Calculus

Why study Calculus? - Now that's a good question!

Will this be on the exam?

The question I am asked most often is, "why do we study this?" (or its variant, "will this be on the exam?"). Indeed, it's not immediately obvious how some of the stuff we're studying will be of any use to the students. Though some of them will eventually use calculus in their work in physics, chemistry, or economics, almost none of those people will ever need prove anything about calculus. They're willing to trust the pure mathematicians whose job it is to certify the reliability of the theorems. Why, then, do we study epsilons and deltas, and all these other abstract concepts of proofs?

Well, calculus is not a just vocational training course. In part, students should study calculus for the same reasons that they study Darwin, Marx, Voltaire, or Dostoyevsky: These ideas are a basic part of our culture; these ideas have shaped how we perceive the world and how we perceive our place in the world. To understand how that is true of calculus, we must put calculus into a historical perspective; we must contrast the world before calculus with the world after calculus.

The earliest mathematics was perhaps the arithmetic of commerce: If 1 cow is worth 3 goats, how much does 4 cows cost? Geometry grew from the surveying of real estate. And so on; math was useful and it grew.

The ancient Greeks did a great deal of clever thinking, but very few experiments; this led to some errors. For instance, Aristotle observed that a rock falls faster than a feather, and concluded that heavier objects fall faster than lighter objects. Aristotle's views persisted for centuries, until the discovery of air resistance.

The most dramatic part of the story of calculus comes with astronomy. In the 1600's the earth was the center of the universe. Each day, the sun rose in the east and set in the west. Each night, the constellations of stars rose in the east and set in the west. The stars were fixed in position, relative to each other, except for a handful of "wanderers," or "planets". The motions of these planets were extremely erratic and complicated. Astrologers kept careful records of the motions of the planets, so as to predict their future motions and (hopefully) their effects on humans.

Newton and Liebnitz

In 1543 Copernicus published his observations that the motions of the planets could be explained more simply by assuming that the planets move around the sun, rather than around the earth -- and that the earth moves around the sun too; it is just another planet. This makes the planets' orbits approximately circular. The Catholic church did not like this idea at all.

During the years 1580-1597, Brahe and his assistant Kepler made many accurate observations of the planets. Based on these observations, in 1596 Kepler published his refinement of Copernicus's ideas. Kepler showed that the movements of the planets are described more accurately by ellipses, rather than circles. Kepler gave three "laws" that described, very simply and accurately, many aspects of planetary motion:

the orbits are ellipses, with the sun at one focus
the velocity of a planet varies in such a way that the area swept out by the line between planet and sun is increasing at a constant rate
the square of the orbital period of a planet is proportional to the cube of the planet's average distance from the sun.

The few people who understood geometry could see that Kepler had uncovered some very basic truths.

In 1609 Galileo took a "spyglass" -- a popular toy of the time -- and used it as a telescope to observe the heavens. He discovered many celestial bodies that could not be seen with the naked eye. The moons of Jupiter clearly went around Jupiter; this gave very clear and simple evidence supporting Copernicus's idea that not everything goes around the earth. The church punished Galileo, but his ideas, once released to the world, could not be halted.

Galileo also began experiments to measure the effects of gravity; his ideas on this subject would later influence astronomy too. He realized that Aristotle was wrong -- that heavier objects do not fall faster than light ones. He established this by making careful measurements of the times that it took balls of different sizes to roll down ramps. There is a story that Galileo dropped objects of different sizes off the Leaning Tower of Pisa, but it is not clear that this really happened.

Some of the most rudimentary ideas of calculus had been around for centuries, but it took Newton and Leibniz to put the ideas together. Independently of each other, around the same time, those two men discovered the Fundamental Theorem of Calculus, which states that integrals (areas) are the same thing as antiderivatives. Though Newton and Leibniz generally share credit for "inventing" calculus, Newton went much further in its applications. A derivative is a rate of change, and everything in the world changes as time passes, so derivatives can be very useful. In 1687 Newton published his "three laws of motion," now known as "Newtonian mechanics"; these laws became the basis of physics.

If no forces (not even gravity or friction) are acting on an object, it will continue to move with constant velocity -- i.e., constant speed and direction. (In particular, if it is sitting still, it will remain so.)
The force acting on an object is equal to its mass times its acceleration.
The forces that two objects exert on each other must be equal in magnitude and opposite in direction.

To explain planetary motion, Newton's basic laws must be combined with his law of gravitation:

the gravitational attraction between two bodies is directly proportional to the product of the masses of the two bodies and inversely proportional to the square of the distance between them.

