sec44 - Several Variables The Calculus of Functions of...

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Unformatted text preview: Several Variables The Calculus of Functions of Section 4.4 Green’s Theorem Green’s theorem is an example from a family of theorems which connect line integrals (and their higher-dimensional analogues) with the definite integrals we studied in Section 3.6. We will first look at Green’s theorem for rectangles, and then generalize to more complex curves and regions in R 2 . Green’s theorem for rectangles Suppose F : R 2 → R 2 is C 1 on an open set containing the closed rectangle D = [ a,b ] × [ c,d ] , and let F 1 and F 2 be the coordinate functions of F . If C denotes the boundary of D , oriented in the clockwise direction, then we may decompose C into the four curves C 1 , C 2 , C 3 , and C 4 shown in Figure 4.4.1. Then C 1 C 2 C 3 C 4 c d b a Figure 4.4.1 The boundary of a rectangle decomposed into four smooth curves α ( t ) = ( t,c ) , a ≤ t ≤ b , is a smooth parametrization of C 1 , β ( t ) = ( b,t ) , 1 Copyright c by Dan Sloughter 2001 2 Green’s Theorem Section 4.4 c ≤ t ≤ d , is a smooth parametrization of C 2 , γ ( t ) = ( t,d ) , a ≤ t ≤ b , is a smooth parametrization of- C 3 , and δ ( t ) = ( a,t ) , c ≤ t ≤ d , is a smooth parametrization of- C 4 . Now Z C F · ds = Z C 1 F · ds + Z C 2 F · ds + Z C 3 F · ds + Z C 4 F · ds = Z C 1 F · ds + Z C 2 F · ds- Z- C 3 F · ds- Z- C 4 F · ds, (4.4.1) and Z C 1 F · ds = Z b a (( F 1 ( t,c ) ,F 2 ( t,c )) · (1 , 0) dt = Z b a F 1 ( t,c ) dt, (4.4.2) Z C 2 F · ds = Z d c (( F 1 ( b,t ) ,F 2 ( b,t )) · (0 , 1) dt = Z c c F 2 ( b,t ) dt, (4.4.3) Z- C 3 F · ds = Z b a (( F 1 ( t,d ) ,F 2 ( t,d )) · (1 , 0) dt = Z b a F 1 ( t,d ) dt, (4.4.4) and Z- C 4 F · ds = Z d c (( F 1 ( a,t ) ,F 2 ( a,t )) · (0 , 1) dt = Z c c F 2 ( a,t ) dt, (4.4.5) Hence, inserting (4.4.2) through (4.4.5) into (4.4.1), Z C F · ds = Z b a F 1 ( t,c ) dt + Z d c F 2 ( b,t ) dt- Z b a F 1 ( t,d ) dt- Z d c F 2 ( a,t ) dt = Z d c ( F 2 ( b,t )- F 2 ( a,t )) dt- Z b a ( F 1 ( t,d )- F 1 ( t,c )) dt. (4.4.6) Now, by the Fundamental Theorem of Calculus, for a fixed value of t , Z b a ∂ ∂x F 2 ( x,t ) dx = F 2 ( b,t )- F 2 ( a,t ) (4.4.7) and Z d c ∂ ∂y F 1 ( t,y ) dy = F 1 ( t,d )- F 1 ( t,c ) . (4.4.8) Section 4.4 Green’s Theorem 3 Thus, combining (4.4.7) and (4.4.8) with (4.4.6), we have Z C F · ds = Z d c Z b a ∂ ∂x F 2 ( x,t ) dxdt- Z b a Z d c ∂ ∂y F 1 ( t,y ) dydt = Z d c Z b a ∂ ∂x F 2 ( x,y ) dxdy- Z b a Z d c ∂ ∂y F 1 ( x,y ) dydx = Z d c Z b a ∂ ∂x F 2 ( x,y )- ∂ ∂y F 1 ( x,y ) dxdy. (4.4.9) If we let p = F 1 ( x,y ), q = F 2 ( x,y ), and ∂D = C (a common notation for the boundary of D ), then we may rewrite (4.4.9) as Z...
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This note was uploaded on 03/13/2010 for the course MATH 0314 taught by Professor Ivoklemes during the Spring '10 term at McGill.

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sec44 - Several Variables The Calculus of Functions of...

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