Cylindrical / Spherical Coordinates

[eurekameh]

I'm trying to convert the below Cartesian coordinate system into cylindrical and spherical coordinate systems. For the cylindrical system, I had r,vector = er,hat + sint(e3,hat).
While I do have a technically correct answer for the spherical coordinate system, I believe, I was wondering if there was a way to simplify the expression 1/cos[tan^-1*(sint)]. Thanks.

 

[mfb]

$r=\sqrt{x^2+y^2+z^2} = \sqrt{1+\sin^2(t)}$?

 

[eurekameh]

rad(1+sin^2(t)) = 1/(cos(tan^-1(sint)))?
Also, am I right in thinking r = rad(x^2 + y^2) = 1 for cylindrical coordinates?

 

[mfb]

rad(1+sin^2(t)) = 1/(cos(tan^-1(sint)))?

Would be interesting to check this in an analytic way.

Also, am I right in thinking r = rad(x^2 + y^2) = 1 for cylindrical coordinates?

If you use (r,theta,z) as coordinates: Right.

 

[rezeew]

Hello,

Thank you for sharing your progress in converting Cartesian coordinates into cylindrical and spherical systems. It seems like you have a good understanding of the concepts so far.

To answer your question, there is a way to simplify the expression 1/cos[tan^-1(sint)]. This expression can be rewritten as sec[tan^-1(sint)]. Using the trigonometric identity sec^2x = 1 + tan^2x, we can simplify it further to 1 + (sint)^2, which can be represented as r^2 in spherical coordinates. Therefore, the simplified expression for the spherical coordinate system would be r,theta,phi = r, r,hat + rsin(theta),cos(phi),hat.

I hope this helps with your conversion process. Keep up the good work!