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 $(r,\; \backslash theta,\; h)$. 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.
• $\backslash theta$ 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 $(r,\; \backslash theta,\; z)$. It is also common in physics to use $(\backslash rho,\; \backslash phi,\; z)$ 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

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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 $dl\backslash \; =\; dr\backslash \; +\; r\backslash ,d\backslash theta\backslash \; +\; dz$.

The volume element is $dV\backslash \; =\; r\backslash ,dr\backslash ,d\backslash theta\backslash ,dz$.

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.

$\backslash nabla\; \backslash equiv\; \backslash mathbf\{\backslash hat\; r\}\backslash frac\{\backslash partial\}\{\backslash partial\; r\}\; +\; \backslash boldsymbol\{\backslash hat\; \backslash theta\}\backslash frac\{1\}\{r\}\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash theta\}\; +\; \backslash mathbf\{\backslash hat\; z\}\backslash frac\{\backslash partial\}\{\backslash partial\; z\}$.

See also

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• List of canonical coordinate transformations
• Cartesian coordinate system
• Spherical coordinate system
• Parabolic coordinate system
• Vector fields in cylindrical and spherical coordinates

Coordinate systems

Válcová soustava souřadnic | Cylindrisk koordinatsystem | Coordenadas cilíndricas | 원통 좌표계