Polar coordinates: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Paul Wormer
mNo edit summary
 
(6 intermediate revisions by 4 users not shown)
Line 1: Line 1:
{{subpages}}
{{subpages}}
[[Image: Polar coordinates .png|right|thumb|250px|Two dimensional polar coordinates ''r'' and &theta; of vector <math>\scriptstyle \vec{\mathbf{r}}</math>]]
 
In [[mathematics]] and [[physics]],  '''polar coordinates''' give the position of a vector <math>\scriptstyle \vec{\mathbf{r}}</math> in two-dimensional real space <math>\scriptstyle \mathbb{R}^2</math>.  A Cartesian system of two orthogonal axes is presupposed. One number (''r'') gives the length of the vector and the other number (&theta;)  gives the angle of the vector with the ''x''-axis of the Cartesian system (measured in the direction of the positive y-axis).  
 
==Definition==
::''For an extension to three dimensions, see [[spherical polar coordinates]].''
The polar coordinates are related to the [[Cartesian coordinates]] ''x'' and ''y''  through
 
{{Image| Polar coordinates .png|right|250px|Two dimensional polar coordinates ''r'' and &theta; of vector <math>\scriptstyle \vec{\mathbf{r}}</math>}}
 
In [[mathematics]] and [[physics]],  '''polar coordinates''' are two [[number]]s&mdash;a [[distance]] and an [[angle]]&mdash;that specify the position of a [[point (geometry)|point]] on a [[plane (geometry)|plane]].
 
In their classical ("pre-vector") definition, polar coordinates give the position of a point ''P''  with respect to  a given point ''O'' (the ''pole'') and a given line (the ''polar axis'') through ''O''. One [[real number]] (''r'' ) gives the distance of ''P'' to  ''O''  and another number (&theta;)  gives the angle of the line  ''O''&mdash;''P'' with the polar axis.  Given ''r'' and &theta;,  one determines ''P'' by constructing  a [[circle]] of [[radius]] ''r'' with ''O'' as origin, and  a line with  angle &theta; measured counterclockwise from the polar axis. The point ''P'' is on the intersection of the circle and the [[line]].
 
In modern [[vector]] language one identifies the plane with a real [[Euclidean space]] <math>\scriptstyle \mathbb{R}^2</math> that has a [[Cartesian coordinates|Cartesian coordinate]] system. The crossing of the  Cartesian axes is on the pole, that is, ''O'' is the origin of the Cartesian system and the polar axis is identified with the ''x''-axis of the Cartesian system.  The line ''O''&mdash;''P'' is generated by the vector
:<math>
\overrightarrow{OP} \equiv \vec{\mathbf{r}}.
</math>
Hence we obtain the figure on the right where  <math>\scriptstyle \vec{\mathbf{r}}</math> is the position vector of the point ''P''.
 
==Algebraic definition==
The polar coordinates ''r'' and &theta; are related to the [[Cartesian coordinates]] ''x'' and ''y''  through
:<math>
:<math>
\begin{align}
\begin{align}
Line 19: Line 33:
\end{cases}
\end{cases}
</math>
</math>
Bounds on the coordinates are: ''r'' &ge; 0  and 0 &le; &theta; < 360<sup>0</sup>.
Coordinate lines are: the circle (fixed ''r'', all &theta;) and a half-line from the origin (fixed direction &theta; all ''r''). The [[slope]] of the half-line is tan&theta; = ''y''/''x''.
==Surface element==
==Surface element==
The infinitesimal surface element in polar coordinates is
The infinitesimal surface element in polar coordinates is
Line 30: Line 46:
\cos\theta &  -r\sin\theta \\
\cos\theta &  -r\sin\theta \\
\sin\theta &  r\cos\theta \\
\sin\theta &  r\cos\theta \\
\end{vmatrix}
\end{vmatrix} = r \cos^2\theta + r\sin^2\theta
= r .
= r .
</math>
</math>
Example: the area ''A'' of a circle of radius ''R'' is given by
Example: the area ''A'' of a circle of radius ''R'' is given by
:<math>
:<math>
A = \int_{0}^{2\pi}  \int_{0}^R r\, dr\, d\theta = \pi R^2
A = \int_{0}^{2\pi}  \int_{0}^R r\, dr\, d\theta = \pi R^2 .
</math>
</math>[[Category:Suggestion Bot Tag]]
 
[[Category: CZ Live]]
[[Category: Physics Workgroup]]
[[Category: Mathematics Workgroup]]

Latest revision as of 12:00, 5 October 2024

This article is developing and not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.


For an extension to three dimensions, see spherical polar coordinates.
CC Image
Two dimensional polar coordinates r and θ of vector

In mathematics and physics, polar coordinates are two numbers—a distance and an angle—that specify the position of a point on a plane.

In their classical ("pre-vector") definition, polar coordinates give the position of a point P with respect to a given point O (the pole) and a given line (the polar axis) through O. One real number (r ) gives the distance of P to O and another number (θ) gives the angle of the line OP with the polar axis. Given r and θ, one determines P by constructing a circle of radius r with O as origin, and a line with angle θ measured counterclockwise from the polar axis. The point P is on the intersection of the circle and the line.

In modern vector language one identifies the plane with a real Euclidean space that has a Cartesian coordinate system. The crossing of the Cartesian axes is on the pole, that is, O is the origin of the Cartesian system and the polar axis is identified with the x-axis of the Cartesian system. The line OP is generated by the vector

Hence we obtain the figure on the right where is the position vector of the point P.

Algebraic definition

The polar coordinates r and θ are related to the Cartesian coordinates x and y through

so that for r ≠ 0,

Bounds on the coordinates are: r ≥ 0 and 0 ≤ θ < 3600. Coordinate lines are: the circle (fixed r, all θ) and a half-line from the origin (fixed direction θ all r). The slope of the half-line is tanθ = y/x.

Surface element

The infinitesimal surface element in polar coordinates is

The Jacobian J is the determinant

Example: the area A of a circle of radius R is given by