Exercises
1. Convert the following points from cylindrical
coordinates ( r, q, z) to Cartesian coordinates (i.e., xyz coordinates):
2. What section of the cylinder x2+y2 = 1 corresponds to
cylindrical points ( 1,q, z) in the range q in [ 0,p] and z in [ -1,1] ?
3. Convert the following points from spherical
coordinates ( r,f,q) to Cartesian coordinates:
4. What section of the unit sphere corresponds to
spherical points ( 1,f,q) for f in [0,p] and q in [ 0,p] ?
Find the pullback of the following surfaces into cylindrical
coordinates. What is a parameterization of the surface?
Find the pullback of the following surfaces into spherical
coordinates. What is a parameterization of the surface?
Find a parameterization of the conic section formed by the
intersection of z = p+ex and the right circular cone.
Then sketch its graph.
31. Discuss the surface of revolution given by
What is its parameterization? Is it a deformation of the sphere?
32. Discuss the surface of revolution given by
What is its parameterization? Is it a deformation of the sphere?
33. The curve formed by the intersection of a sphere centered at
the origin and a plane through the origin is called a great circle. Let's use spherical coordinates to develop a method for parameterizing a
great circle.
- A non-vertical plane through the origin is of the form z = ax+by,
where a and b are constants. Show that spherical coordinates transforms
the equation into
|
cos( f) = sin( f) [ acos(q) + bsin( q) ] |
|
- Show that intersection of the plane with a sphere of radius R
results in the parameterization
|
r( t) = Rsin( f)
á cos( q), sin(q), a cos(q) + b sin(q)
ñ |
|
where tan( f) = acos( q) +bsin(q) . Then use a right triangle to find sin( f) in terms of acos( q) +bsin( q) to finish the parameterization.
34. Show that cylindrical coordinates results in the same
parameterization for a great circle (see exercise 33) as does spherical
coordinates.
35. If a point has a location of ( r,f,q) in spherical coordinates, then its longitude is q and its latitude
is
What is the parameterization of a sphere of radius R in latitude-longitude
coordinates? (be sure to simplify expressions like
|
sin |
æ
è
|
|
p
2
|
-j |
ö
ø
|
and cos |
æ
è
|
|
p
2
|
-j |
ö
ø
|
|
|
36. What is the equation in spherical coordinates of the sphere of
radius R centered at ( 0, 0, R )? What is its parameterization?
37. For A, w, k, and d constants, the function
|
f(( r,t) = |
A
r
|
cos( wt-kr+d) |
|
is a spherical wave about the origin with angular frequency w, wavenumber k, and phase d. Explain why the spherical wave is the
same in all directions. What happens to the spherical wave as the spherical
distance r goes to infinity?
38. (Continues 37) Spherical waves are often studied as they impact
a small region of a plane.
In particular, for R > 0 constant, points (x,y,R) in the plane z = R are at a distance
r from the origin, where
and r2 = x2 + y2.
Show that the Maclaurin series of r as a function of r is of the
form
|
r
= R + |
r2
2R
|
- |
r4
8R3
|
+ |
r6
16R5
|
+ ¼ |
|
so that if r2 << R (i.e., if r2 is much, much, less than R ), then
Finally, explain why for small regions in the z = R plane, a spherical wave
about the origin is approximately the same as
|
f(r, t) = |
A
R
|
cos |
æ
è
|
wt- |
kr2
2R
|
+d1 |
ö
ø
|
|
|
where d1 = d-d-kR is a new phase constant for the spherical
wave. (i.e., a change in phase).
39. The interconnected double helix structure of DNA describes a
helicoid, which is the surface parametrized by
|
r( q,v) =
á vcos(q), vsin(q) , q
ñ |
|
Show that v = q tan( f) in spherical coordinates and that the helicoid in
spherical coordinates is given by r = q sec(f) .
40. What is the parameterization of the helicoid in exercise 39 in cylindrical coordinates?
41. Write to Learn: Explain in a short essay why a torus is
radially symmetric but why there does not exist a postive
continuous function f( f) on [ 0,p] such
that the torus is given by r = f( f) for f in [ 0,p] in spherical coordinates. (i.e., explain why it is
impossible to deform a sphere into a torus).
42. Is an ellipsoid a radially symmetric deformation of the
sphere? What about a paraboloid? Or a hyperboloid? Explain.
Cylindrical and spherical coordinates are examples of
curvilinear coordinate systems. Exercises 43-46 explore some additional
curvilinear coordinates.
43. Ellcylindrical coordinates assign a point P in three
dimensional space the coordinates ( u,q,z) , where
|
x = cosh( u) cos( q) , y = sinh(u) sin( q) |
|
and where z is the usual z-coordinate. What surface corresponds to u = a
for a constant? What surface corresponds to q = c for c constant?
44. Paraboloidal coordinates assign a point P in three
dimensional space the coordinates ( u,v,q) , where
|
x = uvcos( q) , y = uvsin( q) , z = |
u2-v2
2
|
|
|
What surface corresponds to u = a for a constant? What surface corresponds
to v = b for b constant? What surface corresponds to q = c for c
constant?
45. Bispherical coordinates assign a point P in three dimensional
space the coordinates ( v,f,q) , where
|
x = |
sin( f) cos( q)
cosh(v) -cos( f)
|
, y = |
sin( f)sin( q)
cosh( v) -cos( f)
|
, z = |
sinh( v)
cosh( v) -cos(f)
|
|
|
Explain why a surface of the form v = k for k constant is a sphere of radius
csch( k) centered at ( 0,0,coth( k) ).
46. Toroidal coordinates assign a point P in three dimensional
space the coordinates ( v,f,q) , where
|
x = |
sinh( v) cos( q)
cosh(v) -cos( f)
|
, y = |
sinh( v) sin( q)
cosh( v) -cos( f)
|
, z = |
sin( f)
cosh( v) -cos( f) )
|
|
|
Explain why a surface of the form v = k for k constant is a torus, and
explain why if k1 ¹ k2, then v = k1 does not intersect v = k2.
