This video looks at Finding the curl of the vector field and then evaluating the double integral in the parameter domain.
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".
This video covers The beginning of a proof of stokes' theorem for a special class of surfaces. Finding the curl of our vector field.
This video looks at Figuring out a parameterization of our surface and representing ds.
This video looks at Writing our surface integral as a double integral over the domain of our parameters.
This video looks at starting to work on the line integral about the surface.
This video looks at Working on the integrals.
This video looks at the more manipulating the integrals.
This video looks at using Green's Theorem to complete the proof.
This video looks at Parametrizing a surface that can be explictly made a function of x and y.
This video looks at evaluating the surface integral.
This video looks at breaking apart a larger surface into its components.
This video looks at evaluating the surface integral over the outside of the chopped cylinder.
This video looks at Parametrizing the top surface.
This video looks at evaluating the third surface integral and coming to the final answer.
This video looks at Visualizing a suitable parameterization.
This video looks at Taking the cross product to calculate the surface differential in terms of the parameters.
This video looks at using a few trigonometric identities to finally calculate the value of the surface integral.
This series of videos focusing on calculus covers parameterizing a surface, surface integrals, stokes' theorem.
You can parameterize a line with a position vector valued function and understand what a differential means in that context already. This tutorial will take things further by parametrizing surfaces (2 parameters baby!) and have us thinking about partial differentials.