ps3.pdf - MATH 317 PROBLEM SET II SOLUTIONS R 1 Evaluate...

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MATH 317 PROBLEM SET II SOLUTIONS 1) Evaluate the path integral R C f ( x, y, z ) ds for a) f ( x, y, z ) = x cos z , C : t 7! t ˆ ı + t 2 ˆ | , t 2 [0 , 1]. b) f ( x, y, z ) = x + y y + z , C : t 7! ( t, 2 3 t 3 / 2 , t ) , t 2 [1 , 2]. Solution. a) In this case ~ r ( t ) = t ˆ ı + t 2 ˆ | , so that ~ v ( t ) = ˆ ı + 2 t ˆ | and ds dt = p 1 + 4 t 2 . Hence Z C f ( x, y, z ) ds = Z 1 0 x ( t ) cos z ( t ) ds dt dt = Z 1 0 t (cos 0) p 1 + 4 t 2 dt = 1 8 (1+4 t 2 ) 3 / 2 3 / 2 1 0 = 5 3 / 2 - 1 12 b) In this case ~ r ( t ) = ( t, 2 3 t 3 / 2 , t ) , so that ~ v ( t ) = ( 1 , t 1 / 2 , 1 ) and ds dt = p 2 + t . Hence Z C f ( x, y, z ) ds = Z 2 1 x ( t )+ y ( t ) y ( t )+ z ( t ) ds dt dt = Z 2 1 t + 2 3 t 3 / 2 2 3 t 3 / 2 + t p 2 + t dt = (2+ t ) 3 / 2 3 / 2 2 1 = 8 - 3 3 / 2 3 / 2 2) Show that the path integral of f ( x, y ) along a path given in polar coordinates by r = r ( ), 1 2 , is Z 2 1 f ( r ( ) cos , r ( ) sin ) q r ( ) 2 + ( dr d ( ) ) 2 d Compute the arc length of r = 1 + cos , 0 2 . Solution. The path is ~ r ( ) = x ( ı + y ( | with x ( ) = r ( ) cos , y ( ) = r ( ) sin and 0 2 . On this path ~ v ( ) = x 0 ( ı + y 0 ( | = r 0 ( ) cos - r ( ) sin ˆ ı + r 0 ( ) sin + r ( ) cos ˆ | = ) ds d = q r 0 ( ) cos - r ( ) sin 2 + r 0 ( ) sin + r ( ) cos 2 = p r 0 ( ) 2 + r ( ) 2 Hence Z C f ( x, y ) ds =
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