In two dimension some curves are easier to represent in polar coordinates than in Cartesian coordinates. This is true in three dimensions as well. There are two ways to extend polar coordinates to three dimensions. Watch the three videos provided for definitions of the three coordinate systems to be used in this course.
Figure3.1.1.Comparing Cartesian and Cylindrical Coordinates
Figure3.1.2.Comparing Cartesian and Spherical Coordinates
ExercisesExercises
1.
Graph by hand each of the points below. Note sketching in 3D by hand will lead to a better understanding of of each coordinate system and its inherent strengths. Graphing by hand is not a skill for daily use.
(a)
\((x,y,z)=(-2,3,5)\)
(b)
\((r,\theta,z)=(\sqrt{3},\pi/3,4)\)
(c)
\((r,\theta,z)=(\sqrt{3},\pi/3,6)\)
(d)
\((r,\theta,z)=(1,\pi/2,4)\)
(e)
\((\rho,\theta,\phi)=(2,0,0)\)
(f)
\((\rho,\theta,\phi)=(2,0,\pi/6)\)
(g)
\((\rho,\theta,\phi)=(2,\pi/4,0)\)
(h)
\((\rho,\theta,\phi)=(2,\pi/4,\pi/6)\)
2.
Compare and contrast cylindrical coordinates (3D) to polar coordinates (2D).
3.
Compare and contrast spherical coordinates (3D) to polar coordinates (2D).
Subsection3.1.2Conversion of Coordinates
A coordinate system is chosen for a problem, because it is convenient for that problem. However, sometimes it is necessary to convert from one coordinate system to another. One such area is integration which is covered in later sections.
Figure3.1.3.Cartesian and Polar Comparison
Activity48.
(a)
Cylindrical to Cartesian
(i)
How are polar coordinates converted to Cartesian coordinates (2D)? This is illustrated in Figure 3.1.3. List by name the two principles used.
(ii)
How can cylindrical coordinates be converted to Cartesian coordinates?
(iii)
Convert the point \((10,\pi/3,2)\) in cylindrical coordinates to Cartesian.
(iv)
Convert the point \((1,2,7)\) in Cartesian coordinates to cylindrical.
(b)
Spherical to Cylindrical
(i)
How can \(\theta\) be converted?
(ii)
How can \(\rho\) and \(\phi\) be used to calculate \(z\text{?}\)
(iii)
How can \(\rho\) and \(\phi\) be used to calculate \(r\text{?}\)
(iv)
Convert the point \((10,\pi/3,2)\) in cylindrical coordinates to spherical coordinates.
(v)
Convert the point \((12,\pi/3,\pi/4)\) in spherical coordinates to cylindrical coordinates.
(c)
Spherical to Cartesian
(i)
How can spherical coordinates by converted to Cartesian coordinates?
(ii)
Convert the point \((12,\pi/3,\pi/4)\) in spherical coordinates to Cartesian.
(iii)
Convert the point \((1,2,7)\) in Cartesian coordinates to spherical.
Activity49.
(a)
For cylindrical coordinates what range (min to max) of values are needed for \(\theta\) to be able to represent all points?
(b)
For spherical coordinates what range (min to max) of values are needed for \(\theta\) to be able to represent all points?
(c)
For spherical coordinates what range (min to max) of values are needed for \(\phi\) to be able to represent all points?
ExercisesExercises
Graph each function or equation below by hand if directed or using technology. If using technology graphing only part of the \(\theta\) or \(\phi\) ranges may help in studying the graph. Identify each of graphs as a curve or surface.
1.
Cartesian \(x=0\) (by hand)
2.
Cartesian \((0,1,z)\) (by hand)
3.
Cartesian \(x^2+y^2+z^2=1.\)
4.
Cylindrical \(r=1.\) (by hand)
5.
Cylindrical \((1,\theta,0)\) (by hand)
6.
Cylindrical \(r=1+\cos\theta\)
7.
Cylindrical \(r=z(1+\cos\theta)\)
8.
Spherical \(\rho=1\) (by hand)
9.
Spherical \((1,\theta,\pi/2)\) (by hand)
10.
Spherical \((1,0,\phi)\) (by hand)
11.
Spherical \(\rho=1+\cos(4\theta)\)
12.
Spherical \(\rho=\phi\)
13.
Why are cylindrical coordinates called ‘cylindrical’?
