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24 October 2003

A Reality Check and a Look Inside of Mathematics

by George E. Hrabovsky, President, MAST

Where We Have Been

In the last year we have covered a lot of ground.  We have covered a lot of the salient points of set theory, a little bit of algebra, and lots of calculus. Last week I introduced integration.

What Is It All Good For?

As a theoretical physicist I can count on one hand the number of times I have actually had to calculate a derivative in the last year, once. As a mathematician I don't remember the last time. Does that mean that we have wasted our time? Not at all. The real reason for knowing how to calculate derivatives is knowing what the properties of a derivative are.

Last week I introduced the notion of the integral as the limit of the sum of the values underneath a curve. The truth is that understanding that makes integration understandable. One the other hand no one really ever calculates an integral that way. It will become clear how to actually get an integral. The usefulness of the expression,

∫_0^af(t) t = Underscript[lim, Δ t0] Underoverscript[∑, i = 1, arg3] f(t_i) Δ t_i

the right hand side of which is called the Riemann sum, is in proving statements about integrals.

Some time ago I wrote a Mind of a Theorist column on the most effective way to learn mathematics. It was entitled, "Coursing Through Mathematics." I will recreate some of that column here and use some of the details that we have covered over the last year to illustrate what I mean.

How to Approach Mathematics, Revisited

When we study mathematics it is tempting to just think of it as a collection of tools that merge or diverge depending upon the application under study. This seems to be how lower-level math courses are taught, like calculus. Even in higher-level courses we rarely get to see the overall picture until it is too late. Almost always we, as students, are left to motivate the discussions and lecture topics for ourselves. For the non-mathematician this can be stultifying, and makes mathematics harder than it needs to be (and it is pretty hard at the best of times.) Mathematics seems to be a bewildering array of topics that require genius to really understand. The professors/authors look smarter than we could hope to be as they reverse the order of a derivation and seem to work magic (because they know the answer already and can supply it, then show it is true, while giving us no real way to get the answer they got from first principles). It is a pretty dismal state of affairs...

Mathematics does have a flow to it. It is designed to meet certain needs that are unique to the branch of mathematics under study, and at the same time one branch can look eerily similar to other branches of mathematics. Mathematics is concerned with three primary topics: mathematical objects, relationships between these objects, and showing when these objects or relationships are equivalent. Keeping these three points in mind you can learn a lot about any branch of mathematics by answering five questions:

1. What are the objects used in this branch of mathematics? For calculus the objects are variables and new variables created by manipulating these variables through the use of the limit, the derivative, and the integral.

2. What are the principal relationships between the objects? For calculus the relationships are functions. We can think of a derivative as a kind of function because it transforms one set of variables into another set. Integration will also be seen as a kind of function, too.

3. What are the criterion for equivalence? For calculus we establish differential equations and both differential and integral inequalities.

4. What is the central goal of the branch of mathematics? For differential calculus the goal is to calculate the derivative (rate of change) of a function by using a limiting process. For integral calculus the goal is to calculate the integral (area under a curve) of a function (also by using a limiting process). Both of these goals tell us about the behavior of functions.

5. What are the principle methods used to achieve the goal? For differential calculus the method is to take the rate of change of the function as the change in the independent variable tends toward zero. For integral calculus the method is to sum up the areas of vertical slices under a curve as the width of the slices tends to zero.

These capsule answers are not sufficient to develop an understanding of mathematics. When you try to learn mathematics keep them in mind. As you read a paragraph of text ask yourself these questions over and over again. Did the paragraph reveal a new mathematical object? A new relationship between objects? A new equivalence (or condition when things are not equivalent)? A new goal? A new method?

Do not stop with these questions, though. Always demand that any claim be shown to be true clearly. Do you understand the proof of a theorem? Can you explain it to an imaginary person in front of you? Try it! If you don't understand something try looking at it from a different point of view. Try different approaches, see if they solve the difficulty. Try applying the idea to a simple case, to extreme cases, to problems where you already know the answer.

The idea here is to get you to do something other than simply reading or listening to a lecture. Mathematics is not a spectator sport. In the same way that you will never become an athlete by watching athletes (you have to actually do the work), mathematics requires that you develop expertise not just in the ideas, but also the methods.

The Math Challenge from Last Time

The challenge was to prove two theorems.  The first states that if f has an extreme value at a point x_0, then either f is not differentiable at the point or f ' (x_0) = 0.  It seems to me the way to do this proof is to focus on the first -derivative test as a definition of an extreme value. If the first derivative vanishes then we have an extreme value. It seems to me that there are only two circumstances when the derivative vanishes; either f is not differentiable at the point or f ' (x_0) = 0. Thus the theorem is proven by definition.

The second theorem states that if x_0 is a critical point of f and if f'' (x_0) exists and is nonzero, then f has a minimum at the critical point if f'' (x_0) >0 and a maximum if f'' (x_0) <0. This one is a bit more challenging.  Since we have specified that the second derivative exists, we can define it this way,

f'' (x_0) = Underscript[lim, xx_0] (f ' (x) - f ' (x_0))/(x - x_0)

Since f ' (x_0) = 0 since x_0 is a critical point,

f'' (x_0) = Underscript[lim, xx_0] (f ' (x) )/(x - x_0) .

So long as f'' (x_0) >0 then f ' will change from negative to positive at x_0, thus there is a local minimum. So long as f'' (x_0) <0 then f ' will change from positive to negative at x_0, thus there is a local maximum. Thus we have proven the theorem.

The Math Challenge

Speculate on how integration can become relatively easier than using Riemann sums.

Math Resources for Integrals

Online:

http://www.ma.utexas.edu/users/kawasaki/mathPages.dir/

This is a neat site.  For a more challenging page look at this one:

http://www.mathpages.com/home/icalculu.htm

Both sites are very cool.  Another very nice site is:

http://www.ugrad.math.ubc.ca/coursedoc/math101/

Here is an online table of integrals

http://www.math2.org/math/integrals.htm

Here is a site powered by WebMathematica:

http://www.calc101.com/

Of course there is the Wolfram research Integrator:

http://integrals.wolfram.com/

Books:

Richard A. Silverman, 1969, Modern Calculus and Analytic Geometry, MacMillan Company, New York (Dover Publications has reprinted this book with corrections in 2002). This has a rigorous, but readable chapter on derivatives and another very good one on extrema of functions.

Created by Mathematica  (October 23, 2003)