Multi-Variable Calculus Assignment

due Tuesday, April 24, 2001

Exercises

1. Let F be the vector field in the plane that is defined away from the origin by

   F(x, y)   =

   ( {-y}/{SQRT{x^{2} + y^{2}}},     {x}/{SQRT{x^{2} + y^{2}}} )

   .

   Find the path integral of this vector field over the path C that is given parametrically by

   R(t)  =  (r cos t, r sin t)     for    alpha <= t <= beta

   when r > 0 is a given constant.

2. Let C be the path that is given parametrically by

   R(t)  =

   ( (cos(theta))t ,  (sin(theta))t )

   for     0 <= t <= r ,

   where r and theta are constants. Evaluate the path integral

   INT[_{C} F]

   when F is the vector field that is defined by

   1. F(x, y)  =

      ( y e^{x y},   x e^{x y} )

      .

   2. F(x, y)  =

      ( x - y,   x + y )

      .

   3. F(x, y)  =

      ( {x}/{SQRT{x^{2} + y^{2}}},   {y}/{SQRT{x^{2} + y^{2}}} )

      .

   4. F(x, y)  =

      ( {-y}/{x^{2} + y^{2}},  {x}/{x^{2} + y^{2}} )

      .

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