Maple: How to convert Cylindrical coordinates to Cartesian coordinates?

Question

We get some expression in Cylindrical coordinates (r, ϕ, z ) like : expr := r*z^2*sin((1/3)*ϕ) we need to convert it into Cartesian coordinates and than back to Cylindrical coordinates. How to do such thing?

So I found something like this : eval(expr, {r = sqrt(x^2+y^2), z = z,ϕ= arctan(y, x)}) but it seems incorrect, how to correct it and how make eval to convert backwords from Cartesian to Cylindrical?

ϕ == ϕ

So I try:

R := 1; 

H := h; 

sigma[0] := sig0;

sigma := sigma[0]*z^2*sin((1/3)*`ϕ`);

toCar := eval(sigma, {r = sqrt(x^2+y^2), z = z, `ϕ` = arctan(y, x)});

toCyl := collect(eval(toCar, {x = r*cos(`ϕ`), y = r*sin(`ϕ`), z = z}), `ϕ`)

It looks close to true but look:

why arctan(r*sin(ϕ), r*cos(ϕ)) is not shown as ϕ?

Answer 1

I don't speak Maple, but it looks like your eval takes you from Cartesian to cylindrical coordinates. The inverse is $x=r \cos \phi , y=r \sin \phi, z=z$. The Wikipedia link you have gives this, though using $\rho$ instead of $r$