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VG.7 Cylindrical and Spherical Coordinates

 

In plane geometry we used the polar coordinate system to describe points. In three dimensions we have a similar system, however it has to be modified slightly. We call this new 3-D system the cylindrical coordinate system.


Cylindrical Coordinate System- A system of locating a point in 3-D space that uses an ordered triple of the form $(r,\theta ,z),$ where $r$ and $\theta $ are polar coordinates of the projection of the point onto the xy-plane and z is the directed distance from the xy-plane to the point.







Picture of a point with labels of $(r,\theta ,z)$







To convert from cylindrical to rectangular coordinates we use the following equations:


MATH

 

and to convert from rectangular to cylindrical coordinates we use the following equations:


MATH

 

Cylindrical coordinates are useful in problems that involve symmetry about an axis, and the z-axis is chosen to coincide with the axis of symmetry. For instance, the axis of the circular cylinder with Cartesian equation $x^{2}+y^{2}=c^{2}$ is the z axis. In the cylindrical coordinates this cylinder has the very simple equation $r=c.$ This is the reason for the name "cylindrical" coordinates.




picture of r=c




Spherical Coordinates- A system of locating a point in 3-D space that uses an ordered triple of the form MATH. In this notation, $\rho $ is the distance from the origin to the point, $\theta $ is the same angle as in cylindrical coordinates, and $\phi $ is the angle between the positive z-axis and the line segment formed between the point and the origin.










picture explaining spherical coordinates




To convert from spherical coordinates to rectangular coordinates we use the following equations:


MATH

 

Also, the distance formula shows that


MATH

 

We use this equation in converting from rectangular to spherical coordinates.


 

The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point. For example, the sphere with center the origin and the radius $c$ has the simple equation $\rho =c$; 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.




picture of three things described above: $\theta =c$ $\phi =c$ & $p=c$

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