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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2008): Lecture 26 Greens Theorem in normal forms. Greens Theorem: why it works. Examples April 10 Reading Material: From Simmons : 21.3. Lecture Notes : V2, V3, V4. Last time: Greens theorem. Flux. Today: Greens Theorem in normal forms. Greens Theorem: why it works. Examples 2 Elements from Lecture 25 We start by recalling the Greens Theorem: Theorem 1. Assume that ~ F = M i + N j is a continuous differentiable vector field and C is a closed simple curve around a region R . Then Work of ~ F along C = I C F d~ r = I C Mdx + Ndy = Z Z R N x- M y dA. Notation: line integral for work: Z C ~ F d~ r line integral for flux: Z C ~ F b n ds = Z C ~ G d~ r, where ~ G is obtained by rotating ~ F by 90 x . So in particular if ~ F = M i + N j , then ~ G =- N i + M j and Z C ~ F b n ds = Z C ~ G d~ r = Z C- N dx + M dy. 3 Greens Theorem for flux also called Greens Theorem in normal form Since we defined the flux through the work we can now use the Greens Theorem to relate the flux to a double integral. We have 1 Theorem 2. With the same assumption of Greens Theorem we have...
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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