SUNY Geneseo Department of Mathematics
Wednesday, September 7
Math 223 01
Fall 2022
Prof. Doug Baldwin
(No.)
I demonstrated how I make appointments through Google calendar.
Also see this video tutorial from CIT, which is old but still a reasonably good guide — the things that have changed are mostly the look of the user interface, and mostly to get to places you really want to be in fewer clicks:
Geneseo’s student math club is getting started for the semester.
“Welcome Back” party with pizza, soda: next Tuesday (Sept. 13), 5:30 - 6:30, in South 309.
Regular meetings: Mondays, 4:00 - 5:00 PM, South 309.
They have open spots on their e-board if you’re interested in helping out. Go to one of the meetings to learn more, or contact Darien Connolly at prism@geneseo.edu.
Based on “Cylindrical Coordinates” in section 1.7 of the textbook.
What sort of uses do cylindrical (and spherical) coordinates have? Spherical coordinates have lots of uses in geographical and astronomical settings, where the things you’re working with inherently are or at least look like spheres. Cylindrical coordinates probably have fewer obvious uses, although they’re a simple extension of polar coordinates and so often show up when you basically want a polar system but with a third dimension.
Cylindrical coordinates are basically polar coordinates with a 3rd dimension; that third dimension is exactly the z dimension from rectangular coordinates.
Conversions between cylindrical (r,Θ,z) and rectangular (x,y,z):
Or, to go the other way:
Some virtual world is set up so that a tree grows out of the origin of the world’s coordinate system. The tree grows up along the z axis, and has radius 50 cm. Further, suppose our computer animated snake comes along, and wraps itself around the tree, ending with it’s head 100 cm up the tree and directly above the positive y axis:
Define a reasonable cylindrical coordinate system for this situation, and describe the position of the snake’s head in it. What is the position of the head in rectangular coordinates?
The cylindrical coordinate system measured Θ relative to the positive x axis, with positive angles being towards the positive y axis. This is the most common way to connect a cylindrical system to a rectangular one.
In both systems, the snake’s coordinates can be worked out from the facts that the head is 50 units in the r (cylindrical) or y (rectangular) directions, and 100 in the z. You can verify that the conversions between the two coordinate systems work correctly.
Based on “Spherical Coordinates” in section 1.7 of the textbook.
(No.)
Similar to cylindrical or polar coordinates, except information about z is presented via an angle from the positive z axis. Or more or less equivalently, spherical coordinates use 2 angles to specify a direction relative to 2 reference axes, and a third coordinate to specify distance in that direction.
Equations to convert between spherical (ρ,Θ,Φ) and rectangular (x,y,z) coordinates:
Or to go from rectangular to spherical:
Suppose that Earth has a coordinate system whose origin is at the center of the planet, with x and y axes coming out through the equator and the positive z axis coming out through the North Pole. The x axis intersects the equator at longitude 0 degrees:
Geneseo has latitude 48 degrees north, 78 degrees west. The Earth’s radius is about 4000 miles. Define a reasonable spherical coordinate system for Earth, and give Geneseo’s position in this system. Then give Geneseo’s position in the rectangular system.
Here it’s definitely easier to start with spherical coordinates and then convert to rectangular. But notice that latitude and longitude don’t measure angles the same way standard spherical coordinates do. In particular, we want to measure the “Φ” angle, analogous to latitude, down from the pole, not up from the equator, and we want to measure Θ from the positive x axis towards the positive z axis, which corresponds to moving east around the globe, not west. These corrections let us give Geneseo’s coordinates in a standard spherical system, and then we used the spherical-to-rectangular equations to convert them to rectangular:
We now have a basic understanding of geometry and coordinate systems in 3 dimensions, so we can start using it to talk about one of the key building blocks for functions in 3D space: vectors.
Please read “Working with Vectors in R3” in section 1.2 of the textbook.
(Some of what’s in section 1.2 is presented as extensions of 2-dimensional vectors. Use section 1.1 for review of 2D vectors as needed, and/or identify questions about them that you’d like to talk about in Friday’s class.)