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

x = r cos q

y = r sin q

z = z

r ² = x ² + y ²

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:

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.

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.

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:

EXAMPLE 15:

Convert

to rectangular coordinates, then cylindrical coordinates.

SOLUTION:

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 £ 180^o

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