Cylindrical coordinate system
From Wikipedia, the free encyclopedia
The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted h) which measures the height of a point above the plane.
A point P is given as (r,θ,h). In terms of the Cartesian coordinate system:
- r is the distance from O to P', the orthogonal projection of the point P onto the XY plane. This is the same as the distance of P to the z-axis.
- θ is the angle between the positive x-axis and the line OP', measured counterclockwise.
- h is the same as z.
- Thus, the conversion function f from cylindrical coordinates to Cartesian coordinates is f(r,θ,h) = (rcosθ,rsinθ,h).
For use in physical sciences and technology, the recommended international standard notation is ρ, φ, z (ISO 31-11).
Some mathematicians indeed use (r,θ,z).
Cylindrical coordinates are useful in analyzing surfaces that are symmetrical about an axis, with the z-axis chosen as the axis of symmetry. For example, the infinitely long circular cylinder that has the Cartesian equation x2 + y2 = c2 has the very simple equation r = c in cylindrical coordinates. Hence the name "cylindrical" coordinates.
[edit] Line and volume elements
In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.
The line element is 
.
The volume element is 
.
The gradient is 
.
