Intermediate Analysis: Supremums and Infimums
Thursday morning, three students presented homework problems at the board. I followed this with a lecture introducing the concepts of supremum and infimum of a set. Because I got a little sidetracked, I did not quite make it to the Completeness Axiom.
What was the "rabbit chase" for this class? Well, at least a couple of students had commented to me, outside of class, on the difficulty they've had with recent homework assignments. They pointed out that they work and work and often can't make any headway on a few of the proofs. They seemed a little discouraged by the fact that they need help from their professor on every assignment.
I took class time to reassure them that they are not alone. Just about everyone in the class is going through the same thing. I pointed out the fact that I was in their place not that long ago. In fact, since I also did my undergrad here at Wayland, I was almost exactly in their place. I recounted tales of my discouragement as well as the fact that I also spent time in my professor's office get help on almost every assignment. I was largely motivated by a reader of this blog, who is also a blogger I read regularly. He has recently made the point that the students can gain a new level of insight to a subject by seeing the learning process that the instructor, themselves went through to understand certain concepts.
In my mind, although some may disagree, it does not get any harder as an undergrad than a senior level mathematics course. There are many courses that require as much "work" as a course like this, but I can't think of too many that require such an high level of abstract and critical thinking. I've yet to be convinced otherwise. However, I'll admit that my undergraduate Physical Chemistry class may have been close.
Next time, we'll finally cover the completeness axiom.
Linear Algebra: Polynomial Interpolation
Today, in class, I answered some homework questions from Section 1.4, including the following problems:
- Show that an upper triangular matrix with non-zero diagonal entries is nonsingular. (We've not covered determinants yet so we used a row-equivalence to the Identity argument)
- Show that the inverse of an upper triangular matrix is upper triangular. (We used the fact that the same row operations that reduce a matrix to the identity will reduce the identity to the inverse of a matrix. The row operations that reduce an upper triangular matrix to the identity will necessarily change the identity into an upper triangular matrix.)
- Given the matrices,
and 
, solve the matrix equation: 
.
The last homework example was over the Vandermonde matrix system, proving that the Vandermonde system is equivalent to polynomial interpolation. We also proved that if the values of the 
-vector (independent variables) are distinct, then the Vandermonde matrix is non-singular. As you might imagine, I got very excited and energetic about explaining this example. Polynomial interpolation was one of the first topics from Numerical Analysis that I fell "in love" with. It set me down the path to becoming a numerical analyst.
Originally, we had a text scheduled for the next class but because I spent a significant portion of class time on polynomial interpolation, I decided to postpone the exam until after I can answer some questions from the review and last section's homework. We'll do this next time and if there are no questions we'll move on to the next chapter: a short one on determinants.
Calculus IV: The Chain Rule
In class on Tuesday, we stated and used the chain rule for partial derivative:
For 
be a function of 
variables, such that 
is a function of 
variables for each 
then
for each 
We derived the result called the Implicit Differentiation theorem which gives you a shortcut to perform (single-variable) implicit differentiation using partial derivatives.
We just got started talking about directional derivatives. Next time will define the gradient and begin to show some simple useful results involving it. Then, we'll be able to begin studying optimization of functions of several variables.
Intermediate Analysis: Wayne’s World
What does "Wayne's World" have to do with Real Analysis?
Good question. I inadvertently structured my lecture in such a way that it reminded me of the old saturday-night-live-turned-box-office-money-maker. Both in the sketch and the movies, Wayne and Garth would broadcast their rock-n-roll show from Wayne's basement. Anyways, whenever they would do a dream segment or a "back in time" segment, they would mimic the "fuzzy" screen effect and dreamy music my waving their hands frantically and saying something to the effect of "doodle-oodle-oodle-..." (DISCLAIMER: By referring to Wayne's World, I in no way condone some of the associated raunchy humor, even though I probably laughed at some of it - shame on me.)
During class, I started a section of material and after beginning a proof, realized that we had not yet covered some essential material. Thus, we had to do a dream sequence, much like Wayne's World or any other movie where you see the end of the story and then go back in time to see how you got there. How about that for interesting use of lecture style?
I had begun the definition of an 
-neighborhood and in trying to explain exactly what it was, I realized that we had some left some important results of the absolute value had not been established. Those results are well-known facts but we have emphasized the importance of fundamental principles in this class. We use (almost) nothing without first establishing the facts we need through proofs or use of axioms and definitions.
I finished with a few examples demonstrating sets that have the varying properties:
(i) Every point in the set has a an -neighborhood that is contained in the set.
(ii) There is a point in the set that, every -neighborhood of that point contains both elements of the set and elements in its complement.
