Sir Isaac Newton, 1643-1727 | Gottfried Wilhelm von Leibniz, 1646-1716 |
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The goal of the course is to show why calculus has served as the principal quantitative language of science for more than three hundred years. How did Newton and Leibniz transform a bag of tricks into a powerful tool for both mathematics and science? Why is calculus so useful in geometry, physics, probability and economics? Why are mathematicians so concerned with rigor in calculus? Since calculus is about calculating, what is the relationship between calculus and computers? What is the relationship between calculus and new topics like chaos and nonlinearity? If you want to understand what calculus is really about, then this is the course for you.
Ancient peoples, driven by natural curiosity and the demands of applications, confronted the problems of finding areas and volumes of various shapes. Their methods of solving these problems may be regarded as precursors to integration. Outstanding in this regard was the work of the Greeks, exemplified by Archimedes' solutions to numerous problems of quadrature, and the works of the Chinese mathematicians Liu Hui and Zu Chongzhi. Concepts resembling differentiation did not arise until much later. Precursors to differentiation can be recognized in the work of Fermat and Descartes on tangents, and finding maxima and minima.
Greek mathematics separated algebra and geometry. The invention of Cartesian coordinates and modern symbolism allowed Newton and Leibniz to create calculus. Our focus is on how they used the Fundamental Theorem of Calculus to transform a bag of tricks into a powerful tool.
The power of calculus was clear from the start. It achieved spectacular successes in geometry, physics, probability and economics.
The foundations of calculus were not secure at the time of invention, and the limitations of calculus were obvious to many critics. However, mathematicians gradually succeeded in putting calculus on a firm foundation. This required developing a clear understanding of infinity.
Calculus is fundamentally a theory of continuous objects. However, in many applications the theory is extended through the interplay between discreteness and continuity. Discrete problems may be described by continuous models and discrete methods may be applied to solve continuous problems.
Part of the success of calculus comes from the fact that we can simplify problems by linear approximations. For example, we look at the tangent line rather than the whole curve. However, calculus is about calculating and with the advent of computers it has become possible to attack non-linear problems. The contrast between linearity and non-linearity shows up in the new and unexpected world of chaos.
It is recommended that you have seen some calculus already. Like knowing how to differentiate and integrate polynomials. But there are no formal requirements.
There will be two hours of lectures and two hours of tutorials a week.
The final exam counts 60% of your grade. You have to write a project that counts 40%. A lot of topics will only be touched upon in lectures, and I hope that you will explore them further on your own in the projects. I will provide a list of possible topics, but I also encourage you to propose your own topics and send them to me for approval. The project can be a normal paper project, or it can be a web page with animations and graphics.
The tomb of Archimedes, 287-212 B.C. |
Various strands had to come together before “the Calculus” developed into what we recognize it to be today - a coherent method that can be applied algorithmically to solve certain types of problems. These strands include psychological and philosophical shifts no less than the discovery of new mathematical techniques.
The Lorenz Attractor in 3D by Paul Bourke |
Applications of calculus
Problems in economics usually involve discrete variables, but we can treat them as continuous variables and use the tools of calculus. Computers are inherently discrete, but we can still use them to approximate continuous problems. In a similar manner, integration, as in the Greek method of exhaustion, starts out as a discrete approximation before we take a limit.
Calculus is about calculating, and we show how computers have changed mathematics. One of the basic ideas behind calculus is linearization, but thanks to computers, nonlinearity is now a hot topic. This has lead to chaos theory. Finally, we discuss how computers have brought continuous and discrete mathematics closer together.
A nowhere differentiable function by Douglas N. Arnold |
Week | Topic | Tutorial | Project |
---|---|---|---|
1 | Precursors to Integration | ||
2 | Precursors to Integration | ||
3 | Precursors to Differentiation | 1 | |
4 | Symbolism | 2 | |
5 | Calculus: Putting it All Together | 3 | Proposal for Project due |
6 | Recess | ||
7 | The Power of Calculus | 4 | |
8 | The Power of Calculus | 5 | |
9 | Infinity and Infinitesimals | 6 | |
10 | Discreteness and Continuity | 7 | Project due |
11 | Linearity and Nonlinearity | 8 | |
12 | Linearity and Nonlinearity | 9 | |
13 | Revision |
Most of the topics we talk about in lecture can be extended to projects. Here are some suggestions. I also encourage you to propose your own topics and send them to me for approval.
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