Describe the surface in cylindrical coordinates?

in your book, you will probably find equations for x, for y in terms of theta and z. don't have my book here with me.

 

apparently, this is the conversion from cartesian coordinates (x,y,z) to cilindrical (rho, phi, z):

[itex] x = \rho \cos \varphi [/itex]
[itex]y = \rho \sin \varphi [/itex]



[itex] \rho = \sqrt{x^{2}+y^{2}}[/itex] and

[itex] \varphi = \begin{cases} 0 & \mbox{if } x = 0 \mbox{ and } y = 0\\ \arcsin(\frac{y}{\rho}) & \mbox{if } x \geq 0 \\ -\arcsin(\frac{y}{\rho}) + \pi & \mbox{if } x < 0\\ \end{cases}
[/itex]

 

Substitute ##x=r\cos\theta,\, y=r\sin\theta## in the equation and solve it for ##r##.

 

f(theta, z) means that you should equate radius to a function of theta and z. in this case theta is the angle [itex]\phi [/phi] in the equations above. this angle, simply put, is the same as the angle in polar coordinates. the only difference between polar coordinates and cilindrical, is that with cilindrical, you have height (z) as well

 

##\theta## is given by \theta, not \phi.