SUNY Geneseo Department of Mathematics
Monday, October 3
Math 223 01
Fall 2022
Prof. Doug Baldwin
Part A asks you to evaluate a derivative as limh→0 f(t+h) - f(t) / ...
Part B asks you to evaluate the same derivative using scalar differentiation rules on each component (which is what Theorem 2.2.1 allows you to do).
Another reference to Theorem 2.2.1, this time suggesting that you can apply derivatives to the components of vectors, then use scalar rules, e.g., the sum/difference rules, to complete the proof.
The proof is likely to end up being a series of equations involving the functions r(t) and u(t), you don’t need to provide specific functions as examples.
The math department’s annual research weekend is coming up October 14 - 15.
It’s a chance to spend roughly a day getting a taste of mathematics research, led by a visiting mathematician. It’s a great experience if you haven’t done the sort of math that no-one knows the answer to yet, but you might be the one to find it.
This year’s leader is Prof. Allan Knutson, from Cornell.
Prerequisites are Proofs (Math 239) and Linear Algebra 1 (Math 233).
Apply to participate by October 6 — see email you should have gotten from Prof. Ngoma for a link to the application form and more information, or talk to him or Dr. Almomani directly.
Prof. Knutson will also be giving a colloquium talk that gives an introduction to the research weekend topic on Friday, October 14.
Based partly on “Graphing Functions of Two Variables” in section 3.1 of the textbook, with extensions to Mathematica described below.
I won’t ask you to create plots of surfaces or level curves by hand on problem sets. But I might ask you to do related things (e.g., find the equations for certain level curves, find places where 2-variable functions reach maximum or minimum values), and I’ll almost certainly ask you to create plots with Mathematica.
Mathematica’s Plot3D and ContourPlot functions are very useful for plotting surfaces defined by 2-variable functions, and level curves, respectively.
Knowing a function’s domain, or the part of the domain in which it has the values you’re interested in, can be helpful in picking x and y ranges.
You’ve already plotted some 2-variable functions with ContourPlot3D, by writing an equation of the form z == f(x,y) and setting appropriate x, y, and z bounds. But there are other ways of plotting 2-variable functions too…
Mathematica provides a built-in function specifically for plotting 2-variable functions, namely Plot3D. This function takes the 2-variable expression to plot, and ranges of x and y values to plot over, as its arguments. For example
Plotd3D[ x^2 + y^2, {x,-2,2}, {y,-2,2} ]See if you can plot the other 2-variable examples from the book, i.e., g(x,y) = √(9 - x2 - y2) and f(x,y) = 16 - (x-3)2 - (y-2)2.
What does the function h(x,y) = (sin x)(cos y) look like near the origin?
Download our Mathematica notebook to see the commands we used and the plots they produced, along with some explanations.
Mathematica can also plot level curves aka contour curves with the built-in function ContourPlot. Given a 2-variable expression and ranges of values for x and y, this function will plot a set of level curves for the function over the given ranges. For example,
ContourPlot[ x^2 + y^2, {x, -3, 3}, {y,-3,3} ]You can tell Mathematica what constants to associate the level curves with by using the Contours option. In its most interesting form this option takes a list of constants to plot the level curves for, for instance…
ContourPlot[ x^2 + y^2, {x, -3, 3}, {y,-3,3}, Contours->{1,3,5} ]What do some level curves for other functions we’ve looked at (e.g., g(x,y) = √(9 - x2 - y2), f(x,y) = 16 - (x-3)2 - (y-2)2, or h(x,y) = (sin x)(cos y)) look like?
The same Mathematica notebook that has our examples of surface plots also has examples we explored of level curves.
Now that you’ve been introduced to multivariable functions, we’ll start working on limits and derivatives of them. We’ll start with limits, so…
Please read section 3.2 of the textbook. But I plan to talk about the basics of limits and continuity tomorrow and recognizing when limits don’t exist Wednesday, so if you only lightly read material on limits that don’t exist (some unlabeled discussion and examples at the end of the “Limit of a Function of Two Variables” subsection), you’ll have another chance at it for Wednesday.