SUNY Geneseo Department of Mathematics
Tuesday, September 6
Math 223 01
Fall 2022
Prof. Doug Baldwin
(No.)
The library is having an open house Sept. 8 (this Thursday). Stop by the second floor of Fraser any time between 11:00 and 1:30 to get some food, pet the therapy dog, etc. (or just to convince yourself that Geneseo really does have a library).
The Math Learning Center is now open. Roughly, the hours are (most of) the morning and (most of) the afternoon Monday through Friday, and afternoons extending into early evenings Sunday through Thursday.
Based on section 1.6 of the textbook.
(None.)
Definition of “cylinder.” A curve “pushed” or extruded infinitely along a set of parallel lines called “rulings.” This is a weird definition compared to what people usually think a cylinder is.
There’s a lot of discussion of relationships between equations and the appearance of their graphs, particularly the kinds of surfaces that different quadrics produce, and their connections to conic sections.
Using traces (i.e, cross-sections parallel to axes) to determine what graphs look like.
Does the surface from the textbook’s Example 1.6.1b (z = 2y2 - x) meet the technical definition of “cylinder”? If so, what would be an equation for a representative ruling, and what would be an equation for a representative curve those rulings pass through?
It does meet the definition. You can judge what the curve and ruling are by imagining what happens if you hold one of x and y constant while letting the other vary (note that this is basically looking at cross-sections of the surface parallel to pairs of axes, i.e., looking at traces).
Does the equation 2(x-1)2 + (y+3)2 + 8(z-2)2 = 8 define a quadric surface? If so, what sort? Can you put the equation into the canonical form for a quadric?
At first glance, it doesn’t seem to fit the form for a quadric (that form doesn’t have (x±c)2 terms for constants c). But expanding out the terms and simplifying shows that the equation is indeed in the right form, and we were even able to do things like divide by the value on the right, and put coefficients in denominators, to make it look vaguely like some of the common forms from the textbook.
Another way of comparing this equation to common forms is to divide both sides of the original by 8. Then you see something that looks like the common form for an ellipsoid, except for the (x±c)2 terms. If you graph the equation with Mathematica, you do indeed see an ellipsoid, but it’s not centered at the origin.
Broader moral: the common forms don’t make very much use of the “xy,” “xz,” or “yz” terms in the general quadric form, nor of the “x,” “y,” and “z” terms. But what those terms do is translate one of the common forms (the “x,” “y,” and “z”) terms, or rotate them (“xy,” “xz,” and “yz”).
Just like it’s sometimes easier to describe curves or functions in 2 dimensions in polar coordinates, it’s sometimes handy to describe curves or surfaces in 3 dimensions in “cylindrical” or “spherical” coordinates.
To get a sense for what these coordinate systems are like, please read section 1.7 of the textbook for tomorrow.