$1)[5$ pts$]$ COORDINATE TRANSFORMATION

In MATLAB, a user defined function allows for the calling of repeated operations that do not exist as built in functions. Functions are created in their own m$-$files, with the format:

function $[$outputArguments$]=$ functionname$($inputArguments$)$

where the m $-$file must be saved as the name of the function. To call a function in the command window, the first line from the function file without the first word $($function$)$ can be used.

$($a$)$ Write a function car$2$Cyl$(0)$ with the header line

function $[$r$,$phi,z$]=$ car$2$Cyl$(\backslash (\backslash$mathrm$\{$x$\},\backslash$mathrm$\{$y$\},\backslash$mathrm$\{$z$\})\backslash )$

that converts coordinates of a point given in the Cartesian coordinate system to points in the cylindrical coordinate system.

$($b$)$ Write a function car$2$Sph$0$ with the header line function $[$R$,$theta,phi$]=$ car$2$Sph $(\backslash (\backslash$mathrm$\{$x$\},\backslash$mathrm$\{$y$\},\backslash$mathrm$\{$z$\}\backslash ))$

that converts coordinates of a point given in the Cartesian coordinate system to points in the spherical coordinate system.

$($c$)$ Write a function cyl$2$Car$()$ with the header line function $[$x$,$y$,$z$]=$ cyl$2$Car$($r$,$phi,z$)$

that converts coordinates of a point given in the cylindrical coordinate system to points in the Cartesian coordinate system.

$($d$)$ Write a function sph$2$Car$()$ with the header line function $[$x$,$y$,$z$]=$ sph$2$Car$($R$,$theta, phi$)$

that converts coordinates of a point given in the spherical coordinate system to points in the Cartesian coordinate system.

$($e$)$ Verify that the point found by one function will return the original point using the opposite function. In particular, use the point $[$x$,$y$.$z$]=[1,1,1]$ with the functions car$2$Cyl and car$2$Sph$,$ then use the results with the functions cyl$2$Car and sph$2$Car to show that $\backslash ([1,1,1]\backslash )$ is returned.