# Friday Random 10 (late)

Sorry I missed Friday’s Random 10. That makes this, technically, a Monday Random 10 but we’ll overlook it this once.

1. “Chevette” by Audio Adrenaline (Hit Parade)
2. “More Than the Watchmen” by Jeremy Casella/Rebecca St. James (Worship God)
3. “What If I Stumble?” by dc Talk (Jesus Freak) – One of my All-Time Favorites!!!
4. “I Thank You, Lord” by Alicia Williamson (New Orleans Homecoming)
5. “He Understands My Tears” by The Isaacs (New Orleans Homecoming)
6. “He Didn’t Have To But He Did” by Shiloh Quartet (Jesus Knows Who I Am)
7. “Sixteen Candles” by The Crests (Happy Days 50′s And 60′s)
8. “We Will Dance” by David Ruis (Refiner’s Fire: 25 Top Vinyard Worship Songs Disc 2)
9. “Quiet You With My Love” by Matt Bronleewe/Rebecca St. James (Worship God)
10. “Just to be with You” by Third Day – This makes two from the all-time favorites list making it onto the random 10

In light of a couple of results on this list, I am going to have to start keeping track of the Official All Time Favorites List as they appear on the Friday Random 10. Once this list reaches a significant number, it will become an official page on this blog.

# Calculus IV: Directional Derivatives

We finished the concept of directional derivatives, introducing the notation for the gradient of a function of several variables. We proved the formula of the maximum value of the directional derivative, as well as the direction for which it is maximized.

Next time we’ll begin basic optimization theory for functions of several variables.

# 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:

1. 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)
2. 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.)
3. Given the matrices, $A$ and $C$, solve the matrix equation: $XA + C = X$.

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 $\mathbf{x}$-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.