Essential Concepts

• In the cylindrical coordinate system, a point in space is represented by the ordered triple $(r,\theta,z)$, where $(r,\theta)$ represents the polar coordinates of the point’s projection in the $xy$-plane and $z$ represents the point’s projection onto the $z$-axis.
• To convert a point from cylindrical coordinates to Cartesian coordinates, use equations $x=r\cos\theta$, $y=r\sin\theta$, and $z=z$.
• To convert a point from Cartesian coordinates to cylindrical coordinates, use equations $r^2=x^2+y^2$, $\tan\theta=\frac{y}{x}$, and $z=z$.
• In the spherical coordinate system, a point $P$ in space is represented by the ordered triple $(\rho,\theta,\varphi)$, where $\rho$ is the distance between $P$ and the origin $(\rho\ne{0})$, $\theta$ is the same angle used to describe the location in cylindrical coordinates, and $\varphi$ is the angle formed by the positive $z$-axis and line segment $\overline{OP}$, where $O$ is the origin and $0\le\varphi\le\pi$.
• To convert a point from spherical coordinates to Cartesian coordinates, use equations $x=\rho\sin\varphi\cos\theta$, $y=\rho\sin\varphi\sin\theta$, and $z=\rho\cos\varphi$.
• To convert a point from Cartesian coordinates to spherical coordinates, use equations $\rho^{2}=x^2+y^2+z^2$, $\tan\theta=\frac{y}{x}$, and $\varphi=\arccos\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)$
• To convert a point from spherical coordinates to cylindrical coordinates, use equations $r=\rho\sin\varphi$, $\theta=\theta$, and $z=\rho\cos\varphi$.
• To convert a point from cylindrical coordinates to spherical coordinates, use equations $\rho=\sqrt{r^2+z^2}$, $\theta=\theta$, and $\varphi=\arccos\left(\frac{z}{\sqrt{r^2+z^2}}\right)$