Question

I need help understanding cylindrical co-ordinates

Answer 1

for example: \[x \hat{x}+y\hat{y} + z\hat{z} \] becomes what when expressed in polar coordinates and why?

Answer 2

and by polar i mean cylindrical

Answer 3

do you understand polar coordinates?

Answer 4

I don't know anymore, I thought I did, but I seem to be getting confused alot by this hat notation

Answer 5

Cylindrical coordinates are like polar coordinates with a z-component added. They are three dimensional and have coordinates (r, theta, z). (x,y,z) becomes (Rcos(theta),Rsin (theta), Z).
http://tutorial.math.lamar.edu/Classes/CalcII/CylindricalCoords.aspx
the hat thing just reminds you which axis (or direction)your talking about. I think the way you wrote it is confusing, I would write "x (i hat), y (j hat), z (k hat)". Don't know if that helps at all...

Answer 6

no, that honestly doesnt help at all, if you think about changing from a cylindrical shape in cartesian co-ordinates the shape should become rectangular in cylindrical co-ordinates, the directions like r hat, theta hat and z hats should point you in the correct direction which is extremely important when concidering surface integrals on open surfaces when you can't use the divergence theorem.

what you wrote for what the coordinates (x,y,z) become also in (R,theta,z) form. Z=/= Rcostheta. x=Rcos(theta), if you want to do things that way then you haven't changed co-ordinate systems, only co-ordinates, that would become:\[Rcos\theta \hat{x} +Rsin\theta \hat{y} + z\hat{z}\]As you can see you still need to convert the hatted unit vectors in order to complete the transition. But how does one go about doing this? Please, more information is needed.

Answer 7

Hmmm... you seem to be at about the same level as me so I can only try to help, but if you wanted to use different unit vector the obvious choice to me would be
R (in the r-hat) + Z (z-hat),
so you would only need the two standard basis vectors, not three. For instance in spherical coordinates all points just get described with the unit vector r-hat.
As for surface integrals yes that kind of change-of-basis-vector thing comes into play, but I'm not so well versed in it. AL I can refer you to is something called the Jacobian Transformation which is used when you want to change coordinate systems for integrals and wind up basically just changing the differentials you have when you integrate.
http://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables.aspx
Sorry dude, I hope I'm even understanding your question. If you learn more about it please let me know, you've got me curious now.