Next time, we'll give those concepts names after we establish the identifying property of the reals, namely, the completeness axiom. We also have homework due so there will be some oral presentations.
I didn’t lose my cool
Okay, so I almost lost it today. All of my students can attest to the fact that I am generally a very laid back guy. In most cases, my students can give me a hard time in class, try to distract me by encouraging me to chase rabbits in class, they can even jest about how complicated the material is. I generally play along, all in good fun, and even use their comments to springboard into some great illustrations of application.
But every once in a while, I get pushed too far. I don't yell. I don't scream. I don't turn red-faced. I really don't even get angry, nor do I let it distract from the lecture at hand. But I do respond. Today, I responded by making good on an off-hand threat I made last week: "If they continue to deride this subject and make attempts to get me to dismiss class early, I will begin adding problems to their homework." In other classes, I have simply invited the students to leave and drop the course if they are so disinterested in making an effort.
Now, for the confession. I honestly feel bad about "punishing" the entire class for the offense of a couple of students. I put the word in quotes because, honestly, more homework helps them better understand the topic, but I know that in the eyes of the students, it was a punishment. One rationale for increasing the homework load based on the behavior of a couple of students is that it will curb that sort of behavior in the future, both from them and anyone else. But at the same time, there are not very clear boundaries. I allow the regular discussions of "Why is this difficult?" or "How can we use this?". I often play along with their joking about dismissing class early, even though they all know how improbable it is. Nevertheless, after every theorem and every proof, is too often.
Despite by doubts, the current assignment will stand as it is now, after the added problems. If there are future such occurrences, I will recognize the fact it failed to meet the goals of stemming this behavior for the entire class and I will likely direct the "punishment" toward any offending individuals. In the end, I hold no grudges or ill will. Any students involved in today's class have clearly demonstrated their commitment to the mathematics program as a whole so I am not upset with anyone. The primary goal is that the students learn a lesson: Even though there is a casual atmosphere in the classroom (or workplace), we maintain a certain level of respect for the course (or job) and for the professor (or employer).
Linear Algebra: Block Matrix Multiplication
In class today, we finished the section on partitioning matrices. We verified that that block matrices obey the same rules for matrix algebra. In particular, block matrix multiplication works as scalar multiplication as long as the dimensions are appropriate for the sub-matrices to be multiplied together.
We looked at a couple of examples how we can evaluate the structure of larger matrices using block matrices. For example, we prove that the given matrix:
![A= \left[ \begin{array}{cc}A_{11} & O \\ A_{21} & A_{22} \end{array}\right]](/latexrender/pictures/1b62a68571c9e17e7fed4ad84aad5609.png "A= \left[ \begin{array}{cc}A_{11} & O \\ A_{21} & A_{22} \end{array}\right]")
is nonsingular iff 
and 
are nonsingular. We derived that:
![A^{-1} = \left[ \begin{array}{cc}A_{11}^{-1} & O \\[2ex] -A_{22}^{-1}A_{21}A_{11}^{-1} & A_{22}^{-1} \end{array}\right]](/latexrender/pictures/0124ace7f8a4bf56a611ee2e3963ec78.png "A^{-1} = \left[ \begin{array}{cc}A_{11}^{-1} & O \\[2ex] -A_{22}^{-1}A_{21}A_{11}^{-1} & A_{22}^{-1} \end{array}\right]")
I then used a rather manufactured example to demonstrate how we might make use of this fact. Find the inverse of the following matrix:
![A=\left[\begin{array}{cc|ccc} 1 & 2 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 & 0\\ \hline 4 & 3 & 1 & 0 & 0 \\ 1 & 2 & -2 & 1 & 0 \\ 9 & 2 & 0 & 0 & 1 \end{array}\right]](/latexrender/pictures/86f24539b1429418643907aff7157368.png "A=\left[\begin{array}{cc|ccc} 1 & 2 & 0 & 0 & 0 \\ 2 & 3 & 0 & 0 & 0\\ \hline 4 & 3 & 1 & 0 & 0 \\ 1 & 2 & -2 & 1 & 0 \\ 9 & 2 & 0 & 0 & 1 \end{array}\right]")
I made this matrix up so that 
and 
are easily calculable and thus, so is 
.
I just started to touch on outer product expansions and so I'll finish that up next time. The first exam has been scheduled for Monday, February 19. I handed out a review and will post an answer key to the review in Blackboard.
Faith and Science: The Scientific Cycle
In today's class, we were fortunate to have one of my colleagues from our division direct the class. This is the third time this class has been offered at Wayland. The first two times, this faculty member and I co-taught the class. I'll be honest and tell you that being the sole teacher this semester has been an enormous challenge, much greater than the last two. It was extremely helpful to have someone in class bouncing ideas off of.
