圆柱坐标

Recall that in rectangular (Cartesian) coordinates, shapes such as rectangles and planes are very easy to describe.

Cylindrical coordinates are nothing but a simple extension of polar coordinates.

The three coordinates are:

the distance from the $z$-axis, called $r$;

the angle measured counterclockwise from the positive $xz$-plane, called $\theta$;

the distance from the $xy$-plane, called $z$.

Notice that $r$ and $\theta$ are exactly the same as in polar coordinates.

Also see that $z$ is the same in cylindrical and Cartesian coordinates.

Due to the similarities to the polar and Cartesian coordinate system, one can easily derive the conversion formulas between cylindrical and 3-D Cartesian coordinates:

$$x=r\cos\theta$$ $$y=r\sin\theta$$ $$r^2=x^2+y^2$$ $$\tan\theta=\frac{y}{x}$$ $$z=z$$

Keep the cylindrical coordinate system in mind whenever one is dealing with cylinders or figures with good axial symmetry. Then, one may position the axis of symmetry along the z-axis to simplify calculations.

Be warned that every point in cylindrical coordinates has infinitely many representations. Also, points on the $z$-axis have no well-defined value of $\theta$. These are exactly the same problems that we encountered in polar coordinates, so apply the same discretion when dealing with cylindrical coordinates.