**Introduction**

The usual Cartesian coordinate system can be quite difficult to use in certain situations. Some of the most common situations when Cartesian coordinates are difficult to employ involve those in which circular, cylindrical, or spherical symmetry is present. For these situations it is often more convenient to use a different coordinate system.

In polar coordinates, a point in the plane is determined by its distance
*r* from the origin and the angle *theta* (in radians) between
the line from the origin to the point and the *x*-axis (see the figure
below).

It is common to represent the point by an ordered pair
(*r*,*theta*). Using standard trigonometry we can find
conversions from Cartesian to polar coordinates

and from polar to Cartesian coordinates

**Example**

The point with rectangular coordinates (-1,0) has polar coordinates (1,pi) whereas the point with rectangular coordinates (3,-4) has polar coordinates (5,-0.927).

Note that a point does not have a unique polar representation. The points

are equivalent for any integer n.

The curves *r*=constant and *theta*=constant are a circle and
a half-ray, respectively.

Cylindrical coordinates are obtained by replacing the *x* and
*y* coordinates
with the polar coordinates *r* and *theta* (and leaving the
*z* coordinate unchanged).

Thus, we have the following relations between Cartesian and cylindrical coordinates:

From cylindrical to Cartesian:

From Cartesian to cylindrical:

As an example, the point (3,4,-1) in Cartesian coordinates would have polar coordinates of (5,0.927,-1).Similar conversions can be done for functions. Using the first row of conversions, the function

in Cartesian coordinates would have a cylindrical coordinate representation of

Cylindrical coordinates are most convenient when some type of cylindrical
symmetry is present. The surfaces *r*=constant, *theta*=constant,
and *z*=constant
are a cylinder, a vertical plane, and a horizontal plane, respectively.

The coordinates used in spherical coordinates are *rho*,
*theta*, and *phi*.
*Rho* is the distance from the origin to the point.
*Theta* is the
same as the angle used in polar coordinates. *Phi* is the angle
between the
*z*-axis and the line connecting the origin and the point.

The following are the relations between Cartesian and spherical coordinates:

From spherical to Cartesian:

From Cartesian to spherical:

Relations between cylindrical and spherical coordinates also exist:

From spherical to cylindrical:

From cylindrical to spherical:

The point (5,0,0) in Cartesian coordinates has spherical coordinates of
(5,0,1.57). The surfaces *pho*=constant, *theta*=constant,
and *phi*=constant
are a sphere, a vertical plane, and a cone (or horizontal plane), respectively.
Spherical coordinates are of course very useful when any type of spherical
symmetry is present.

**Example: **Many physical situations have spherical symmetry.
The gravitational field of a single body and the electric field of
a point charge exhibit spherical symmetry. Formulas used in these
situations almost always involve *r, theta*, and *phi *and not
*x*, *y*, and *z*.

It should be noted that formulas for objects like the gradient, curl, and divergence have a different form in different coordinate systems. Always be careful to use the proper formula when dealing with these objects.

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