Math 412, Winter 2004

Introduction and Expectations

When Sir Isaac Newton and the German mathematician Gottfried Leibniz, working independently, invented calculus the germ of one of the most profound accomplishments of the human intellect was present. But their science was imprecise, founded on vague quantities such as the "infinitesimal" and lacking any solid foundation from which those of us without the genious of Newton might hope to learn.

In the early nineteenth century emerged a brash, sometimes nasty, mathematician by the name of Augustin-Louis Cauchy. Cauchy is a name you may have heard before (Cauchy-Euler differential equations, Cauchy sequence, Cauchy-Reimann equations) and his picture graces the front of your syllabus. Among his many achievements (he wrote 789 mathematical papers), Cauchy may be most remembered for putting mathematical analysis back on solid ground. The rigor of Cauchy brought in a new era of structure and precision, while at the same time confirming the genious of Newton more than a century and a half later. As Abel (of abelian group fame) put it:

Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.

Many people term this course "Advanced Calculus". While we will certainly follow Cauchy, rigorously going through much of the classical calculus you already know, this class is so much more than that. This is a first serious course in mathematical analysis. Along with the classics we will visit many new ideas, enjoy many new vistas and think most deeply. As you can imagine, such rewards come only at great effort. For some of you this may be the most difficult math class you have ever taken.

Should you be intimidated by the material? No. But each of us should maintain a healthy respect for the centuries of work that have brought us to this moment. And each of us should realize that this course will take an immense amount of dedication and concentration. To those hearty souls who accept the challenge with a smile, I say welcome. Now on with it.