# Green's Theorem

(1.7 hours to learn)

## Summary

Green's Theorem is a theorem relating the integrals of the curl and the divergence of a vector field over a closed region to a line integral along its boundary.

## Context

This concept has the prerequisites:

- vector fields (Green's Theorem is a theorem about vector fields.)
- line integrals (The statement of Green's Theorem involves a line integral.)
- multiple integrals (The statement of Green's Theorem requires a multiple integral.)
- conservative vector fields (Green's Theorem shows that conservative vector fields have curl zero.)

## Goals

- Know the definitions of curl and divergence

- Prove two versions of Green's Theorem:
- relating line integrals to curl
- relating flux to divergence

- Be able to apply Green's Theorem to compute:
- the area of a closed region
- line integrals
- flux across a curve

- Use Green's Theorem to show that conservative vector fields have zero curl

## 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.

Location:
Section 8.3.3, "Green's Theorem," pages 358-362

Additional dependencies:

- differential forms
- exterior derivative

→ Multivariable Calculus

An introductory multivariable calculus textbook.

- Section 15.1, "Vector fields," subsections "The divergence of a vector field" and "The curl of a vector field," pages 1016-1018
- Section 15.4, "Green's Theorem," pages 1037-1045

## See also

-No Additional Notes-