let t= theta
x= rcos(t) y = rsin(t) z = z
x^2 + y^2 + (z-1)^2= 1
becomes r^2 +(z-1)^2 = 1
how to you transform cartesian equations into cylindrical coordinates?
for example, write the following cartesian equation in cylindrical coordinates and graph:
x^2 + y^2 + (z-1)^2= 1
this seems like it is really simple, so i'm sorry for posting it, but there aren't any examples in my book so i'm not sure what to do. thanks so much
It is still a sphere centered at (0,0,1)
the idea of the exercise is to recognize the from a sphere takes in cylindrical coordinates
For eg a sphere centered at the origin is r^2 + z^2 = 1
as another eg the saddle z = 1- x^2 + y^2
takes the form z= 1 - r^2cos^2(t) + r^2sin^(t)
z = 1 -r^2cos(2t)
and so on.
the thing to remember is that all the surfaces you learned in rectangular coordinates don't change when you go to cylindrical and spherical coordinates.
The form of the equations change in the new coordinates but you still are describing the same exact surfaces.
You may hate it now but believe me when you get to triple integrals you'll thank your lucky stars for the new coordinate systems
Hang in there