Do you have a basic knowledge of the spherical polar coordinate system?

A coordinate system provides a way to describe and plot math functions using two or three variables. If there are two variables the graph is 2D. If there are three variables, the graph is 3D.

## The Cartesian Coordinate System

The most familiar coordinate system is the Cartesian coordinate system. Typical variable names are x and y in 2D (although variables can have any name), and x, y, and z in 3D. Every point of every 2D function has a unique value in (x, y). Every 3D function similarly has a unique value in (x, y, z).

## The Polar Coordinate System

This 2D coordinate system uses one distance coordinate and one angular coordinate, typically r and theta (θ). A circle can be drawn in this coordinate system simply by making the distance coordinate a constant and the angle completely variable. Make the angle a constant and the distance completely variable, and you have an equation for a line. Each point of each 2D function has a value in (r, θ).

Although it is to be expected that a function can be graphed on any of a number of coordinate systems, it is interesting to derive conversion formulas. For instance, a circle can also be graphed in a strictly Cartesian coordinate system and the equation in that system can be converted to its corresponding equation in the polar coordinate system.

## The Spherical Polar Coordinate System

The spherical polar coordinate system is like the polar coordinate system, except an additional angle variable is used, frequently labeled as phi (φ). A point in a 3D function graphed in this coordinate system is then assigned a value (r, θ, φ).

Angles can be given in *degree* units – a complete rotation amounting to 360º – or it can be given in *radians*.

360^{o} = 2π radians.

If both angles are allowed to be of any real value whatsoever, whereas the value of r is constant, one obtains the equation of a sphere. For example, r = 8 is a sphere of radius 8 in the spherical polar coordinate system.

If one angle is held constant and one angle and the distance are variable, a cone results. If both angles are constant and the distance variable may assume any value, one again has the equation for a line.

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