phi
   B. Area
   C. 2 phi
   D. 0
   Answer: D. A = B. (Key point: Closed curve)

3. phi defined as?
   A. Integral of F.dr from fixed A
   B. div F
   C. curl F
   D. |F|
   Answer: A. Well defined. (Key point: Converse construction)

4. curl grad phi?
   A. F
   B. 0
   C. phi
   D. 1
   Answer: B. Identity. (Key point: Why curl zero)

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