Cylindrical coordinate system
The cylindrical coordinate system is a three-dimensional
coordinate system which essentially extends
circular polar coordinates by adding a third coordinate which measures the height of a point above the plane.
A point P is given as . In terms of the
Cartesian coordinate system:
* 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.
* is the angle between the positive x-axis and the line OP', measured counterclockwise.
* is the same as .
Some mathematicians indeed use .
Encyclopedia
The
cylindrical coordinate system is a three-dimensional
coordinate system which essentially extends
circular polar coordinates by adding a third coordinate which measures the height of a point above the plane.
A point P is given as . In terms of the
Cartesian coordinate system:
- 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.
- is the angle between the positive x-axis and the line OP', measured counterclockwise.
- is the same as .
Some mathematicians indeed use . It is also common in physics to use 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 .
The volume element is .
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.
.
See also
