Green's, Stokes', and the divergence theorems

This course is a part of Multivariable calculus, a 5-course Topic series from Khan Academy.

Here we cover four different ways to extend the fundamental theorem of calculus to multiple dimensions. Green's theorem and the 2D divergence theorem do this for two dimensions, then we crank it up to three dimensions with Stokes' theorem and the (3D) divergence theorem.

This course contains 11 segments:

Formal definitions of div and curl (optional reading)

Here you will learn how line integrals and surface integrals are used to give the definitions of divergence and curl. Aside from getting more practice with line integrals, surface integrals, divergence and curl, these definitions help establish the intuition for Green's theorem and Stokes' theorem.

Green's theorem (videos)

It is sometimes easier to take a double integral (a particular double integral as we'll see) over a region and sometimes easier to take a line integral around the boundary. Green's theorem draws the connection between the two so we can go back and forth. This tutorial proves Green's theorem and then gives a few examples of using it. If you can take line integrals through vector fields, you're ready for Mr. Green.

Green's theorem (articles)

Green's theorem ties together double integrals, curl, and line integrals into one very pretty (and useful!) little statement.

2D divergence theorem

Using Green's theorem (which you should already be familiar with) to establish that the "flux" through the boundary of a region is equal to the double integral of the divergence over the region. We'll also talk about why this makes conceptual sense.

Stokes' theorem (videos)

Stokes' theorem relates the line integral around a surface to the curl on the surface. This tutorial explores the intuition behind Stokes' theorem, how it is an extension of Green's theorem to surfaces (as opposed to just regions) and gives some examples using it. We prove Stokes' theorem in another tutorial. Good to come to this tutorial having experienced the tutorial on "flux in 3D".

Stokes' theorem (articles)

You've worked so hard to get to this point, and finally, you're here. Stokes' theorem. The granddaddy of multivariable calculus. Curl, surface integrals and line integrals, all beautifully smooshed into one theorem.

3D divergence theorem (videos)

An earlier tutorial used Green's theorem to prove the divergence theorem in 2-D, this tutorial gives us the 3-D version (what most people are talking about when they refer to the "divergence theorem"). We will get an intuition for it (that the flux through a close surface--like a balloon--should be equal to the divergence across it's volume). We will use it in examples. We will prove it in another tutorial.

Divergence theorem (articles)

Both the 2D and 3D divergence theorems connect two separate measures of fluid flow: Divergence and flux. This can be very handy for solving higher-dimensional integration problems.

Proof of Stokes' theorem

You know what Stokes' theorem is and how to apply it, but are craving for some real proof that it is true. Well, you've found the right tutorial!

Types of regions in three dimensions

This tutorial classifies regions in three dimensions. Comes in useful for some types of double integrals and we use these ideas to prove the divergence theorem.

Divergence theorem proof

You know what the divergence theorem is, you can apply it and you conceptually understand it. This tutorial will actually prove it to you (references types of regions which are covered in the "types of regions in 3d" tutorial).