1. Evaluate the line integral
    ó
    õ

    C 
    		 xdy-ydx
    where C is the curve r( t) = á2t,3t ñ , t in [ 0,1] .
  2. Test for exactness. If exact, find its potential: F(x,y) = á x2+y2,xy ñ
  3. Test for exactness. If exact, find its potential: F(x,y) = á sin( x+y) ,sin( x+y) ñ
  4. Test for exactness. If exact, find its potential: F(x,y,z) = á yex,ex+1,ez ñ
  5. Evaluate the integral below using the fundamental theorem for line integrals
    ó
    õ     		( 1,1,1)

    ( 0,0,0)  
    		 ( x+y+z) (dx+dy+dz)
  6. Explain why the integral ò( 0,0,0) (1,1,1) xdy+ydx+zdz is independent of path. Then calculate the integral along two different paths from ( 0,0,0) to (1,1,1) .
  7. Let R be the unit square. Use Green's theorem to evaluate the line integral


    R 
    		 y2dx+x2dy
  8. Let R denote the upper half of the unit disk. Evaluate using Green's theorem:



    R 
    		 ( xy) ( dx+dy)
  9. Evaluate by using Green's theorem to convert to a line integral over the boundary (D is the unit disk): 
    ó
    õ     		ó
    õ

    D 
    		  -x
    ( x2+y2+1) 3/2
    		    dA
  10. Find the area enclosed by the curve r( t) = á cos2( t) ,cos( t) sin(t) ñ , t in [ 0,p] , using Green's theorem.
  11. Calculate the surface area of the surface S parameterized by r( u,v) = á ucos( v) ,usin( v) ,u2 ñ for u in [ 0,1] and v in [ 0,2p] .
  12. Compute the flux of the vector field F( x,y,z) = á y,x,z ñ through the surface S parameterized by
    r( u,v) = á ucos( v) ,usin( v) ,u2 ñ ,        u  in  [ 0,1],    v  in  [ 0,2p]
  13. Show that if F( x,y,z) = áxy+2z,yz+2x,xz+2y ñ , then curl( F) = á 2-y,2-z,2-x ñ . Then evaluate
    ó
    õ     		ó
    õ
    S    		  curl( F) ·dS
    when S is the surface of the pyramid with vertices ( 2,0,0) , ( 2,2,0) , ( 0,2,0) , ( 0,0,0) , and ( 1,1,2) that is not contained in the xy-plane.
  14. Use Stoke's theorem for differential forms to calculate
    ó
    õ    		 ó
    õ
    S    		 xy  dy^dz-z2  dy^dx
    when S is the solid cube [ 0,1] ×[ 0,1]×[ 0,1] .
  15. Compute the flux of the vector field F( x,y,z) = á x,y,z ñ through the surface of a sphere S with radius R centered at the origin. Then show that the divergence theorem produces the same result.