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15_4_24 - w equals 1-1 and 0 respectively for the curves in...

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24. If is a closed, piecewise smooth curve in 2 having equation r = r ( t ) , a t b , and if does not pass through the origin, then the polar angle function θ = θ ( x ( t ), y ( t ) ) = θ( t ) can be defined so as to vary continuously on . Therefore, θ( x , y ) t = b t = a = 2 π × w( ), where w( ) is the number of times winds around the origin in a counterclockwise direction. For example,
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Unformatted text preview: w( ) equals 1,-1 and 0 respectively, for the curves in parts (a), (b) and (c) of Exercise 22. Since θ = ∂θ ∂ x i + ∂θ ∂ y j =-y i + x j x 2 + y 2 , we have 1 2 π I x dy-y dx x 2 + y 2 = 1 2 π I θ • d r = 1 2 π θ( x , y ) ± ± ± ± t = b t = a = w( )....
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