Cylindrical coordinate system
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The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted <math>h</math>) which measures the height of a point above the plane.
A point P is given as <math>(r, \theta, h)</math>. In terms of the Cartesian coordinate system:
- <math>r</math> 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.
- <math>\theta</math> is the angle between the positive x-axis and the line OP', measured counterclockwise.
- <math>h</math> is the same as <math>z</math>.
- Thus, the conversion function <math>f</math> from cylindrical coordinates to Cartesian coordinates is <math>f(x,y,z)=(r\cos\theta,r\sin\theta,h)\,</math>.
- The conversion function <math>f</math> from Cartesian coordinates to cylindrical coordinates is <math>f(r,\theta,h)=(\sqrt{x^{2}+y^{2}},\operatorname{atan2}(y, x),z)\,</math>.
For use in physical sciences and technology, the recommended international standard notation is ρ, φ, z (ISO 31-11).
Some mathematicians indeed use <math>(r, \theta, z)</math>.
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
- See multiple integral for details of volume integration in cylindrical coordinates, and Del in cylindrical and spherical coordinates for vector calculus formula.
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 <math>dl = dr\,\mathbf{\hat r} + r\,d\theta\,\boldsymbol{\hat\theta} + dz\,\mathbf{\hat z}</math>.
The volume element is <math>dV = r\,dr\,d\theta\,dz</math>.
The gradient is <math>\nabla = \mathbf{\hat r}\frac{\partial}{\partial r} + \boldsymbol{\hat \theta}\frac{1}{r}\frac{\partial}{\partial \theta} + \mathbf{\hat z}\frac{\partial}{\partial z}</math>.
[edit] See also
- List of canonical coordinate transformations
- Cartesian coordinate system
- Spherical coordinate system
- Parabolic coordinate system
- Vector fields in cylindrical and spherical coordinatescs:Válcová soustava souřadnic
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