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ch14_04 - P1 OSO/OVY GTBL001-14-nal P2 OSO/OVY QC OSO/OVY...

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1156 CHAPTER 14 . . Vector Calculus 14-42 14.4 GREEN’S THEOREM HISTORICAL NOTES George Green (1793–1841) English mathematician who discovered Green’s Theorem. Green was self-taught, receiving only two years of schooling before going to work in his father’s bakery at age 9. He continued to work in and eventually took over the family mill while teaching himself mathematics. In 1828, he published an essay in which he gave potential functions their name and applied them to the study of electricity and magnetism. This little-read essay introduced Green’s Theorem and the so-called Green’s functions used in the study of partial differential equations. Green was admitted to Cambridge University at age 40 and published several papers before his early death from illness. The significance of his original essay remained unknown until shortly after his death. In this section, we develop a connection between certain line integrals around a closed curve in the plane and double integrals over the region enclosed by the curve. At first glance, you might think this a strange and abstract connection, one that only a mathematician could care about. Actually, the reverse is true; Green’s Theorem is a significant result with far-reaching implications. It is of fundamental importance in the analysis of fluid flows and in the theories of electricity and magnetism. Before stating the main result, we briefly define some terminology. Recall that for a plane curve C defined parametrically by C = { ( x , y ) | x = g ( t ) , y = h ( t ) , a t b } , C is closed if its two endpoints are the same, i.e., ( g ( a ) , h ( a )) = ( g ( b ) , h ( b )). A curve C is simple if it does not intersect itself, except at the endpoints. We illustrate a simple closed curve in Figure 14.28a and a closed curve that is not simple in Figure 14.28b. y x C y x C FIGURE 14.28a FIGURE 14.28b Simple closed curve Closed, but not simple curve We say that a simple closed curve C has positive orientation if the region R enclosed by C stays to the left of C , as the curve is traversed; a curve has negative orientation if the region R stays to the right of C . In Figures 14.29a and 14.29b, we illustrate a simple closed curve with positive orientation and one with negative orientation, respectively. y x C R y x C R FIGURE 14.29a FIGURE 14.29b Positive orientation Negative orientation
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14-43 SECTION 14.4 . . Green’s Theorem 1157 We use the notation C F ( x , y ) · d r to denote a line integral along a simple closed curve C oriented in the positive direction. y a b x A R B C 2 : y g 2 ( x ) C 1 : y g 1 ( x ) FIGURE 14.30a The region R We can now state the main result of the section. THEOREM 4.1 (Green’s Theorem) Let C be a piecewise-smooth, simple closed curve in the plane with positive orientation and let R be the region enclosed by C , together with C . Suppose that M ( x , y ) and N ( x , y ) are continuous and have continuous first partial derivatives in some open region D , with R D . Then, C M ( x , y ) dx + N ( x , y ) dy = R N x M y dA .
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