# 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.

# Calculus IV: The Chain Rule

In class on Tuesday, we stated and used the chain rule for partial derivative:

For $$z=z(x_1, x_2, \ldots, x_n)$$ be a function of $$n$$ variables, such that $$x_i = x_i(t_1, t_2, \ldots, t_m)$$ is a function of $$m$$ variables for each $$i = 1, \ldots, n$$ then

$$\displaystyle \frac{\partial z}{\partial t_j} = \sum_{i=1}^n \frac{\partial z}{\partial x_i} \frac{\partial x_i}{\partial t_j}$$ for each $$j=1,\ldots, m$$

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.

# 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.

# Calculus IV: Partial Derivatives

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: Can’t we have infinitely many partial derivatives by going in any direction, not just the x or the y direction? Answer: yes, but . . .

We will be covering directional derivatives in couple of sections down the road, and we’ll see that even for those cases we only need the two directions to find all the rest. Next time, we’ll cover linearization, tangent planes, and differentials. Another one of my favorite topics is coming up: the Implicit Differentiation Theorem.

# Calculus IV: CT Scans and the Super Bowl

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. Typically, any function of $$n$$ variables will need $$n+1$$ dimensions to represent the graph of the function. For example, for a function of 2 variables, $$f(x,y)$$ we define its graph to be the set $$\left\{ (x,y,z) \ | \ z=f(x,y) \ \mbox{s.t.} \ (x,y) \in D \right\}$$ where $$D$$ is the domain of $$f$$. Thus, the graph of a function of two variables is a subset of $$\mathbb{R}^3$$.

I discussed a few ways for visualizing these graphs. The techniques I demonstrated went a little beyond what the textbook covers because I have done quite a bit of computer modeling for these types of data sets, especially in my early days at the High Performance Computing Center at Texas Tech. I showed them how to use level surfaces which are the generalization of level curves, or contour maps. I, then, demonstrated the used of animation to help get a better feel for the 4th dimension when using level surfaces. Finally I showed them examples of CT scans where slices are used to show in interior of a volume in three space. I recounted a project that I helped with as a graduate student where we were building a three dimensional computer model of a prosthetic arm using only 2D images which were “slices” of the arm. The primary challenge of that project was that the images were not properly aligned so that we had to apply an optimization technique to translate and rotate the image to minimize the change from image to the next. The result was a decent representation but a little “bumpy” due to the misaligned images.

It reminded me of the Super Bowl several years back where a researcher had been employed to set up cameras around the top of the stadium and have them filming the same point on the field. They they tried to use the cameras to rotate around a play in a given instant, producing the “Matrix” effect, where some player is frozen in mid-air and the camera rotates around them. The end result was kind of cool but still real jumpy. The reason was there was too much computation needed to align the images for rotation. It required too much processing power to do in real time. I wonder if the progress in computer hardware would make that more realistic today. I know they didn’t bring it back after that because it just looked hokey.

Next time, we will finish up the study of limits of functions of several variables and get to start doing partial derivatives. Yeehoo!

# Calculus IV: Exam I

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. I start by writing a review for the test. I generate a solution guide for the review. Then, I come back and make a few minor modifications to the review, as well as adding a couple of new extensions of the problems on the review. The end result is that those minor modifications generally result in something not simplifying where it did before the mod.

With all that said, I was pleased with the Exam I that was given in Calculus IV on Tuesday. It tested over the set of material I was most concerned about. It was just the right length. All the students finished completely and handed in the test within the last 5 – 10 minutes of class. A couple problems were sufficiently difficult to challenge them but they walked away feeling like they had done fairly well. Of course, I have not graded them yet and may have a completely different perspective after that. But for the sake of confidentiality, you will never know. Sorry. But you can look at the test here: calculus-iv-exam-i.pdf

We will begin looking at Partial Derivatives in the next class. Yeehoo!

# Calculus IV: Functions of Several Variables

Having just finished a very short chapter on vector functions, we began the chapter that will cover Partial Derivatives. Before we started that lecture, I took a little class time to answer a homework question. It was one of my favorites. Having covered the concept of curvature, the students were asked to find a polynomial, $$P(x)$$ such that the function

$$$f(x) = \left\{ {\begin{array}{*{20}c} 0 \hfill & {,\,x \le 0} \hfill \\ {P(x)} \hfill & {,\,0 < x < 1} \hfill \\ 1 \hfill & {,\,x \ge 1} \hfill \\ \end{array}} \right.$$$

is continuous, smooth and has continuous curvature. Not hard but still fun to do. Feel free to try it yourself and post the answer in the comments.

We, then, began functions of several variables talking about four ways to describe a function of two variables and then looking at examples of each. (Verbally, Numerically, Graphically, Algebraically) We plotted a few interesting surfaces in Maple 10 and then began discussing level curves (contour maps). Using examples such as weather maps (temperature, pressure, etc.) and topographical maps helped the students get a good feel for how we can look at a 3 dimensional graph in a two dimensional plane (without necessarily resorting to perspective drawings). We’ll start looking at level surfaces, or iso-surfaces, to generalize to functions of three variables. For example, how might you visualize a 4D hypersphere by using iso-surfaces?

For future versions of this lecture, I could definitely improve the lecture by bringing resources to class like video clips of a weather forecast or actual topographical maps of this area. Also, if I have time I’d like to have a demo of the software Viz5d so that I can visualize some data sets with animated iso-surfaces. I used to run a demo of a weather simulation when I worked for the High Performance Computer Center of Texas Tech University. I animated iso-surfaces of water vapor levels in the atmosphere. The result was a three-dimensional model resembling cloud movement.