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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!
The beginning of a proof of Stokes' theorem for a special class of surfaces. Finding the curl of our vector field.
Figuring out a parameterization of our surface and representing dS
Writing our surface integral as a double integral over the domain of our parameters
Starting to work on the line integral about the surface
Working on the integrals...
More manipulating the integrals...
Using Green's Theorem to complete the proof