As you recall (lecture 30), we justified Stokes' Theorem in terms of intuitive arguments based on our original definition of curl (see lecture 14) as the swirl per unit area. Using Green's Theorem, we now prove Stokes' Theorem more rigorously [Transparancies 2, 3, and 4]. The proof hinges on paramaterizing the surface by u and v, and then using Green's theorem in the uv plane.
Now that we've proved Stoke's theorem, we may use it to justify our original definition of curl in terms of swirl per unit area. (See page 299 of the textbook.)
We finished with a short discussion of the final exam [Transparancy 5]. Check these web pages on the long weekend for further updates!
(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 70 and 95 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 240 K, which takes a bit longer to download.)
