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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: Greens theorem. Flux. 2 Greens 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 Greens 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 ( Greens 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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