100hw7soln

100hw7soln - Math 100 Homework 7 fall 2010 1. (a) C can be...

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Unformatted text preview: Math 100 Homework 7 fall 2010 1. (a) C can be parametrized by r ( t ) = ((1- t ) a + tc, (1- t ) b + td ) for ≤ t ≤ 1. Thus, R C- ydx + xdy = R 1 [[- (1- t ) b + td ]( c- a ) + [(1- t ) a + tc ]( d- b )] dt = ad- bc. (b) Let C 1 , C 2 , C 3 be the line segments joining ( x 1 , y 1 ) to ( x 2 , y 2 ), the line segments joining ( x 2 , y 2 ) to ( x 3 , y 3 ), and the line segments joining ( x 3 , y 3 ) to ( x 1 , y 1 ) repectively. C is the curve so that going along C is the same as going along C 1 , then along C 2 and then along C 3 . According to a corollary of Green’s Theorem, A = 1 2 R C- ydx + xdy = 1 2 ( R C 1 (- ydx + xdy ) + R C 2 (- ydx + xdy ) + R C 3 (- ydx + xdy )) = 1 2 [( x 1 y 2- x 2 y 1 ) + ( x 2 y 3- x 3 y 2 ) + ( x 3 y 1- x 1 y 3 )] by (a). (c) For each 1 ≤ i ≤ n- 1, let C i be the line segments joining ( x i , y i ) to ( x i +1 , y i +1 ). C n is the line segment joining ( x n , y n ) to ( x 1 , y 1 ) and finally C is the curve so that going along C is the same as going...
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100hw7soln - Math 100 Homework 7 fall 2010 1. (a) C can be...

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