-
Most educated people have heard of Physicists such as Einstein, Heisenberg, Fermi, etc., and have some rough ideas about their works. More would be familiar with at least the names of the the major composers, painters, and writers not only of the past but of the present. Yet how many can name a couple of first-rank Mathematicians in the previous century or identify even a single concept associated with such great geniuses of the past as Gauss, Cauchy, or Riemann?
 The Giants of Calculus PDF document
 Mathematics is a study of structures and structure preserving maps, and after a preliminary introduction to some basic "abstract mathematics" --- say an undergraduate course in abstract algebra, mathematical analysis, topological spaces, one comes to see the meaning behind this statement. Category Theory makes it possible to understand this very effectively and economically and applies this to a coherent understanding of mathematics. On studying categories one very soon sees the barriers between the various branches disappearing in the oblivion and then sees a beautiful harmony and coherence in the mathematical processes and thought. This is a basic course which introduces to the five basic entities of Category Theory: universality, representability, adjunction, limits and Kan extensions. Furthermore, in this course we shall apply these notions to diverse branches, sometimes providing beautiful proofs of well known results, so that the reader may get convinced of the goal of Category Theory. To conclude: the dream of Category Theory is to produce a coherent presentation of all of mathematics so that mathematical development would seem unified and not broken up in small pieces and bits everywhere around!
Analysis is one of the largest divisions of modern mathematics (Algebra, Geometry, Topology, are some others). It is the study of limits. Anything in mathematics which has limits in it uses ideas of analysis. The idea of taking limits is very old, and just as old are some of the problems which limits can raise. The ancient Greeks calculated the area of a circle using a limiting argument. But they also knew of Zeno's Paradox, which uses a subtle misunderstanding of limits to "prove'' that all motion is impossible. The modern study of analysis grew out of a dissatisfaction with the early intellectual basis of calculus. Because the idea of limits was notorious for leading to paradoxes (like Zeno's Paradox), or else to two different answers, with equally plausible justifications, analysis was developed to put calculus and limits on a firm footing. The heritage of analysis, for this reason, is rigor. The way you can be confident that you are on the right path, is that every little step can be checked carefully, and every assumption can be traced back to a few simple rules (axioms).
- Multiple Integrals in the sense of Riemann, Revision (The concept of Definite Integral, Reduction to Repeated Integrals, Change of the Variables, Volumes and Areas).
Integration in Scalar and Vector Fields (Scalar and Vector Fields, Paths, Line Integrals of I and II order, Relations between Line Integrals and Multiple Integrals, Applications). - Integral Functions (Improper Integrals — brief introduction, Integrals depending on parameter, Calculating Improper Integrals, Euler Integrals).
Sequences and Series of Functions ( Series of Numbers, Uniform and Conditional Convergence of Sequences and Series of Functions, Power Series, Fourier Series).
| Skip Calendar
| Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |
|---|
| 1 |
2 |
3 |
4 |
5 |
6 |
7 |
| 8 |
9 |
10 |
11 |
12 |
13 |
14 |
| 15 |
16 |
17 |
18 |
19 |
20 |
21 |
| 22 |
23 |
24 |
25 |
26 |
Today Friday, 27 January 27 |
28 |
| 29 |
30 |
31 |
| | | |
|
