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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 35 Stokes Theorem continued: applications May 5 Reading Material: From Simmons : 21.5. From Lecture Notes : V13 and V14. Last time: Stokes Theorem Today: Stokes Theorem continued: applications. 2 A consequence of Stokes Theorem We recall the Stokes Theorem: Theorem 1 ( Stokes Theorem). Let S be a bounded piecewise smooth surface, non self intersect- ing, with boundary a closed and simple curve C . Assume S is oriented and use the Right Hand Rule to then orient C . Assume ~ F = M i + N j + P k is a continuously differentiable vector field in a region containing S . Then Z C ~ F d~ r = ZZ S ~ ~ F n dS. Exercise 1. Use Stokes Theorem to evaluate the line integral Z C ( x 2 + z 2 ) dx + y dy + z dz where C is the curve parametrized by ~ r ( t ) = (cos t, sin t, cos 2 t- sin 2 t ) , t 2 . 1 Solution: 3 Idea of the proof of Stokes Theorem Take the surface S with boundary C in the statement of Stokes Theorem. Assume an orientation given by n is set on S and a compatible (by Right Hand Rule) orientation on C is fixed. We can tile S with infinitesimal tiles S , of boundaries C , such that on each of them one can think that n is constant....
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This note was uploaded on 03/08/2010 for the course MATH 18.022 taught by Professor Hartleyrogers during the Spring '06 term at MIT.
- Spring '06
- Multivariable Calculus