11.7 Cylindrical and Spherical Coordinates

Cylindrical Coordinates

Cylindrical coordinates is nothing more than adding a z axis above the xy plane when xy is listed in polar coordinates rq.  The cylindrical coordinates of a point in R³ are (r,q,z) where (r,q) is the points projection onto the xy plane given in polar coordinates, and z is the distance from the xy plane to the point.

Recall conversion from polar to rectangular coordinates, it's the same:

Converting from rectangular to cylindrical (Same as rectangular to polar):

In both types of conversion, the z remains the same, in fact we could say that cylindrical coordinates are just polar coordinates in R² with a z axis orthogonal to the rq plane.

Ex 1 Convert to rectangular, and sketch: (5, 4p/3, 4)

Ex 2 Change from rectangular to cylindrical coordinates and sketch:

Note for ex 2, there are really an infinite number of choices here for q, and r = -2 is also a choice, with q = p/3.

Cylindrical coordinates, like polar coordinates, give us a way to simplify the equations of some surfaces.

Ex 2,3,4,5

2.
This is a sphere of radius 2, substitute x² + y² for r² to see why.

3.
Mult iply by r to get r² = 4r*sin(q), then x² + y² - 4y + 4 = 4.  This is a right circular cylinder, centered at (0,2,z) for any z.

4.
Really z = x. The plane formed by the generator z = x projected along the y axis.

5.
Arctan (y/x) = p/4  so y/x = 1 or y = x. This is the plane using as generator the line y = x and projecting along the z axis.

Spherical Coordinates

A point in space can be notated in this system in the following way: (r, q, f) where r is the distance from the origin to the point, q is the angle from the positive x axis to the projection of the point onto the xy plane, and f is the angle from the 'North Pole' (positive z axis) to the point. See the picture: (green part)

Notes:

1. The r  is generally positive, but could be negative. Our text considers only positive r.
2. The q has a representative in [0, 2p) but there are an infinite number of possible q for each point. This is the same q as in cylindrical coordinates.
3. The f is generally in [0,p], p being the angle to the negative z axis.

 Converting spherical to rectangular:(see the drawing to figure these out)

Converting rectangular to spherical: (again the picture is helpful)

Here think of the distance across the diagonal of a box with sides x,y,and z

Converting spherical to cylindrical:

Converting cylindrical to spherical:

Ex 4,5,6 Some neat simple equations in spherical coordinates:

4. is a sphere of radius c.

5.see above abut cylindrical coordinates. It's the same: the plane y = x.

6.upper half of a cone.