Polar coordinates
The numbers 
and 
(see ) related to rectangular Cartesian coordinates 
and 
by the formulas:
 |
where 
, 
. The coordinate lines are: concentric circles ( 
) and rays (
).
Figure: p073410a
The system of polar coordinates is an orthogonal system. To each point in the 
-plane (except the point 
for which 
and 
is undefined, i.e. can be any number 
) corresponds a pair of numbers 
and vice versa. The distance 
between a point 
and 
(the pole) is called the polar radius, and the angle 
is called the polar angle. The Lamé coefficients (scale factors) are:
 |
The surface element is:
 |
The fundamental operations of vector analysis are:
 |
 |
 |
The numbers 
and 
related to Cartesian rectangular coordinates 
and 
by the formulas:
 |
where 
, 
, 
, 
, are called generalized polar coordinates. The coordinate lines are: ellipses (
) and rays (
).
References
| [1] | G.A. Korn, T.M. Korn, "Mathematical handbook for scientists and engineers" , McGraw-Hill (1961) |
Comments
The generalization of polar coordinates to 3 dimensions are the spherical coordinates.
By viewing a point 
as a complex number 
, the polar coordinates 
correspond to the representation of 
as 
.
See also Complex number.
References
| [a1] | H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. 103 |
| [a2] | K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 216 |
Polar coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_coordinates&oldid=48226