Exercises

Test each vector field to determine if it is conservative. If it is, find its potential.

1. F( x,y) = á y,x ñ 2. F( x,y) = á y,-x ñ 3. F( x,y) = á ex+xex, ey+yey ñ 4. F(x,y) = á excos( y) ,-e-xsin(y) ñ 5. F( x,y) = á cosxcosy, -sinxsiny+1 ñ 6. F( x,y) = á cosxcosy,sinxsiny ñ 7. F( x,y) =  1 2 ln(x2+y2),  tan-1 æ è  y x ö ø 8. F( x,y) = á xsec2( xy),ysec2( xy) +1 ñ 9. F( x,y) = á exsin( y), excos( y) ñ 10. F(x,y) = á cos( x) cosh( y) ,sin( x) sinh( y) ñ 11. F( x,y,z) = á 0,0,-32 ñ 12. F( x,y,z) = á 0,0,-9.8 ñ 13. F( x,y,z) = áx+z,y2,x+z3 ñ 14. F( x,y,z) = á z-y,x-z,y-x ñ 15. F( x,y,z) = ázex, yey, xez ñ 16. F(x,y,z) = á eyz,xzeyz,xyeyz+1 ñ

Identify the vector field, show it is conservative, and find its potential. Then evaluate the integral using theorem 2.

17. ó õ ( 1,1) ( 0,0)   xdx+ydy 18. ó õ ( 1,1) ( 0,0)   x2dx-ydy 19. ó õ ( 1,1) ( 0,0)   2xydx+x2dy 20. ó õ ( 1,1) ( 0,0)   2xy3dx+3x2y2dy 21. ó õ ( 1,1) ( 0,0)   á2xy,x2+2y ñ · á dx,dy ñ 22. ó õ ( 1,p/4) ( 0,0)   ysec2(xy) dx+xsec2( xy) dy 23. ó õ ( 1,2,1) ( 0,0,0)   xdx+y2dy+z3dz 24. ó õ ( 0,p,1) ( 0,0,0)   sin(x) dx+cos( y) dy+dz 25. ó õ ( 1,2,1) ( 0,0,0)   yzdx+xzdy+xydz 26. ó õ ( 1,2,1) ( 0,0,0)   yzdx+xzdy+xydz

Show that the following vector fields are conservative. Then find their potentials and use them to determine the amount of work done in moving an object from point A to point B through the given vector field.

27. F( x,y,z) = áx2,y2,z2 ñ 28. F( x,y,z) = á x2,y2,z2 ñ from A( 0,0,0) to B( 1,2,3) from A( 0,0,0) to B( 1,1,1) 29. F( x,y,z) = á yzsin( xy),xzsin( xy) ,-cos( xy) ñ 30. F( x,y,z) = áyze-xyz,xze-xyz,xye-xyz ñ from A( 0,0,2) to B( 1,p,1) from A( 0,0,0) to B( ¥,¥,¥)

31. In the following, F( x,y) = áe^y+ye^x, xe^y+e^x ñ .

1. Evaluate ò_CF·dr along the curve C parametrized by r( t) = á1-t,t² ñ , t in [ 0,1]
2. Evaluate ò_CF·dr along the curve C parametrized by

   r( t) = á 1-t2 , t ñ,  t in [0,1]

     Ö{}

3. Show that F( x,y) is conservative and find its potential.
4. Use theorem 2 and the potential in step c to evaluate

   ó õ ( 0,1) ( 1,0)   ( ey+yex)dx+( xey+ex) dy

   Is the result the same as the result in (a) and (b)? Explain.

32. For q a constant, the following is the electric field in the xy-plane of a charge q placed at the origin:

E( x,y) =  qx (x2+y2) 3/2 ,  qy ( x2+y2) 3/2

1. Evaluate ò_CE·dr along the curve C parametrized by

   r( t) = á cos( t) ,sin(t) ñ ,t  in   é ë 0,  p 2 ù û

2. Evaluate ò_CE·dr along the curve C parametrized by r( t) = á 1-t,t ñ , t in [ 0,1]
3. Show that E( x,y) is conservative in the upper half plane and find its potential.
4. Use theorem 2 and the potential in step c to evaluate

   ó õ ( 0,1) ( 1,0)   E·dr

   Is the result the same as the result in (a) and (b)? Explain.

