Natural Blogarithms

On Wednesday, last week, we completed our material over elementary matrices, using them to derive the inverse of a matrix. Upon proving that a matrix is non-singular (i.e., invertible) if and only if it is row equivalent to the identity, we noticed that the same row operations that change a matrix A into the identity will change the identity into the inverse of A.

We have finally started taking partial derivatives in Calculus IV. This is definitely one of my favorite parts of the entire Calculus sequence. Today, we finished up talking about limits of functions of several variables and the concept of continuity. We, then, introduced the concept of holding one variable fixed and finding the derivative of the function with respect to the remaining variable. After seeing the picture drawn on the board of two curves, or slices of a surface, and identifying the partial derivatives as the “slope” of tangent lines to those two curves, one of the students made a keen observation: [..]

During this morning’s class I surprised the students by letting them know that today was my turn to answer homework questions at the board. As I have mentioned in previous posts, the students are being randomly selected to present some of the homework problems at the board. For many of them this is a considerably stressful event and with today’s assignment being particularly challenging, I noticed that many were very relieved when I offered to answer some of the questions at the board. Plus, it saves me needing to grade those particular problems. [...]

I don’t think it is possible for me to cover a whole section of this linear algebra book in one class. Today, we started the section on Elementary Matrices and just a little ways in, I knew it was going to be a difficult section. [...]

Today in class, the rest of the students finished presenting the short oral reports over the classical arguments for the existence of God. Following this, we delved into the basics of Logic and Reason, covering the basic structures of logic: [...]

Today in Faith and Science we went over a couple of interesting articles concerning how we might integrate Faith into our Science curriculum at Christian universities. One author proposes that scientific research can be directed by the understanding that God is the cause of natural law. The second author responds to the first author, maintaining that the creation must surely have some level of autonomy so that not all things are directly caused by God. They are worth your time if you are interested: [...]

We picked up where we left off prior to the Exam on Tuesday. We are discussing function of several variables. I began by recapping the techniques we use for visualizing the graphs of functions of two variables, namely, surfaces and contour maps, or level curves. We used this as a springboard to move up to functions of 3 variables. [...]

On Thursday, we began the chapter on Real Numbers. We are taking the axiomatic approach to defining the set of real numbers. I demonstrated the difficulty of defining Real Numbers in the techniques that have been typically used in their earlier class, such as by decimal expansion or as points on the real number line. I alluded to the method of actually constructing Real Numbers through the method of Dedekind cuts but I have chosen to follow the text by describing the properties that uniquely define the set. [..]

It took three classes but I finally finished all of the section over Matrix Algebra. During today’s lecture we walked through the concepts of matrix inverse and then matrix transposes. Its amazing how long these lectures stretch out when you choose to demonstrate the concepts with examples involving matrix operations. As far as I can tell, they are all following along very closely, except when I start going off on a tangent and trying to draw connections between linear algebra and higher level mathematics. [..]

During Calculus IV on Tuesday, we had Exam I. I’ve decided that I’m not generally very good at producing tests at the appropriate level for the course. Usually they are much too hard (1 or 2 problems that require a LOT of tedious calculation) or too long (too many problems). The reason is not that I am particularly mean-spirited but that the my writing process is flawed. [..]