Anything You Want to Talk About?
(No.)
Problem Set 1
Remember that today is the day to finish it by, then meet with me next week to grade it.
Cylindrical and Spherical Coordinates
Based on section 1.7 in the textbook.
Key Ideas
Definitions:
- Cylindrical coordinates basically use polar form to locate a point in the xy plane, then a linear z axis to add the third dimension.
- Spherical coordinates use polar form to locate a point in the xy plane, then a second angle (specifically, the angle between the vertical axis and the line to the point) to locate it in the third dimension:
Conversion equations: see the book, particularly the box “HOWTO: Converting among Spherical, Cylindrical, and Rectangular Coordinates,” for the exact equations. This is one of those pieces of knowledge where the important thing is not to memorize all the equations, but to remember that they can be looked up.
It’s interesting that rectangular and cylindrical coordinates use the same idea of a z coordinate. This is probably because both solve the problem of a third dimension the same way, namely by adding a linear third axis to a coordinate system for 2 dimensions.
Which coordinate system is best to use can be problem-specific.
Questions
How to figure out the shapes of surfaces from their equations? For the equations the book gives, which hold 1 variable constant and consider what happens when you vary the others, I find it easiest to imagine where points can go as the unconstrained variables change. For example, the book gives r = c for some constant c as an equation that produces a cylinder. If you think about how this allows you to move anywhere in a circle (thanks to unconstrained Θ) around the z axis, and anywhere up or down (unconstrained z), but always staying a specific distance away from the z axis (r can only have one value), you start to see how the equation defines a cylinder.
For more complicated plots, the next question is relevant…
Does Mathematica have a way of plotting in spherical or cylindrical coordinates? I
didn’t know, but a little searching showed that it does: the
SphericalPlot3D function. We tried this function out, and you can
download the notebook with our experiments. In
doing this, we looked at Mathematica’s online documentation (available via the
booklet icon in the top right part of a Mathematica window) and its palettes of special
characters and mathematical symbols (available via the “Palettes” menu). One
thing I discovered after class, when I had a chance to read the documentation more
carefully, is that Mathematica switches the meanings of coordinates Θ and φ
relative to our textbook: Θ is the angle down from the vertical axis, and φ is
the angle around that axis. I changed the examples in the notebook to abide by this rule.
How to figure out what point a set of spherical coordinates describes? The usual coordinate order is ρ (distance from origin) then Θ (angle in the horizontal plane) and finally φ (angle from vertical axis). So you can work out where a point is by interpreting each coordinate in turn.
Examples
Estimate the cylindrical coordinates of one or more points on a cylinder.
Consider a point on the outside of a vertical paper cylinder of radius 0.1, perched on the origin of a set of axes. The point is about 0.5 units above the y axis. Since the point is on a radius 0.1 cylinder, its r coordinate is that radius. The y axis is 90 degrees, or π/2 radians, counterclockwise from the x axis, which is the reference for cylindrical angles. So the Θ coordinate is π/2. Finally the z coordinate is just the point’s height above the origin, or 0.5. So we end up with coordinates (0.1, π/2, 0.5).
Convert to rectangular form: Find the conversion equations for cylindrical to rectangular, and plug r, Θ, and z into them:
Problem Set
Problem set 2, on surfaces and coordinate systems, is now available.
Work on it through next week, and grade it the week after.
See the handout for details.
Next
Now that we can talk about 3 dimensional coordinates, we can talk about vectors, i.e., distances and directions, in 3 dimensions.
Please read “Working with Vectors in R3” in section 1.2 of the textbook for Monday.
(Section 1.1 may also be helpful, if you want some background on vectors.)