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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Cylindrical and Spherical Coordinates">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 13:32:27 1997}
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\section{Cylindrical and Spherical Coordinates\label%
{Cylindrical and Spherical Coordinates}}

In the \textbf{cylindrical coordinate system}, a point $P$ in
three-dimensional space is represented by the ordered triple $\left(
r,\theta ,z\right) $, where $r$ and $\theta $ are polar coordinates of the
projection of $P$ onto the $xy$-plane and $z$ is the directed distance from
the $xy$-plane to $P.$

To convert from cylindrical to rectangular coordinates use the equations 
\begin{eqnarray*}
x &=&r\cos \theta \qquad y=r\sin \theta \qquad Z=z \\
&&
\end{eqnarray*}
whereas to convert from rectangular to cylindrical coordinates use 
\[
r^{2}=x^{2}+y^{2}\qquad \tan \theta =\frac{y}{x}\qquad Z=z 
\]
In the \textbf{spherical coordinates system}, a point $P$ in
three-dimensional space is represented by the ordered triple $\left( \rho
,\theta ,\phi \right) ,$ where $\rho =\left| OP\right| $ is the distance
from the origin to $P$, $\theta $ is the same angle as in cylindrical
coordinates, and $\phi $ is the angle between the positive $z$-axis and the
line segment $OP$. Note that 
\[
\rho \geq 0\qquad 0\leq \phi \leq \pi 
\]
The spherical coordinate system is especially useful in problems where there
is symmetry about the origin. For example, the sphere with center the origin
and radius $c$ has the simple equation $\rho =c$ and this is the reason for
the name ``spherical'' coordinates. The graph of the equation $\theta =c$ is
a vertical half-plane, and the equation $\phi =c$ represents a half-cone
with the $z$-axis as its axis.

The relationship between rectangular and spherical coordinates are 
\begin{eqnarray*}
z &=&\rho \cos \phi \qquad r=\rho \sin \phi  \\
&&
\end{eqnarray*}
But $x=r\cos \theta $ and $y=r\sin \theta ,$so 
\[
x=\rho \sin \phi \cos \theta \qquad y=\rho \sin \phi \sin \theta \qquad
z=\rho \cos \phi 
\]
Also, the distance formula shows that 
\[
\rho ^{2}=x^{2}+y^{2}+z^{2}
\]

\subsection{Related Reference Topics}

{\small \hyperref{Cylindrical coordinates}{}{}{GE06_01.tex#Cylindrical
coordinates}}

{\small \hyperref{Distance formula in the plane}{}{}{Al03_01.tex#Distance
Formula in the Plane}}

{\small \hyperref{Relations between rectangular, cylindrical, and spherical
coordinates}{}{}{GE06_01.tex#Relations between coordinate systems}}

{\small \hyperref{Relations between rectangular and polar coordinates}{}{}{%
GE06_01.tex#Relations between rectangular and polar coordinates}}

{\small \hyperref{Spherical coordinates}{}{}{GE06_01.tex#Spherical
coordinates}}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Plot 3D, cylindrical}{}{}{../../Help/DM6-6.tex#Cylindrical
coordinates}}

{\small \hyperref{Plot 3D, spherical}{}{}{../../Help/DM6-6.tex#Spherical
coordinates}}

{\small \hyperref{Plot 3D, tube}{}{}{../../Help/DM6-6.tex#Curves in space}}

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