Exercises

Use a graphing calculator or computer algebra system to sketch each surface and then find the level surface representations of each of the following parametric equations. Also, calculate r_u and r_v and determine if the parameterization is orthogonal.

1. r = á vsin( u) ,vcos( u),v ñ 2. r = á vsin( u),v,vcos( u) ñ 3. r = á sin( u) cos( v),cos( u) ,sin( u) sin( v) ñ 4. r = á sin( v) sin( u) ,cos( v) sin( u) ,cos(u) ñ 5. r = á 2sin( u) cos( v),3cos( u) ,2sin( u) sin( v) ñ 6. r = á 2sin( u)cos( v) ,2cos( u) ,sin( u) sin( v) ñ 7. r = á sin( u) cosh( v),sinh( v) ,cos( u) cosh( v) ñ 8. r = á sin( u)cosh( v) ,sin( u) sinh( v) ,cos( u) ñ 9. r = á sec( u) sin( v),sec( u) cos( v) ,tan( u) ñ 10. r = á vsin( u),vcos( u) ,v2sin( 2u) ñ 11. r = á evsin( u) ,evcos(u) ,e-v ñ 12. r = á sin( v) cos( u) ,sin( v) sin(u) ,sin2( v) cos( 2u) ñ

Verify that each of the following parameterizes the unit sphere. Then calculate r_u and r_v. Is the parameterization orthogonal?

13. r = á cos( u) sin( v),cos( v) ,sin( u) sin( v) ñ 14. r = á sin( v)sin( u) ,cos( v) sin( u) ,cos(u) ñ 15. r = á sin( v) ,cos( u)cos( v) ,sin( u) cos( v) ñ 16. r = á cos( u)cos( v) ,sin( u) cos( v) ,sin(v) ñ 17. r =   u,  1-u2 sin( v) , 1-v2 cos( v) 18. r = 1-u2 ,usin( v) ,ucos( v) 19. r =    2u u2+v2+1 ,  2v u2+v2+1 ,  u2+v2-1 u2+v2+1     20. r =    2v u2+v2+1 ,  2u u2+v2+1 ,  u2+v2-1 u2+v2+1

Find the surface of revolution obtained by revolving the following surfaces about the x-axis. Then find r_u and r_v and determine if the parametrization is orthogonal.

21. y = x,  x in [ 0,1] 26. y = x+1,  x in [ 0,1] 23. y = x-x2, x in [ 0,1] 28. y = x-x3, xin [ 0,1] 25. y = cosh( x) , x in [ -1,1] 30. y = sin( x) , x in [ 0,p] 27. r( u,0) = á u2,0,u ñ , uin [ 0,1] 28. r( u,0) = á e-u ,0,u ñ , u in [ 0,1] 29. r( u,0) = á 2sin( u),0,3+2cos( u) ñ 30. r(u,0) = á sin( u) ,0,5+cos( u) ñ        u in [ 0,2p]        u in [0,2p]

31. Find another parameterization of the sphere of radius R centered at the origin by revolving the upper half circle

y = R2-x2 ,    x  in  [ -R,R]

about the x-axis. Is the parameterization orthogonal?

32. Show that every parametric equation of the form

r( u,v) = á f( v) cos(u) ,f( v) sin( u) ,f( v) ñ

is a parameterization of a section the cone x²+y² = z². Is the parameterization orthogonal?

33. A Mobius strip is a surface parameterized by

r( u,v) = cos( u) +vcos æ è  u 2 ö ø cos( u) ,sin( u) +vcos æ è  u 2 ö ø sin( u) ,vsin æ è  u 2 ö ø

for u in [ 0,2p] and v in [ -0.3,0.3] . Graph the Mobius strip with either a graphing calculator or a computer. Also find r_u and r_v. Is the parameterization of the Mobius strip orthogonal?

34. Show that r( t,u,v) = á sin( t) cos( v), cos( t) cos(v), sin( u) sin( v), cos( u)sin( v) ñ is a parameterization of the sphere in 4 dimensions given by

x2+y2+z2+w2 = 1

35. Mercator: Use an identity for the hyperbolic trigonometric functions to prove that

sech2( A) +tanh2( A) = 1

Then show that the Mercator parameterization

r( q,m) = á R sech( m) cos( q) ,R sech( m) sin( q) ,Rtanh( m) ñ

is indeed a parameterization of the sphere of radius R centered at the origin.

36. Mercator: Find r_q and r_m for the Mercator parameterization in exercise 35. Is the parameterization orthogonal? What does that imply about a map constructed as a Mercator projection? Also, what happens to the length of r_m(0,m) as m approaches ¥? What does this imply about the Mercator projection?

37. Mercator: Show that the Mercator projection in exercise 35 is a surface of revolution (for m in ( -¥,¥) ). How is the half circle in the xz-plane related to the cylinder x²+y² = R² as z approaches ¥ on the cylinder? (More in the Maple worksheet)

38. Cylindrical: The ray from the origin through the point P(x,y,z) on a sphere of radius R intersects a right circular cylinder of radius R at only one point, which we call Q.

Click and drag P to see more projections

If Q is of the form ( Rcos(u), Rsin(u), Rv), then what are the coordinates of P in terms of u and v? What is the resulting parameterization of the sphere? Is it the Mercator projection?

39. Find the parameterization of the torus which results from revolving the circle

r( u) = á rsin( u) ,0,R+rcos( u) ñ

for u in [ 0,2p] about the x-axis, where R > r > 0 are constants. *What is a level surface representation of the torus?

41. What is the level surface representation of a sphere of radius R centered at a point ( h,k,l) ? Explain.

42. Show that every cone whose horizontal cross-sections (i.e., level curves) are circles is of the form

x2+y2 = m2z2

What is the significance of the m in this equation?

43. Write to Learn: Determine the longitude q₀ and latitude j₀ of your present location, and then use (2) to find r_q and r_j at your present location. In a short essay, present your results and determine the direction (north, south, east, west) that r_q and r_j are pointing in.

44. Write to Learn: Derive the parameterization of the surface obtained by revolving the curve y = f( x) , x in [ a,b] , about the y-axis. Present and explain your derivation in a short essay.

45.  Write to Learn:  Stereographic projection assigns to each point (u,v,0) the point ( x,y,z) on the unit sphere that is on the line from the point (0,0,1) through the point (u,v,0) .

Use similar right triangles to show that stereographic projection leads to the following parameterization of the sphere.

r( u,v) =  2u u2+v2+1 ,  2v u2+v2+1 ,  u2+v2-1 u2+v2+1