# 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.

# 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 $\varepsilon$-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 $\varepsilon$-neighborhood that is contained in the set.
(ii) There is a point in the set that, every $\varepsilon$-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.

# Online LaTEX Renderer

LaTeX Equation Editor

This little tool could come in real handy. I chose to use WordPress as my blog engine so that I could make use of the LatexRender plugin, but there have been occasions where I’d have liked an additional tool like this. Seems to work quite well.

(And for those of you that aren’t aware, LaTEX is the tool most mathematicians make use of for typesetting)