# conservative vector fields

(1.4 hours to learn)

## Summary

A vector field is conservative if it can be expressed as the gradient of a function (called the potential function), or equivalently, if all line integrals are path independent. The notion is useful because dynamical systems are easier to analyze if they can be described in terms of a potential function.

## Context

This concept has the prerequisites:

- vector fields
- line integrals (Line integrals are independent of the path for a conservative vector field.)
- gradient (Conservative vector fields are gradients of functions.)

## Goals

- Know the Fundamental Theorem of Calculus for line integrals

- Know what a conservative vector field is and why it is useful

- Show the equivalence of two definitions of a conservative vector field:
- can be expressed as the gradient of a function
- line integrals are independent of the path

- Know the definition of the curl of a vector field

- Be able to tell if a vector field is conservative by determining if the curl is 0

- Be able to determine the potential function for a conservative vector field

## Core resources (read/watch one of the following)

## -Free-

→ MIT Open Courseware: Multivariable Caclulus (2010)

Video lectures for MIT's introductory multivariable calculus class.

## -Paid-

→ Multivariable Mathematics

A textbook on linear algebra and multivariable calculus with proofs.

- Section 8.3.1, "The Fundamental Theorem of Calculus for line integrals," pages 351-353
- Section 8.3.2, "Finding a potential function," pages 353-357

Additional dependencies:

- differential forms
- exterior derivative
- pullback

Other notes:

- This resource gives a more formal treatment than the others.

→ Multivariable Calculus

An introductory multivariable calculus textbook.

Location:
Section 15.3, "The Fundamental Theorem and independence of path," pages 1030-1036

## See also

-No Additional Notes-