green_theorem

green_theorem - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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V4. Green's Theorem in Normal Form 1. Green's theorem for flux. Let F = M i + N j represent a two-dimensional flow field, and C a simple closed curve, positively oriented, with interior R. According to the previous section, (1) flux of F across C = Notice that since the normal vector points outwards, away from R, the flux is positive where the flow is out of R; flow into R counts as negative flux. We now apply Green's theorem to the line integral in (1); first we write the integral in standard form (dx first, then dy): This gives us Green's theorem in the normal form Mathematically this is the same theorem as the tangential form of Green's theorem - all we have done is to juggle the symbols M and N around, changing the sign of one of them. What is different is the physical interpretation. The left side represents the flux of F across the closed curve C. What does the right side represent? 2. The two-dimensional divergence. Once again, let F = Mi + N j . We give a name to and a notation for the integrand of the double integral on the right of (2): dM div F = - + -, dN the divergence of F . dx dy Evidently div F is a scalar function of two variables. To get at its physical meaning, look at the small rectangle pictured. If F is continuously differentiable, then div F is a continuous function, which is therefore approximately constant if the rectangle is small enough. We apply (2) to the rectangle; the double integral is approximated by a product, since the integrand is approximately constant: (4) flux across sides of rectangle z (E - + --- Y) AA , AA = area of rectangle. Because of the importance of this approximate relation, we give a more direct derivation
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green_theorem - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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