Conservative Vector Fields and Potential Functions

Overview

A vector field F is conservative if it is the gradient of some scalar function f (the potential): F = gradient(f). For conservative fields, line integrals are path-independent: integral_C F dot dr depends only on endpoints. A field is conservative iff its curl is zero (in simply-connected regions). The fundamental theorem for line integrals: integral_C F dot dr = f(B) - f(A).

Intuition

If you push a box on a flat floor, the work done depends on the path (friction). If you lift a box in a gravitational field, the work done depends only on how high you lift it (endpoints), not the path -- gravity is conservative. A conservative field has a 'height function' (potential f) such that F = gradient(f). Going around a closed loop does zero work: all 'uphill' work is returned 'downhill.'

Formal Definition

Definition

F: D -> R^n is conservative if F = gradient(f) for some C^1 scalar f (the potential or potential energy function). Equivalent conditions (on a simply connected domain): (1) curl(F) = 0 for F: R^3 -> R^3, or dF_1/dy = dF_2/dx for F: R^2 -> R^2. (2) Line integral is path-independent. (3) Closed-path integrals vanish: contour integral F dot dr = 0. Fundamental theorem: integral_C F dot dr = f(B) - f(A).

\mathbf{F} = \nabla f \Rightarrow \int_C \mathbf{F}\cdot d\mathbf{r} = f(B) - f(A)

Fundamental theorem for line integrals

\text{curl}\,\mathbf{F} = \mathbf{0} \Leftrightarrow \mathbf{F} \text{ conservative (simply connected)}

Curl test

\oint_C \mathbf{F}\cdot d\mathbf{r} = 0 \text{ for any closed } C

Closed loop is zero

Notation

Notation	Meaning
$f$	Potential function: F = gradient(f)
$\nabla \times \mathbf{F}$	Curl of F; zero iff F conservative (simply connected domain)

Theorems

Theorem 1: Fundamental Theorem for Line Integrals

If F = gradient(f) and C is any smooth curve from A to B, then integral_C F dot dr = f(B) - f(A). This reduces path integrals to evaluation of f at endpoints.

Worked Examples

1
Check: dF_1/dy = 2x + 2y, dF_2/dx = 2x + 2y. Equal, so F is conservative.
$\frac{\partial F_1}{\partial y} = \frac{\partial F_2}{\partial x} = 2x + 2y$
2
Find f: f_x = 2xy + y^2, so f = x^2y + xy^2 + g(y).
$f = x^2 y + xy^2 + g(y)$
3
f_y = x^2 + 2xy + g'(y) = x^2 + 2xy, so g'(y)=0, g = const.
$f(x,y) = x^2 y + xy^2$

✓ Answer

f(x,y) = x^2*y + xy^2 (+ constant).

Practice Problems

Mediumapplication

Use the potential to evaluate integral_C (2xy)dx + x^2 dy from (0,0) to (1,2).

Common Mistakes

Common Mistake

Concluding a field is conservative from curl F = 0 on a non-simply-connected domain.

curl F = 0 implies conservative only on simply connected domains (no holes). The field F = (-y,x)/(x^2+y^2) has zero curl except at the origin but is not conservative on R^2 minus the origin.

Quiz

For a conservative field F, the line integral from A to B along two different paths C1 and C2:

May differ in signAre always equalDiffer by a constantAdd to zero

Historical Background

Conservative forces appear in classical mechanics -- gravity and electrostatics are conservative. The connection between conservative vector fields, path independence, and potential functions was developed by Gauss, Green, and Stokes in the 19th century. The fundamental theorem for line integrals is the direct multivariable analogue of the single-variable fundamental theorem of calculus.

1800s
Gauss and Green develop potential theory for conservative fields
Carl Friedrich Gauss, George Green

Summary

F conservative iff F = gradient(f) for some potential f.
On simply connected domains: F conservative iff curl(F) = 0.
Fundamental theorem: integral_C F dot dr = f(B) - f(A) (path independent).
Closed loops give zero work for conservative fields.

References

BookStewart, J. Multivariable Calculus. 8th ed. Brooks/Cole, 2015.