The **Cartesian
coordinate system** is the most commonly used
coordinate system. In two dimensions, this system consists of a pair of
lines on a flat surface or plane, that intersect at right angles. The lines
are called **axes** and the point at which they intersect is called the
**origin**. The axes are usually drawn horizontally and vertically and
are referred to as the x- and y-axes, respectively.

A
point
in the plane with coordinates (a, b)
is a units to the
right of the y axis and b
units up from the x axis if a
and b are positive
numbers. If a
and b are both
negative numbers, the point is a units
to the left of the y axis and b
units down from the x axis. In the figure above point P_{1} has
coordinates (3, 4), and point P_{2} has coordinates (-1, -3).

In
three-dimensional Cartesian coordinates, the z axis is added so that there are
three axes all perpendicular to each other.

In
the ** polar coordinate system**, each point in the plane is assigned coordinates (r,
f) with respect to a fixed line in the plane called
the ** axis ** and a point on that line called the
**pole**. For a point in the
plane, the r-coordinate is the distance from the point to the pole, and the f-coordinate is the counterclockwise angle between the axis and a line joining the
origin to the point, The r-coordinate is always positive and
the range of f
is from 0 to 2p (360^{o}). To
be able to transform from Cartesian to polar coordinates and vice versa, we let
the axis of the polar coordinate system coincide with the x-axis of the
Cartesian coordinate system and the pole coincide with the origin.

In
the figure above he Point P_{1} has polar coordinates (r_{1}, f_{1})
= (5, 53.1^{o}), and the point P_{2} has polar coordinates (r_{2},
f_{2})
= (3.16, 251.6^{o} ).

The Transformation equations are:

x
= r cosf, y = r sinf

r
= (x^{2 }+ y^{2})^{1/2}, f
= tan^{-1}(y/x)

Cylindrical
coordinates and spherical coordinates are two different extensions of polar
coordinates to three dimensions.

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