# Stokes' theorem

11 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 intuition

VIDEO
12:12 minutes

Conceptual understanding of why the curl of a vector field along a surface would relate to the line integral around the surface's boundary

### Green's and Stokes' theorem relationship

VIDEO
6:55 minutes

Seeing that Green's Theorem is just a special case of Stokes' Theorem

### Orienting boundary with surface

VIDEO
3:29 minutes

Determining the proper orientation of the boundary given the orientation of the surface

### Orientation and stokes

VIDEO
4:26 minutes

Determining the proper orientation of a boundary given the orientation of the normal vector

### Conditions for stokes theorem

VIDEO
4:44 minutes

Understanding when you can use Stokes. Piecewise-smooth lines and surfaces

### Stokes example part 2: Parameterizing the surface

VIDEO
4:02 minutes

### Stokes example part 3: Surface to double integral

VIDEO
8:05 minutes

Converting the surface integral to a double integral

### Stokes example part 4: Curl and final answer

VIDEO
6:54 minutes

Finding the curl of the vector field and then evaluating the double integral in the parameter domain

### Evaluating line integral directly - part 1

VIDEO
7:44 minutes

Showing that we didn't need to use Stokes' Theorem to evaluate this line integral

### Evaluating line integral directly - part 2

VIDEO
6:23 minutes

Finishing up the line integral with a little trigonometric integration