# Cylindrical coordinate system – 원통 좌표계

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(x,y,z) = (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.

Line and volume elements

The line element is $dl = dr\,\mathbf{\hat r} + r\,d\theta\,\boldsymbol{\hat\theta} + dz\,\mathbf{\hat z}$.

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

The gradient is $\nabla = \mathbf{\hat r}\frac{\partial}{\partial r} + \boldsymbol{\hat \theta}\frac{1}{r}\frac{\partial}{\partial \theta} + \mathbf{\hat z}\frac{\partial}{\partial z}$.

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