A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.

5: Example in cylindrical coordinates: The area of the curved surface of a cylinder. (CC BY SA 4.0; K. Kikkeri). The differential surface vector in this case is ds=ˆρ(ρ0dϕ)(dz)=ˆρρ0 dϕ dz.

Differential Volume

In the cylindrical coordinate system, a point in space is represented by the ordered triple (r,θ,z), where (r,θ) represents the polar coordinates of the point’s projection in the xy-plane and z represents the point’s projection onto the z-axis.

Likewise, if we have a point in Cartesian coordinates the cylindrical coordinates can be found by using the following conversions. Let’s take a quick look at some surfaces in cylindrical coordinates. Example 1 Identify the surface for each of the following equations. In two dimensions we know that this is a circle of radius 5.

When we convert to cylindrical coordinates, the z -coordinate does not change. Therefore, in cylindrical coordinates, surfaces of the form are planes parallel to the xy -plane. Now, let’s think about surfaces of the form The points on these surfaces are at a fixed distance from the z -axis.

A differential volume element in the rectangular coordinate system is generated by making differential changes dx, dy, and dz along the unit vectors x, y and z, respectively, as illustrated in Figure 2.18a.