سااااعدونـ ـ ـ ـ ـي في Polar coordinates

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لو سمحتم هل أجد لديكم روابط لشرح موضوع Polar coordinates

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[السيف البتار]

السلام عليكم :
أخي ماهينور
أنا درست polar coor..وهي موجودة على الرابط التالي في ملف pdf (صفحة30 )وهو عبارة عن محاضرات لمادة كهرومغناطيسية بأحدى الجامعات الأمريكية..

http://www.4shared.com/file/51007013/a2a8dfe7/LectureNotesChapter1.html
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[ابن العراق]

polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed in terms of angles and distance; in the more familiar Cartesian or rectangular coordinate system, such a relationship can only be found through trigonometric formulation.
As the coordinate system is two-dimensional, each point is determined by two polar coordinates: the radial coordinate and the angular coordinate. The radial coordinate (usually denoted as r) denotes the point's distance from a central point known as the pole (equivalent to the origin in the Cartesian system). The angular coordinate (also known as the polar angle or the azimuth angle, and usually denoted by θ or t) denotes the positive or anticlockwise (counterclockwise) angle required to reach the point from the 0° ray or polar axis (which is equivalent to the positive x-axis in the Cartesian coordinate plane).[1]
Contents

1 History
2 Plotting points with polar coordinates
2.1 Converting between polar and Cartesian coordinates
3 Polar equations
3.1 Circle
3.2 Line
3.3 Polar rose
3.4 Archimedean spiral
3.5 Conic sections
4 Complex numbers
5 Calculus
5.1 Differential calculus
5.2 Integral calculus
5.2.1 Generalization
5.3 Vector calculus
6 Three dimensions
6.1 Cylindrical coordinates
6.2 Spherical coordinates
7 Applications
7.1 Position and navigation
7.2 Modeling
8 See also
9 References
10 External links​

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[ابن العراق]

Polar Coordinates​

​

The polar coordinates

(the radial coordinate) and

(the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by

(1)

(2)
where

is the radial distance from the origin, and

is the counterclockwise angle from the x-axis. In terms of

and

,

(3)

(4)
The equation of a curve expressed in polar coordinates is known as a polar equation, and a plot of a curve in polar coordinates is known as a polar plot.
The arc length of a polar curve given by

is

(5) ​

The line element is given by

(6) ​

and the area element by

(7) ​

The area enclosed by a polar curve

is

(8) ​

The slope of a polar function

at the point

is given by

(9) ​

The angle between the tangent and radial line at the point

is

(10) ​

A polar curve is symmetric about the x-axis if replacing

by

in its equation produces an equivalent equation, symmetric about the y-axis if replacing

by

in its equation produces an equivalent equation, and symmetric about the origin if replacing

by

in its equation produces an equivalent equation.
In Cartesian coordinates, the position vector

is

(11) ​

giving derivative

(12) ​

Its unit vector is

(13) ​

giving derivative

(14) ​

In polar coordinates, the position vector is given by

(15) ​

giving derivatives

(16)

(17)

(18)
The unit vectors are

(19)

(20)
giving derivatives

(21)

(22)
http://www.maths.abdn.ac.uk/~igc/tch/ma1002/diff/node46.html​

 

[ابن العراق]

and also try this link it contain Pauls Online Notes : Calculus II - Polar CoordinatesI
http://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx​