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Unformatted text preview: MA1104 Multivariable Calculus Lecture 17 Dr KU Cheng Yeaw Monday March 23, 2009 Recap Last Lecture ... • we considered integral of a vector field. This type of line integrals appear frequently in physics. • It turns out that the computation of this line integral simplifies dramatically (in particular, it depends only on the initial and terminal points of a curve) provided the vector field is conservative. This is the Fundamental Theorem for Line Integrals. Overview In this lecture ... • we study Green’s Theorem and see how to apply it to compute line integrals. • Green’s Theorem can be regarded as the counterpart of the Fundamental Theorem of Calculus for double integral • at first glance, you might think Green’s Theorem is strange and abstract, one that only mathematician could invent to torture students of Calculus. Regardless of what you think, it is of fundamental importance in the analysis of fluid flows and in the theories of electricity and magnetism. Green’s Theorem Whenever we mention a line integral R C F · d r = R C P dx + Qdy along a smooth curve C , C must be given by a parametrization. In particular, C has an orientation. This line integral depends on the orientation of C . However, the line integral is the same for any two parametrization of C with the same orientation. If this parametrization is explicit, then we can evaluate the line integral easily. But if it is not given, then we have to write down one (which is not easy at times) We use C to denote the curve consisting the same point of C but with opposite orientation, that is, from the terminal point of C to the initial point of C . In the case when F is (continuous) conservative, that is O f = F for some scalar field F , the Fundamental Theorem for Line Integral simplifies the evaluation of a line integral: Z C F · d r = f ( r ( b )) f ( r ( a )) where C is given by r ( t ) , a ≤ t ≤ b . That is really amazing because it says that for conservative vector field, the line integral depends just on the initial and terminal points! Moreover, if F is continuous conservative with domain D , then Z C F · d r = 0 for every closed path C in D . Now, let us restrict our discussion to oriented, piecewisesmooth, simple closed curve on R 2 and twodimensional vector fields. The question remains: Given F = P i + Q j , how can we evaluate R C F · d r along an oriented, piecewisesmooth, simple closed curve C ? Green’s Theorem helps us to achieve this. To state Green’s Theorem, we require to use the convention that the positive orientation of a simple closed curve C refers to a single counterclockwise traversal of C . We can think of this being given by the righthandrule. Thus, if the positively oriented C is given by the vector function r ( t ) , a ≤ t ≤ b , then the region D enclosed by C is always on the left as the point r ( t ) traverses C . Green’s Theorem Let C be a positively oriented, piecewisesmooth, simple closed curve in the plane and let...
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 Spring '08
 Kuchengyeaw
 Integrals, Multivariable Calculus

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