In the current scenario, I appear much more as the "expert" and less of the facilitator in the discussion. In an applied math course, that would be fine, but in this course, I am reluctant to accept that role. Today's class was like the "good ole days".
The two main topics we covered in class today were the scientific method and the way in which scientific knowledge develops. In discussing the scientific method, we went back over some of the original discussions of the class, centering on the issue of just "how" faith can affect or inform science.
The class proposed that faith can serve much like a feedback loop to weigh the conclusions of science against. Most of them were uncomfortable thinking of faith as a filter through which we choose to accept or deny the claims of science. Instead, it was determined that our faith and/or worldview helps us to evaluate the conclusions reached by science. In most cases, excepting bad experimental design, we don't throw out the results (or data) but we can choose to re-evaluate the conclusions as long as they are rational and justifiable.
This is the true challenge in letting one's faith affect scientific endeavor. The real difficulty is how you draw the line between allowing for, say, supernatural explanations (if at all) and allowing for paranormal explanations. In a classroom full of Christians with a strong conservative backgrounds, most, if not all, of us are comfortable in accepting the supernatural's involvement in our world. However, far fewer of us would be willing to accept the paranormal.
Thus, in evaluating scientific conclusion we must answer the question of what provides the "best" explanation. Some answers to that question have been posed such as Occam's Razor or the fact that the natural trumps the supernatural for all cases. We did not arrive at an answer that satisfied everyone.
We discussed the scientific method in greater detail. We followed that by a short discussion on the nature of scientific development/revolution as proposed by Thomas Kuhn in his book, The Structure of Scientific Revolutions
Next time, we will begin to cover some specific case studies of the interaction of faith and science throughout the history of science.
We will also discuss some of the claims made in Eric Snow's paper: "Christianity: A Cause for Modern Science". In this article, he gives a summary of a couple of papers in which the authors contend that either Christianity helped to create modern science through its worldview, or at the very least, aided in its development. They also contend that other cultures' worldviews stifled the development of modern science, giving examples from China, India, Islam and others.
Faith and Science: Symbolic Logic
Almost all of today's class was spent using logical equivalences and rules for inference in their symbolic form to verify the validity of various arguments. One of my favorites was the following:
If the Mosaic account of cosmogony (the account of the creation in Genesis) is strictly correct, the sun was not created till the fourth day. And if the sun was not created till the fourth day, it could not have been the cause of the alternation of day and night for the first three days. But either the word “day” is used in Scripture in a different sense from that in which it is commonly accepted now or else the sun must have been the cause of the alternation of day and night for the first three days. Hence it follows that either the Mosaic account of the cosmogony is not strictly correct or else the word “day” is used in Scripture in a different sense from that in which it is commonly accepted now.
We label the various statements that make up this argument by 
. Thus the argument takes the form:
The proof goes like this:
Next time, a colleague from the Division of Mathematics and Sciences will take over for a couple classes, helping us to understand the scientific method cycle and the historical development of modern science.
Calculus IV: Tangent Planes and Linearization
In Calculus IV on Thursday, February 8th, we covered the derivation of tangent planes. We also showed how the tangent plane for functions of 2 variables generalizes to functions of several variables, a process we call linearization.
We also used this concept of linearization to define differentiability for functions of several variables. We also covered the concept of differentials.
I know that in this day and age, with technology so available, I should be utilizing some tools on the computer to draw these pictures that help us visualize these concept. Honestly, I just really enjoy drawing the pictures by hand. Having taught this course for several years now, I think I'm getting pretty good at it. I'd post a picture, but I wouldn't want anyone to disagree with me and hurt my feelings. 
Next time, we cover the chain rule and will start directional derivatives.
Intermediate Analysis: More order principles and absolute values
We had no homework assignments due on Thursday, so I dove right into lecture. We proved a few more results following from our ordering of the reals. Upon completing these theorems, we were then able to do the basic sorts of solving of inequalities we teach in our lowest level of algebra. It is interesting that we get to one of our final senior undergraduate courses and start doing things we cover in the earliest mathematics courses, but this time, we've built our understanding of these rules from the ground up. Have no fear, analysis students, we will go much, much beyond those mathematical tools we used intermediate algebra.
Next, we defined the concept of absolute value and some simple results of that definition. The next big thing after that was to use absolute values to create -neighborhoods which will provide us the necessary structures to define open and closed sets, cluster points, limits, etc.
Before we get to those concepts, however, we will divert to another low-level algebra topic of solving absolute value equations and inequalities. That will happen next time at the very beginning of class.