Newton's laws were simpler and more intuitive as Kepler's, but they yielded Kepler's laws as corollaries, i.e., as logical consequences.

Newton's universe is sometimes described as a "clockwork universe," predictable and perhaps even deterministic. We can predict how billiard balls will move after a collision. In principle we can predict everything else in the same fashion; a planet acts a little like a billiard ball.

Suddenly the complicated movements of the heavens were revealed as consequences of very simple mathematical principles. This gave humans new confidence in their ability to understand -- and ultimately, to control -- the world around them. No longer were they mere subjects of incomprehensible forces. The works of Kepler and Newton changed not just astronomy, but the way that people viewed their relation to the universe. A new age began, commonly known as the "Age of Enlightenment"; philosophers such as Voltaire and Rousseau wrote about the power of reason and the dignity of humans. Surely this new viewpoint contributed to

portable accurate timepieces, developed over the next couple of centuries, increasing the feasibility of overseas navigation and hence overseas commerce
the steam engine, developed over the next century, making possible the industrial revolution
the overthrow of "divine-right" monarchies, in America (1776) and France (1789).

Perhaps Newton's greatest discovery, however, was this fact about knowledge in general, which is mentioned less often: The fact that a partial explanation can be useful and meaningful. Newton's laws of motion did not fully explain gravity. Newton described how much gravity there is, with mathematical preciseness, but he did not explain what causes gravity. Are there some sort of "invisible wires" connecting each two objects in the universe and pulling them toward each other? Apparently not. How gravity works is understood a little better nowadays, but Newton had no understanding of it whatsoever. So when Newton formulated his law of gravity, he was also implicitly formulating a new principle of epistemology (i.e., of how we know things): we do not need to have a complete explanation of something, in order to have useful (predictive) information about it. That principle revolutionized science and technology.

That principle can be seen in the calculus itself. Newton and Leibniz knew how to correctly give the derivatives of most common functions, but they did not have a precise definition of "derivative"; they could not actually prove the theorems that they were using. Their descriptions were not explanations. They explained a derivative as a quotient of two infinitesimals (i.e., infinitely small but nonzero numbers). This explanation didn't really make much sense to mathematicians of that time; but it was clear that the computational methods of Newton and Leibniz were getting the right answers, regardless of their explanations. Over the next couple of hundred years, other mathematicians -- particularly Weierstrass and Cauchy -- provided better explanations (epsilons and deltas) for those same computational methods.

It may be interesting to note that, in 1960, logician Abraham Robinson finally found a way to make sense of infinitesimals. This led to a new branch of mathematics, called nonstandard analysis. Its devotees claim that it gives better intuition for calculus, differential equations, and related subjects; it yields the same kinds of insights that Newton and Leibniz originally had in mind. Ultimately, the biggest difference between the infinitesimal approach and the epsilon-delta approach is in what kind of language you use to hide the quantifiers:

The numbers epsilon and delta are "ordinary-sized", in the sense that they are not infinitely small. They are moderately small, e.g., numbers like one billionth. We look at what happens when we vary these numbers and make them smaller. In effect, these numbers are changing, so there is motion or action in our description. We can make these numbers smaller than any ordinary positive number that has been chosen in advance.
The approach of Newton, Leibniz, and Robinson involves numbers that do not need to change, because the numbers are infinitesimals -- i.e., they are already smaller than any ordinary positive number. But one of the modern ways to represent an infinitesimal is with a sequence of ordinary numbers that keep getting smaller and smaller as we go farther out in the sequence.

To a large extent, mathematics -- or any kind of abstract reasoning -- works by selectively suppressing information. We choose a notation or terminology that hides the information we're not currently concerned with, and focuses our attention on the aspects that we currently want to vary and study. The epsilon-delta approach and the infinitesimal approach differ only slightly in how they carry out this suppression.

A college calculus book based on the infintesimal approach was published by Keisler in 1986. However, it did not catch on. I suspect the reason it didn't catch on was simply because the ideas in it were too unfamiliar to most of the teachers of calculus. Actually, most of the unfamiliar ideas were relegated to an appendix; the new material that was really central to the book was quite small.

Adapted from an essay by Eric Schechter
version of August 23, 2006

About the class

The book we will be using is Calculus and Analytical Geometry by Larson, Hostetler and Edwards. This will be the same book that is used for Calculus I also. First semester covered differentiation, integration, and their applications to our world around us.

calculus text

This class will meet M,W,F from 10:00 - 10:50 in my office. The chapters that will be covered in second semester are:

1. L'Hopital's rule
2. Inverse and hyperbolic functions
3. Techniques of integration
4. Applications of integration
5. Differential Equations
6. Sequences and Series