# 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:

## 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

## -Free-

MIT Open Courseware: Multivariable Caclulus (2010)
Video lectures for MIT's introductory multivariable calculus class.
Author: Denis Auroux