Khan Academy on a Stick

Line integrals for scalar functions

With traditional integrals, our "path" was straight and linear (most of the time, we traversed the x-axis). Now we can explore taking integrals over any line or curve (called line integrals).

Position vector functions and derivatives

In this tutorial, we will explore position vector functions and think about what it means to take a derivative of one. Very valuable for thinking about what it means to take a line integral along a path in a vector field (next tutorial).

Line integrals in vector fields

You've done some work with line integral with scalar functions and you know something about parameterizing position-vector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.

Green's theorem

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.

2-D 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.