lecture35-09 - 18.02 Multivariable Calculus (Spring 2009):...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

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.

Page1 / 7

lecture35-09 - 18.02 Multivariable Calculus (Spring 2009):...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online