lecture25-09

lecture25-09 - 18.02 Multivariable Calculus(Spring 2009...

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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 25 Fundamental Theorem of Line Integrals. Grad Test. Potential Functions April 9 Reading Material: From Simmons : 21.3. Lecture Notes : V2, V3 and V4. Last time: Fundamental Theorem of Line Integrals. Grad Test. Potential Functions Today: Green’s theorem. Flux. 2 Green’s Theorem Recall: If the FTLI says that if ~ F = ~ ∇ f (gradient vector field) then Z ~ F · d~ r = Z Mdx + Ndy = f (end)- f (start) . If C is a loop (closed curve) then Z C ~ F · d~ r = 0 [start = end] . Question: What can we say if the vector field is not a gradient? The answer is the the Green’s Theorem! We start with the set up for the theorem. This is what we need: • Simple Closed Curves: In this case simple means that the curve is not self intersecting • It has an interior part that we call R • By convention the curve is oriented counterclockwise 1 Notation I C for line integral if the path C is simple and closed. Theorem 1 ( Green’s Theorem). If C is a simple closed curve (oriented ) that encloses a bounded region R and if ~ F is everywhere continuously differentiable, then...
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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.

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lecture25-09 - 18.02 Multivariable Calculus(Spring 2009...

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