Cylindrical coordinate system

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The cylindrical coordinate system is a three-dimensional 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, \theta, 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.
[\theta] is the angle between the positive x-axis and the line OP', measured anti-clockwise.
[h] is the same as [z].

Some mathematicians indeed use [(r, \theta, z)]. It is also common in physics to use [(\rho, \phi, z)] to denote these coordinates.

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.

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 [dl\ = dr\ + r\,d\theta\ + dz].

The volume element is [dV\ = r\,dr\,d\theta\,dz].

It is also important in many cases to be able to find the gradient of a vector field in cylindrical polar coordinates. The gradient can be worked out from first principals, if one knows theta, r and z in terms of cartesian coordinates, but the general equation is given below.

[\nabla \equiv \mathbf\frac + \boldsymbol\frac\frac + \mathbf\frac].

Cylindrical coordinate system

Line and volume elements

See also