2.7 Cylindrical and spherical coordinates  (Page 7/11)

Which coordinate system is most appropriate for creating a star map, as viewed from Earth (see the following figure)?

How should we orient the coordinate axes?

Spherical coordinates with the origin located at the center of the earth, the z -axis aligned with the North Pole, and the x -axis aligned with the prime meridian

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Key concepts

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

Use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems.

For the following exercises, the cylindrical coordinates $\left(r,\theta ,z\right)$ of a point are given. Find the rectangular coordinates $\left(x,y,z\right)$ of the point.

For the following exercises, the rectangular coordinates $\left(x,y,z\right)$ of a point are given. Find the cylindrical coordinates $\left(r,\theta ,z\right)$ of the point.

For the following exercises, the equation of a surface in cylindrical coordinates is given.

Find the equation of the surface in rectangular coordinates. Identify and graph the surface.

For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates.

For the following exercises, the spherical coordinates $\left(\rho ,\theta ,\phi \right)$ of a point are given. Find the rectangular coordinates $\left(x,y,z\right)$ of the point.