MATH 220 SUPPLEMENTAL NOTES 7

CYLINDRICAL AND SPHERICAL COORDINATES

CYLINDRICAL COORDINATES

Cylindrical coordinates combine the polar coordinates in the xy-plane with the usual z-axis.

FACT:

Cylindrical coordinates represent a point P in space by an ordered triple (r, q , z) in which (1) r and q are polar coordinates for the vertical projection of P on the xy-plane, and (2) z is the rectangular vertical coordinate. (See figure 1)

figure 1

EQUATIONS RELATING RECTANGULAR (x, y, z) COORDINATES AND CYLINDRICAL (r, q , z) COORDINATES

x = r cos q

y = r sin q

z = z

r 2 = x 2 + y 2

SPECIAL EQUATIONS

r = a

This is not just a circle in the xy-plane, it a right circular cylinder about the z-axis.

q = q 0

This equation describes the plane that contains the z-axis and makes an angle q 0 with the positive x-axis.

EXAMPLE 1:

Convert (1, 3, 4) in rectangular coordinates to cylindrical coordinates.

SOLUTION:

First of all, let us determine r.

Now to find q .

z stays the same.

EXAMPLE 2:

Convert (0, 0, 1) in rectangular coordinates to cylindrical coordinates.

SOLUTION:

First determine r.

Now to determine q .

Therefore, q = p /2.

The z value stays the same, so the point is (0, p /2,1).

EXAMPLE 3:

Convert

in cylindrical coordinates to rectangular coordinates.

SOLUTION:

z stays the same, so I need to find the values for x and y. To do this, I will use r and q .

EXAMPLE 4:

Convert (2, p /6, 6) in cylindrical coordinates to rectangular coordinates.

SOLUTION:

z = 6, so all I need to find is the values for x and y.

EXAMPLE 5:

Convert x 2 + y 2 = 5 into cylindrical coordinates.

SOLUTION:

Remember that r 2 = x 2 + y 2, so r 2 = 5 or

EXAMPLE 6:

Convert

into cylindrical coordinates.

SOLUTION:

EXAMPLE 7:

Convert x 2 + y 2 + z 2 = 9 into cylindrical coordinates.

SOLUTION:

x 2 + y 2 + z 2 = 9 ® r 2 + z 2 = 9

EXAMPLE 8:

Convert r = -3sec q to rectangular coordinates.

SOLUTION:

EXAMPLE 9:

Convert

to rectangular coordinates.

SOLUTION:

SPHERICAL COORDINATES

One thing that you should keep in mind when working with spherical coordinates is that r is never negative, j is the angle that OP makes with the positive z-axis, therefore its range is [0, p ], and q is measured as in cylindrical coordinates. (See figure 2)

figure 2

FACT:

Spherical coordinates represent a point P in space by the ordered triple (r , j , q ) in which

1.

r is the distance from P to the origin.

2.

j is the angle OP makes with the positive z-axis. ( 0 j p )

3.

q is the angle from cylindrical coordinates.

SPECIAL EQUATIONS

r = a

This equation describes a sphere of radius a centered at the origin.

j = j 0

This equation describes a single cone whose vertex lies at the origin and whose axis lies along the z-axis.

If j = p /2, then we have the xy-plane.

If j > p /2, then the cone j = j 0 opens downward.

EQUATIONS RELATING SPHERICAL COORDINATES TO CARTESIAN AND CYLINDRICAL COORDINATES

z = r cos j

r = r sin j

x = r cos q = r sin j cos q

y = r sin q = r sin j sin q

EXAMPLE 10:

Convert (1, 3, 4) in rectangular coordinates to spherical coordinates.

SOLUTION:

First, find r.

Now find q .

Now determine j. To do this, I will use z = r cos j and solve for j.

EXAMPLE 11:

Convert (0, 0, 1) in rectangular coordinates to spherical coordinates.

SOLUTION:

Therefore, q = p /2.

1 = cos j ® j = 0.

So the point is (1, 0, p /2).

EXAMPLE 12:

Convert

in cylindrical coordinates to spherical coordinates.

SOLUTION:

q stays the same, so I need to find j .

EXAMPLE 13:

Convert (2, p /6, 6) in cylindrical coordinates to spherical coordinates.

SOLUTION:

q = p /6 = 30 o

EXAMPLE 14:

Convert

to rectangular coordinates, then cylindrical coordinates.

SOLUTION:

Rectangular coordinates

Cylindrical coordinates

EXAMPLE 15:

Convert

to rectangular coordinates, then cylindrical coordinates.

SOLUTION:

Rectangular coordinates

Cylindrical coordinates

EXAMPLE 16:

Convert x 2 + y 2 = 5 into spherical coordinates.

SOLUTION:

EXAMPLE 17:

Convert x 2 + y 2 + z 2 = 9 into spherical coordinates.

SOLUTION:

EXAMPLE 18:

Convert r = -3sec q into spherical coordinates.

SOLUTION:

r = -3sec q ® r sin j = -3sec q ® r = -3csc j sec q

EXAMPLE 19:

Convert

into spherical coordinates.

SOLUTION:

j = 135 o

Here are the spherical equations.

r = 2

135 o j 180o

EXAMPLE 20:

Convert

into cylindrical and rectangular coordinates.

SOLUTION:

The above equation is both the cylindrical form of the given equation and the rectangular form of the given equation.

EXAMPLE 21:

Convert

into cylindrical and rectangular coordinates.

SOLUTION:

Let us convert to rectangular coordinates first.

® x 2 + y 2 - z 2 = 0

The rectangular equations are the following.

Now the cylindrical equations.

x 2 + y 2 - z 2 = 0 ® r 2 - z 2 = 0 ® r = -z or r = z, but r ³ 0 and z £ 0 ® r = -z

Your mission, if you choose to accept it, is to work through these examples. You should realize from these examples that cylindrical coordinates work great for equations of cylinders, and spherical coordinates work well for equations of spheres. You will probably find that converting from rectangular coordinates to cylindrical coordinates easier than converting from either rectangular or cylindrical coordinates to spherical. It is just the nature of the beast (spherical coordinates) to be hard to convert to and from. If you have any questions on any of these conversions, please feel free to contact me.

RETURN TO INDEX