View source for 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 } $).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p073410a.gif" />

Figure: p073410a

The system of polar coordinates is an [[Orthogonal system|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|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 } $).

====Comments====
The generalization of polar coordinates to 3 dimensions are the [[Spherical coordinates|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 } $.

See also [[Complex number]].

{{OldImage}}

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.A. Korn,   T.M. Korn,   "Mathematical handbook for scientists and engineers" , McGraw-Hill  (1961)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Triebel,   "Analysis and mathematical physics" , Reidel  (1986)  pp. 103</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 216</TD></TR></table>