Cylindrical Coordinates

A 3-dimensional coordinate transformation is a mapping of the form

T( u,v,w) = á x(u,v,w), y(u,v,w), z(u,v,w) ñ

Correspondingly, a 3-dimensional coordinate transformation T maps a solid W in the uvw-coordinate system to a solid T(W) in the xyz -coordinate system (and similarly, T maps curves in uvw to curves in xyz, surfaces in uvw to surfaces in xyz, and so on). In this section, we introduce and explore two of the more important 3-dimensional coordinate transformations.

To begin with, the cylindrical coordinates of a point P are cartesian coordinates in which the x and y coordinates have been transformed into polar coordinates (and the z-coordinate is left as is). Not surprisingly, to convert to cylindrical coordinates, we simply apply x = r cos(q) and y = r sin(q) to the x and y coordinates. That is, the cylindrical coordinate transformation is

T( r,q,z) = á r cos(q) ,r sin(q) ,z ñ

Cylindrical coordinates get their name from the fact that the surface ''r = constant'' is a cylinder. For example, the cylinder

r( q, z) = á cos( q),sin( q) ,z ñ

is obtained by setting r = 1 in the cylindrical coordinate transformation.

Graphic of Directional Derivative

EXAMPLE 1    Find a parametrization of the right circular cone

z2 = x2+y2

by pulling back into cylindrical coordinates.       

Solution: Transforming x and y into polar coordinates yields

z2 = r2,    z = r

Letting z = r in the cylindrical coordinate transformation yields

r( r,q) = á rcos( q),rsin( q) ,r ñ

which is a parametrization of the right circular cone.

Graphic of Directional Derivative

       

EXAMPLE 2    Parameterize the surface z = x²-y² by pulling back into cylindrical coordinates       

Solution: Setting x = rcos(q) and y = rsin(q) leads to

z = r2cos2( q) -r2sin2( q) = r2cos( 2q)

Thus, the parametrization is

r( r,q) = á rcos( q),rsin( q) ,r2cos( 2q) ñ

Maple Graphics Export