# Stokes' Theorem (three dimensions)

(1.3 hours to learn)

## Summary

Stokes' Theorem is a theorem relating a line integral along the boundary of a surface to the integral of curl over the surface. It can be seen as a three-dimensional generalization of Green's Theorem.

## Context

This concept has the prerequisites:

- surface integrals (Stokes' Theorem is a theorem about surface integrals.)
- line integrals (Stokes' Theorem is a theorem about line integrals.)
- Green's Theorem (Stokes' Theorem is a generalization of Green's Theorem.)
- determinant (The determinant is used as a mnemonic for the definition of curl.)

## Goals

- Know the definition of curl in three dimensions

- Know the statement of Stokes' Theorem (in three dimensions)

- Prove Stokes' Theorem (in three dimensions)

- Be able to use Stokes' Theorem to compute line integrals and surface integrals

## 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 Calculus

An introductory multivariable calculus textbook.

Location:
Section 15.7, "Stokes' Theorem," pages 1065-1070

## See also

-No Additional Notes-