# Polar coordinates

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The numbers $\rho$ and $\phi$ related to rectangular Cartesian coordinates $x$ and $y$ by the formulas:

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

where $0 \leq \rho < \infty$, $0 \leq \phi < 2 \pi$. The coordinate lines are: concentric circles ($\rho = \textrm{ const }$) and rays ( $\phi = \textrm{ const }$).

Figure: p073410a

The system of polar coordinates is an orthogonal system. To each point in the $Oxy$- plane (except the point $O$ for which $\rho = 0$ and $\phi$ is undefined, i.e. can be any number $0 \leq \phi < 2 \pi$) corresponds a pair of numbers $( \rho , \phi )$ and vice versa. The distance $\rho$ between a point $P$ and $( 0 , 0 )$( the pole) is called the polar radius, and the angle $\phi$ is called the polar angle. The Lamé coefficients (scale factors) are:

$$L _ \rho = 1 ,\ L _ \phi = \rho .$$

The surface element is:

$$d \sigma = \rho d \rho d \phi .$$

The fundamental operations of vector analysis are:

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

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

$$\Delta f = \frac{1} \rho \frac \partial {\partial \rho } \left ( \rho \frac{\partial f }{\partial \rho } \right ) + \frac{1}{\rho ^ {2} } \frac{ \partial ^ {2} f }{\partial \phi ^ {2} } = \frac{\partial ^ {2} f }{\partial \rho ^ {2} } + \frac{1} \rho \frac{\partial f }{\partial \rho } + \frac{1}{\rho ^ {2} } \frac{\partial ^ {2} f }{\partial \phi ^ {2} } .$$

The numbers $r$ and $\psi$ related to Cartesian rectangular coordinates $x$ and $y$ by the formulas:

$$x = a r \cos \psi ,\ \ y = b r \sin \psi ,$$

where $0 \leq r < \infty$, $0 \leq \psi < 2 \pi$, $a, b > 0$, $a \neq b$, are called generalized polar coordinates. The coordinate lines are: ellipses ( $r = \textrm{ const }$) and rays ( $\psi = \textrm{ const }$).

The generalization of polar coordinates to 3 dimensions are the spherical coordinates.

By viewing a point $( x, y)$ as a complex number $z = x+ iy$, the polar coordinates $( \rho , \phi )$ correspond to the representation of $z$ as $z = \rho e ^ {i \phi }$.