33. The electric field of a single charge q placed at the origin is of the form

E( x,y,z) =  qx (x2+y2+z2) 3/2 ,  qy ( x2+y2+z2)3/2 ,  qz ( x2+y2+z2) 3/2

Compute the work done in moving a test charge from the point A(1,0,0) to the point B( 1,0,2p) along two different routes.

1. Along the helix r( t) = á cos(t) ,sin( t) ,t ñ ,    t  in  [ 0,2p] .
2. Along the line r( t) = á1,0,t ñ , t in [ 0,2p] .

Then show that E( x,y,z) is conservative and explain why we would expect the same value in both (a) and (b).

34. A 1 kg object is propelled from the earth's surface from the origin ( 0,0,0) to a point ( 0,490,0) . How much work is done by the object if the gravitational force field is F( x,y,z) = á 0,0,-9.8 ñ ?

35. Show that the vector field

F( x,y) =  x+1 ( x+1)2+y2 ,  y ( x+1) 2+y2

is conservative in the right half-plane. Then write its potential as an integral using the fact that

U( x,y) = ó õ ( x,y) ( 0,0)   F·dr

along the curve r( t) = á xt,yt ñ for t in [ 0,1]

36. Show that the vector field

F( x,y) =  1 2 ln(x2+y2) ,tan-1 æ è  x y ö ø

is conservative in the right half-plane. Then write its potential as an integral using the fact that

U( x,y) = ó õ ( x,y) ( 1,0)   F·dr

along the curve r( t) = á sin(t) x+cos( t) ,sin( t) y ñ for t in [ 0,p/2] .

37. Evaluating Definite Integrals: In this exercise, we use path independence to evaluate the improper integral

ó õ 1 -1  sin-1( x) 1-x2   dx

1. Show that F( x,y) = á sin^-1(x) ( 1-x²) ^-1/2,0 ñ is conservative
2. Let C be the upper half of the unit circle with parametrization r( t) = á sin( t) ,cos(t) ñ , t in [ -p/2,p/2] . Evaluate

   ó õ C  F·dr

3. Use path independence to explain why

   ó õ 1 -1  sin-1( x) 1-x2     dx =  ó õ   C  F·dr

38. Evaluating Definite Integrals: In this exercise, we use path independence to evaluate the improper integral

ó õ 1 -1   1 ( x2+1) 3/2   dx

1. Show that F( x,y) = á (x²+1) ^-3/2,0 ñ is conservative
2. Let C be the curve parametrized by r( t) = á tan( t) ,1-2sec²( t) ñ , t in [ -p/4,p/4]

   ó õ C  F·dr

3. Use path independence to explain why

   ó õ 1 -1   1 ( x2+1) 3/2   dx = ó õ C  F·dr

39. Show that if F( x,y,z) = áM( x) ,N( y) ,P( z) ñ , then

U( x,y,z) = ó õ M( x) dx+ ó õ N( y) dy+ ó õ P( z) dz

40. Show that if F( x) and G( y) are antiderivatives of f( x) and g( y) , respectively, then

ó õ ( x2,y2) ( x1,y1)   f(x) dx+g( y) dy = F( x2) -F( x1)+G( y2) -G( y1)

41. Write to Learn: Suppose that A and B are two points on the surface of a spherical planet whose force of gravity is of the form

F( x,y,z) =  -kx (x2+y2+z2) 3/2 ,  -ky ( x2+y2+z2)3/2 ,  -kz ( x2+y2+z2) 3/2

where k is a constant. Explain why F·dr = 0 on any curve from A to B on the surface itself, and then use this to explain why there is no work done in moving from A to B along any curve in R³ that remains outside of the planet.

42. Write to Learn: In a short essay, explain why the definite integral introduced in single variable calculus

ó õ b a  g( x) dx

is actually a line integral along the curve r( t) = á t,0 ñ , t in [ a,b] , through a vector field F( x,y) = á g( x),0 ñ . In addition, explain why F( x,y) is conservative and what the potential for F( x,y) is given that G( x) is an antiderivative of g( x) (that is, G^¢( x) = g( x) ).