# Cylinder coordinates

cylindrical coordinates

Numbers $\rho , \phi$ and $z$ connected with the Cartesian coordinates $x, y$ and $z$ by the formulas:

$$x = \rho \cos \phi ,\ \ y = \rho \sin \phi ,\ \ z = z,$$

where $0 \leq \rho < \infty$, $0 \leq \phi < 2 \pi$, $- \infty < z < \infty$. The coordinate surfaces (see Fig.) are: circular cylinders $( \rho = \textrm{ const } )$, half-planes $( \phi = \textrm{ const } )$ and planes $( z = \textrm{ const } )$.

Figure: c027600a

The system of cylinder coordinates is orthogonal.

The Lamé coefficients are:

$$L _ \rho = \ L _ {z} = 1,\ \ L _ \phi = \rho .$$

The area element of a surface is:

$$ds = \ \sqrt {\rho ^ {2} ( d \rho d \phi ) ^ {2} + ( d \rho dz) ^ {2} + \rho ^ {2} ( d \phi dz) ^ {2} } .$$

The volume element is:

$$dV = \rho d \rho d \phi dz.$$

The differentiation operations of vector analysis are given by:

$$\mathop{\rm grad} _ \rho f = \ \frac{\partial f }{\partial \rho } ,\ \ \mathop{\rm grad} _ \phi f = \ { \frac{1} \rho } \frac{\partial f }{\partial \phi } ,\ \ \mathop{\rm grad} _ {z} f = \ \frac{\partial f }{\partial z } ;$$

$$\mathop{\rm div} \mathbf a = { \frac{1} \rho } a _ \rho + \frac{\partial a _ \rho }{\partial \rho } + { \frac{1} \rho } \frac{\partial a _ \phi }{\partial \phi } + \frac{\partial a _ {z} }{\partial z } ;$$

$$\mathop{\rm curl} _ \rho \mathbf a = { \frac{1} \rho } \frac{\partial a _ {z} }{\partial \phi } - \frac{\partial a _ \phi }{\partial z } ,\ \mathop{\rm curl} _ \phi \mathbf a = \ \frac{\partial a _ \rho }{\partial z } - \frac{\partial a _ {z} }{\partial \rho } ;$$

$$\mathop{\rm curl} _ {z} \mathbf a = { \frac{1} \rho } a _ \phi + \frac{\partial a _ \phi }{\partial \rho } - { \frac{1} \rho } \frac{\partial a _ \rho }{\partial \phi } ;$$

$$\Delta f = \frac{\partial ^ {2} f }{\partial \rho ^ {2} } + { \frac{1} \rho } \frac{\partial f }{\partial \rho } + { \frac{1}{ \rho ^ {2} } } \frac{\partial ^ {2} f }{\partial \phi ^ {2} } + \frac{\partial ^ {2} \phi }{\partial z ^ {2} } .$$

Generalized cylinder coordinates are numbers $u , v$ and $w$ connected with Cartesian coordinates $x, y$ and $z$ by the formulas

$$x = au \cos v ,\ \ y = bu \sin v ,\ \ z = cw,$$

where $0 \leq u < \infty$, $0 \leq v < 2 \pi$, $- \infty < w < \infty$, $a > 0$, $b > 0$, $c > 0$, $a \neq b$. The coordinate surfaces are: elliptic cylinders $( u = \textrm{ const } )$, half-planes $( v = \textrm{ const } )$ and planes $( w = \textrm{ const } )$.