MAT 224
Johns – SCCC
Name
Alternate Coordinate Systems in R3 – Notes
Cylindrical Coordinate System
Spherical Coordinate System
z
z
(r, , z)

(, , )

z
y
y
r

x

x
z
x
y
Conversion Formulas
Cartesian to Cylindrical :
2
2
2
r =x +y
Cartesian to Spherical :
2
2
2 2
 =x +y +z
 x2  y2 

tan  = 


z


y
tan  =
x
y
tan  =
x
z=z
Cylindrical to Cartesian :
x = r cos 



Spherical to Cartesian :
x =  sin  cos  
 sin  
z=z
z =  cos 
Nuts and Bolts of Cartesian, Cylindrical, and Spherical Coordinate Systems


y = r sin 



y =  sin
Cartesian
Cylindrical
Point (x,y,z) = (-2,3,1) is the intersection of planes
x = -2, y = 3, and z =1.
Point (r,,z) = (3,
of the circular cylinder r = 3, the half-plane
=
Point (, , ) = (2,

.
12

, and the plane z = 2.
12
 
,
) is the intersection
6 12

of the sphere  = 2, the cone  =
half-plane  =

,2) is the intersection
12
6
, and the
Unit Vectors in Cartesian, Cylindrical, and Spherical Coordinates
iˆ is a unit vector orthogonal to the plane x = −2
r̂ is a unit vector orthogonal to the
cylinder r = 3 pointing in the direction of
increasing r.
pointing in the direction of increasing x.
ˆ
ĵ is a unit vector orthogonal to the plane y = 3
is a unit vector orthogonal to the half-

12
pointing in the direction of increasing y.
plane  =
k̂ is a unit vector orthogonal to the plane z = 1
pointing in the direction of increasing z.

 


increasing .
ẑ is a unit vector orthogonal to the plane
z = 2 in the direction of increasing z.


in the direction of
z
x
̂ is a unit vector orthogonal to the sphere  = 2 in the
direction of increasing .

ˆ is a unit vector orthogonal to the cone  = in the
6
direction of increasing .
ˆ
is a unit vector orthogonal to the half-plane
=

12
in the direction of increasing .
MAT 224
Johns – SCCC
Name
y
Coordinates Systems Worksheet
Do all of the following plotting on 3D cartesian graph paper.
1.a) On the same set of cartesian axes, draw and label (with their letter) the following points:
A: (x, y, z) = (0, -2, 0)
B: (x, y, z) = (0, 3, -3)
C: (x, y, z) = (-1, 0, 1)
D: (x, y, z) = (3, -4, 0)
E: (x, y, z) = (-2, 2, 1)
b) Then draw the cylindrical unit vectors r̂ ,
ˆ , and ẑ at each of the points in part 1a).
c) Find the equivalent cylindrical coordinates (r, z) for each of the points in part 1a). First, give exact
answers. Then check your exact answers by stating approximate answers to the nearest tenth.
d) By inspection, state the cylindrical unit vectors r̂ , ˆ , and ẑ in terms of the cartesian unit
vectors
iˆ , ĵ , and k̂ at each of the points in part 1a). Give exact answers.
2.a) On the new set of cartesian axes, draw and label (with their letter) the following points:
A: (x, y, z) = (0, -2, 0)
B: (x, y, z) = (0, 3, -3)
C: (x, y, z) = (-1, 0, 1)
D: (x, y, z) = (3, -4, 0)
E: (x, y, z) = (-2, 2, 1)
b) Then draw the spherical unit vectors
̂ , ˆ , and ˆ at each of the points in part 2a).
c) Find the equivalent spherical coordinates (, , ) for each of the points in part 2a). First, give exact
answers. Then check your exact answers by stating approximate answers to the nearest tenth.
d) By inspection, state the spherical unit vectors
vectors
̂ , ˆ , and ˆ
in terms of the cartesian unit
iˆ , ĵ , and k̂ at each of the points in part 2a). Give exact answers.
3. a) Find the cylindrical coordinates and cylindrical unit vectors at the point (x, y, z) = (-3, 4, 5).
b) Find the spherical coordinates and spherical unit vectors at the point (x, y, z) = (-3, 4, 5).
4. Pick any point where the cylindrical unit vectors r̂ , ˆ , and ẑ are defined. Using the right-hand rule for
the cross products, find the following:
rˆ  rˆ =
rˆ  zˆ =
ˆ  ẑ =
5. Pick any point where the spherical unit vectors
the cross products, find the following:
ˆ  ˆ =
ˆ  ˆ =
̂ , ˆ , and ˆ
ˆ  ˆ =
rˆ  rˆ =
ẑ  ˆ 
are defined. Using the right-hand rule for
ˆ  ˆ
=
ˆ  ˆ 
6. The formulas for the cylindrical unit vectors r̂ , ˆ , and ẑ in terms of cartesian unit vectors
k̂ are :
rˆ  cos iˆ  sin ˆj
ˆ   sin iˆ  cos ˆj
iˆ , ĵ , and
zˆ  kˆ
a) Using dot product, show that these three unit vectors are mutually orthogonal.
b) Show the following derivative properties of cylindrical unit vectors:
rˆ 
0
r
ˆ 
0
r
zˆ 
0
r
7. The formulas for the spherical unit vectors
k̂ are :
rˆ 
0
z
 
0
z
zˆ 
0
z
rˆ ˆ


ˆ
  r̂

zˆ 
0

̂ , ˆ , and ˆ
in terms of cartesian unit vectors
iˆ , ĵ , and
ˆ  sin  cosiˆ  sin  sin ˆj  cos kˆ
ˆ  cos  cosiˆ  cos  sin ˆj  sin kˆ
ˆ   sin iˆ  cos ˆj
a) Using dot product, show that these three unit vectors are mutually orthogonal.
b) Show the following derivative properties of spherical unit vectors :
ˆ 
0

ˆ 
0

ˆ 
0

ˆ ˆ


ˆ
  ˆ

ˆ 
0

ˆ
  sin  sin iˆ  sin  cos ˆj

ˆ
 cos ˆ

ˆ
  cos iˆ  sin ˆj

8. Find cylindrical coordinate equations for the following features :
a) a right circular cylinder with its axis along the z-axis and a radius of 4
b) the plane y = x
c) the plane z = 3
d) the plane x = 5
2
2
2
e) the sphere x + y + z = 2
9. Find spherical coordinate equations for the following features:
2
2
2
a) the sphere x + y + z = 2
b) the plane y = -x
c) the line z = 3
d) the plane x = 5
e) the xy-plane (a very flat cone!)
2
2
10. By any means, show that the spherical coordinates equation of the right circular cylinder x + y = 16 is
given by  sin   4 .