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 1

VIDEO 3:10 minutes
Starting to apply Stokes theorem to solve a line integral

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