It is an elementary fact that we are able to change coordinates system to new one. for example in Cartesian coordinates $x^2+y^2=1$ illustrates a circle. Changing to polar coordinates, this equation becomes $r^2=1$ with the same graph.
But the graph of $z=x^3-3xy^2$ in Cartesian coordinates and Cylindrical coordinates seems to be different?!
$(x,y,z): z=x^3-3xy^2$: see its graph
$(t,r,w): w=r^3\sin(t)(-3+4\sin^2(t))$: see its graph
What's wrong? Am I misunderstanding the concept of coordinates?
Indeed the surface should be the same regardless of the choice of coordinates. In particular, setting $x = r \cos t$ as more commonly done, or alternatively $x = \sin t$ necessarily yield the same result.
Programming-wise, there's so built-in cylindrical plot in Wolfram, as they do it in the more general PareametricPlot3D.
What matters is to have the correct expression for the height of the surface accordingly (be it called $z$ or $w$).
For $\{x, y \} = \{r \cos t, r \sin t\}$, we have the height $z = r^3 \cos t \left( \cos^2 t - 3 \sin^2 t \right)$
For $\{x, y \} = \{r \sin t, r \cos t\}$, we have the height $w = r^3 \sin t \left( \sin^2 t - 3 \cos^2 t \right)$
The expression in the question post is a different surface and cannot be obtained via trigonometric substitution of any kind. (no matter how you swap the coordinates: this, or that)