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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 32 Divergence Theorem: Applications April 28 Reading Material: From Simmons : 21.4. From Lecture Notes : V8, V9. Last time: Surface Area and flux across cylinders and spheres. Today: Divergence Theorem (or Gauss Theorem). 2 Divergence Theorem The Divergence Theorem (or Gauss Theorem) is the 3D analogue of the of the Greens Theorem in normal form that we studied in 2D. Recall in fact Theorem 1 (Greens Theorem in Normal Form) . Assume that C is a simple, closed curve oriented counterclockwise on the plane. Assume that R is the region on the plane enclosed by C . Assume that ~ F = M i + N j is a continuously differentiable vector field on the region R : Then Flux of ~ F Across C = ZZ R ( M x + N y ) dA. We recall here that we defined the divergence of ~ F at ( x,y ) to be div ~ F ( x,y ) = M x ( x,y ) + N y ( x,y ) and it represents the net amount of the outgoing fluid at ( x,y ) (source) if it is positive, or the net amount of the incoming fluid at ( x,y ) (sink) if it is negative. Notation : If we think of ~ = x i + y j , then it is easy to justify the notation div ~ F ( x,y ) = M x ( x,y ) + N y ( x,y = ~ ~ F....
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- Spring '06
- Multivariable Calculus