Section 1.9 Cylindrical and Spherical Coordinates

1.9 Cylindrical and Spherical Coordinates

In this section, we introduce cylindrical and spherical coordinates system. This is the extension of the polar coordinate system in the 2-dimensional space. Recall that in 2-dimensional space, the cartesian coordinate point $P(x,y)$ can be converted into polar coordinate by setting $x=r\text{cos}(\theta )$ and $y=r\text{sin}(\theta )$ and we use $P=(r,\theta )$ to present the point. 

Definition 

In the cylindrical coordinate system, a point in space is represented by the ordered triple $(r,\theta ,z),$ where $(r,\theta )$ are the polar coordinates of the point\textquoteright s projection in the $xy$-plane and $z$ is the usual $z$-coordinate in the Cartesian coordinate system.

Theorem: Conversion between Cylindrical and Cartesian Coordinates

The rectangular coordinates $(x,y,z)$ and the cylindrical coordinates $(r,\theta ,z)$ of a point are related as follows: $x=r\text{cos}(\theta ),y=r\text{sin}(\theta ),z=z$ equations that are used to convert from cylindrical coordinates to rectangular coordinates. On the other hand, $r^2=x^2+y^2,\text{tan}(\theta )=\frac{y}{x},z=z$ are used to convert from rectangular coordinates to cylindrical coordinates.

Example 1: Plot the point with cylindrical coordinates $(2,\frac{\pi }{3},-2)$ and express its location in rectangular coordinates. 

Exercise 1: Plot the point with cylindrical coordinates $(3,\frac{2\pi }{3},1)$ and express its location in rectangular coordinates. 

Example 2: Convert the rectangular coordinates $(-\sqrt{2},\sqrt{2},-3)$ to cylindrical coordinates. Express the point on the cylindrical system.

Exercise 2: Convert the rectangular coordinates $(1,-\sqrt{3},4)$ to cylindrical coordinates. express its location in cylindrical coordinates.

Example 3: Identifying Surfaces in the Cylindrical Coordinate System. Describe the surfaces with the given cylindrical equations. 

a. $\theta =\pi /4$

b. $r^2-z^2=9$

c. $z-r=2$

Exercise 3: Describe the surfaces with the given cylindrical equations. 

a. $\theta =\pi /3$

b. $r^2+z^2=1$

c. $z=r$ 

Definition:

In the spherical coordinate system, a point $P$ in space is represented by the ordered triple $(\rho ,\theta ,\phi )$ where

(a) $\rho$ (the Greek letter rho) is the distance between $P$ and the origin;

(b) $\theta$ is the same angle used to describe the location in cylindrical coordinates $0\le \theta \le 2\pi$;

(c) $\phi$ (the Greek letter phi) is the angle formed by the positive $z$-axis and line segment $\overline{OP}$, where $O$ is the origin and $0\le \phi \le \pi$. 

Theorem: Converting among Spherical, Cylindrical, and Rectangular Coordinates

(a) Rectangular coordinates $(x,y,z)$ and spherical coordinates $(\rho ,\theta ,\phi )$ of a point are related as follows: $x=\rho \text{sin}(\phi )\text{cos}(\theta )$, $y=\rho \text{sin}(\phi )\text{sin}(\theta )$, $z=\rho \text{cos}(\phi )$ are equations that are used to convert from spherical coordinates to rectangular coordinates. ${\rho }^2=x^2+y^2+z^2$, $\text{tan}(\theta )=\frac{y}{x}$ and $\text{cos}(\phi )=\frac{z}{\rho }=\frac{z}{\sqrt{x^2+y^2+z^2}}$ are equations that are used to convert from rectangular coordinates to spherical coordinates.

(b) Cylindrical coordinates $(r,\theta ,z)$ and spherical coordinates $(\rho ,\theta ,\phi )$ of a point are related as follows: $r=\rho \text{sin}(\phi )$, $\theta =\theta$, $z=\rho \text{cos}(\phi )$ are equations that are used to convert from spherical coordinates to cylindrical coordinates. ${\rho }^2=r^2+z^2$, $\theta =\theta$ and $\text{cos}(\phi )=\frac{z}{\rho }=\frac{z}{\sqrt{r^2+z^2}}$ are equations that are used to convert from cylindrical coordinates to spherical coordinates.

Example 4: Plot the point with spherical coordinates $(6,\frac{\pi }{4},\frac{2\pi }{3})$ and express its location in both rectangular and cylindrical coordinates. 

Exercise 4: Plot the point with spherical coordinates $(4,\frac{\pi }{3},\frac{\pi }{6})$ and express its location in both rectangular and cylindrical coordinates. 

Example 5: Describe the surfaces with the given spherical equations.

a. $\phi =\frac{\pi }{6}$ 

b. $\rho =4$ 

c. $\rho =\text{sin}(\phi )\text{cos}(\theta )$

Exercise 5: Describe the surfaces with the given spherical equations.

a. $\phi =\frac{\pi }{3}$ 

b. $\rho =3$ 

c. $\rho =\text{sin}(\phi )\text{sin}(\theta )$

Example 6: Find the equation of the surface in spherical coordinates. Identify the surface.

$$x^2+y^2+z^2-9x=0$$

Example 7: Find the equation of the surface in rectangular coordinates.

Identify and graph the surface.

$$\rho =2\text{cos}(\phi )$$

Group work:

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

$$r=3\text{cos}(\theta )$$

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

$$\rho =\text{cos}(\phi )$$

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

$$\rho =3\text{csc}(\phi )$$

4. Find the equation of the surface in cylindrical coordinates. 

$$x^2+y^2+z^2=4$$

5. Find the equation of the surface in spherical coordinates. Identify the surface.

$$x^2+y^2+z^2-4z=0$$