The number of variables necessary to describe the position in space. For a two dimensional space, we need two variables to describe position. We can use the cartesian system of coordinates, using the $x$ and $y$ variables, or we can use the polar coordinates, which are an angle $\theta$ and the radius $r$. The cartesian and polar coordinate systems are related to each other according to the equations


\begin{align}x = & r\cos\theta, \notag \\
y = & r\sin\theta, \notag
\end{align}
if the origin of both coordinate system is the same. For three dimensional space, examples of coordinate systems are the cartesian, the cylindrical and the spherical coordinate systems.

Cartesian $(x,y,z)$ coordinate system.


 
Figure 1: cartesian coordinate system
\begin{figure}
\centering\epsfig{file=lfig106.eps,height=1.8in,clip=}\centering \notag
\end{figure}

Cylindrical $(r,\theta,z)$ coordinate system.


 
Figure 2: Cylindrical coordinate system
\begin{figure}
\centering\epsfig{file=lfig107.eps,height=1.8in,clip=}\centering\notag
\end{figure}


\begin{align}r^2 & = x^2 + y^2, \notag \\
\theta & = \tan ^{ - 1}(y/x) \notag
\end{align}

Spherical $(r,\theta, \varphi)$ coordinate system.


 
Figure 3: Spherical coordinate system.
\begin{figure}
\centering\epsfig{file=lfig108.eps,height=1.8in,clip=}\centering\notag
\end{figure}


\begin{align}r^2 & = x^2 + y^2 + z^2, \notag \\
\theta & = \cos ^{ - 1}(x/r) \m...
...cos \theta } \right) \notag \\
\varphi & = \tan ^{ - 1}(z/y) \notag
\end{align}



Timothy S Choe
2003-03-15