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chap12 - greens theorem

chap12 - greens theorem - Greens theorem 1 Chapter 12...

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Green’s theorem 1 Chapter 12 Green’s theorem We are now going to begin at last to connect differentiation and integration in multivariable calculus. In addition to all our standard integration techniques, such as Fubini’s theorem and the Jacobian formula for changing variables, we now add the fundamental theorem of calculus to the scene. In fact, Green’s theorem may very well be regarded as a direct application of this fundamental theorem. A. The basic theorem of Green Consider the following type of region R contained in R 2 , which we regard as the x - y plane. We use the standard orientation, so that a 90 counterclockwise rotation moves the positive x -axis to the positive y -axis. Then we assume the existence of two continuous functions a ( y ) and b ( y ), defined for c y d , where a ( y ) < b ( y ) for c < y < d and R = { ( x, y ) | a ( y ) x b ( y ) } . x y d c Given a function R F -→ R of class C 1 , we then compute by means of Fubini’s theorem ZZ R ∂F ∂x dxdy = Z d c Z b ( y ) a ( y ) ∂F ∂x dxdy FTC = Z d c F ( x, y ) fl fl fl fl x = b ( y ) x = a ( y ) dy = Z d c F ( b ( y ) , y ) dy - Z d c F ( a ( y ) , y ) dy. We now write the right side of this equation as Z bd R Fdy,

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2 Chapter 12 where we mean by this notation the counterclockwise integration of F , restricted to bd R , with respect to y . (The actual definition is given in the formula.) x y Notice that on a horizontal portion of bd R , y is constant and we thus interpret dy = 0 there. B. Line integrals We have now met an entirely new kind of integral, the integral along the counterclockwise bd R seen above. Before proceeding further, we need to discuss this sort of oriented integral. It will prove useful to do this in more generality, so we consider a curve γ in R n which is of class C 1 . Thus, [ a, b ] γ -→ R n is of class C 1 . We typically denote the independent variable (the “parameter”) as t . If f is a continuous real-valued-function defined on R n , we then define the line integral Z γ fdx i = Z b a f ( γ ( t )) γ 0 i ( t ) dt. Here of course 1 i n . The notation is intended to be very suggestive and to lead us to the appropriate substitutions x = γ ( t ), x i = γ i ( t ), and dx i = γ 0 i ( t ) dt . It is an easy matter to imagine some useful properties of this sort of integral, and even easier to prove them. INDEPENDENCE OF PARAMETER CHANGE Suppose first that γ ( t ), a t b , is represented instead as γ ( h ( s )), c s d , where h is a C 1 function such that h ( c ) = a and h ( d ) = b . Then using the parametrization γ ( h ( s )) leads to the line integral Z d c f ( γ ( h ( s ))) d ds γ i ( h ( s )) ds = Z d c f ( γ ( h ( s ))) γ 0 i ( h ( s )) h 0 ( s ) ds t = h ( s ) = Z b a f ( γ ( t )) γ 0 i ( t ) dt, which is the original line integral.
Green’s theorem 3 s t b a c d This proves the desired independence. On the other hand, if instead h ( c ) = b and h ( d ) = a , then we obtain Z d c f ( γ ( h ( s ))) d ds γ i ( h ( s )) ds = - Z b a f ( γ ( t )) γ 0 i ( t ) dt, so we get the anticipated change of sign.

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