Section 6.6 Other Coordinate Systems
ΒΆExample 6.20. Rectangular to Polar Coordinates.
Convert the point (x,y) = (1, \sqrt3) into polar coordinates.
First calculate \(r\text{:}\)
Now find \(\theta\) such that \(\tan \theta = \frac{\sqrt{3}}{1}\text{.}\) The required \(\theta\) is \(\frac{\pi}{3}\text{.}\)
The polar coordinates are \((2, \frac{\pi}{3})\text{.}\)
Example 6.21.
Find an equation for the sphere of radius 2 in spherical coordinates.
If we start with the Cartesian equation of the sphere and substitute, we get the spherical equation:
Therefore, in spherical coordinates, a sphere of radius \(2\) is expressed \(\rho=2\text{.}\)
Example 6.22. Cylinder Equation in Spherical Coordinates.
Find an equation for the cylinder x^2+y^2=4 in spherical coordinates.
Proceeding as in the previous example:
Exercises for Section 6.6.
Exercise 6.6.1.
Convert the following points in rectangular coordinates to cylindrical and spherical coordinates:
-
\((1,1,1)\)
Answer\(\ds (\sqrt2,\pi/4,1)\text{,}\) \(\ds (\sqrt3,\pi/4,\arccos(1/\sqrt3))\) -
\((7,-7,5)\)
Answer\(\ds (7\sqrt2,7\pi/4,5)\text{,}\) \(\ds (\sqrt{123},7\pi/4,\arccos(5/\sqrt{123})\) -
\((\cos(1),\sin(1),1)\)
Answer\((1,1,1)\text{,}\) \(\ds (\sqrt2,1,\pi/4)\) -
\((0,0,-\pi)\)
Answer\((0,0,-\pi)\text{,}\) \((\pi,0,\pi)\)
Exercise 6.6.2.
Find an equation for the sphere \(x^2+y^2+z^2=4\) in cylindrical coordinates.
Answer\(r^2+z^2=4\)
Exercise 6.6.3.
Find an equation for the \(y\)-\(z\)-plane in cylindrical coordinates.
Answer\(r\cos\theta=0\)
Exercise 6.6.4.
Find an equation equivalent to \(x^2+y^2+2z^2+2z-5=0\) in cylindrical coordinates.
Answer\(r^2+2z^2+2z-5=0\)
Exercise 6.6.5.
Suppose the curve \(z=e^{-x^2}\) in the \(x\)-\(z\)-plane is rotated around the \(z\)-axis. Find an equation for the resulting surface in cylindrical coordinates.
Answer\(z=e^{-r^2}\)
Exercise 6.6.6.
Suppose the curve \(z=x\) in the \(x\)-\(z\)-plane is rotated around the \(z\)-axis. Find an equation for the resulting surface in cylindrical coordinates.
Answer\(z=r\)
Exercise 6.6.7.
Find an equation for the plane \(y=0\) in spherical coordinates.
Answer\(\sin\theta=0\)
Exercise 6.6.8.
Find an equation for the plane \(z=1\) in spherical coordinates.
Answer\(1=\rho\cos\phi\)
Exercise 6.6.9.
Find an equation for the sphere with radius 1 and center at \((0,1,0)\) in spherical coordinates.
Answer\(\rho=2\sin\theta\sin\phi\text{.}\)
Exercise 6.6.10.
Find an equation for the cylinder \(x^2+y^2=4\) in spherical coordinates.
Answer\(\rho\sin\phi=2\)
Exercise 6.6.11.
Suppose the curve \(z=x\) in the \(x\)-\(z\)-plane is rotated around the \(z\)-axis. Find an equation for the resulting surface in spherical coordinates.
Answer\(\cos\phi=1/\sqrt2\)
Exercise 6.6.12.
Plot the polar equations \(r=\sin(\theta)\) and \(r=\cos(\theta)\) and comment on their similarities. (If you get stuck on how to plot these, you can multiply both sides of each equation by \(r\) and convert back to rectangular coordinates).
Exercise 6.6.13.
Extend Exercises 6.6.6 and 6.6.11 by rotating the curve \(z=mx\) around the \(z\)-axis and converting to both cylindrical and spherical coordinates.
Answer\(z=mr\text{;}\) \(\cot\phi=m\) if \(m\neq0\text{,}\) \(\phi=0\) if \(m=0\)
Exercise 6.6.14.
Convert the spherical formula \(\rho=\sin \theta \sin \phi\) to rectangular coordinates and describe the surface defined by the formula (Hint: Multiply both sides by \(\rho\text{.}\))
AnswerA sphere with radius \(1/2\text{,}\) center at \((0,1/2,0)\)
Exercise 6.6.15.
We can describe points in the first octant by \(x >0\text{,}\) \(y>0\) and \(z>0\text{.}\) Give similar inequalities for the first octant in cylindrical and spherical coordinates.
Answer\(0\lt \theta\lt \pi/2\text{,}\) \(0\lt \phi\lt \pi/2\text{,}\) \(\rho>0\text{;}\) \(0\lt \theta\lt \pi/2\text{,}\) \(r>0\text{,}\) \(z>0\)