Polar coordinates
The numbers and
(see ) related to rectangular Cartesian coordinates
and
by the formulas:
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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:
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The surface element is:
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The fundamental operations of vector analysis are:
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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=17546