CAAM 401-402 Syllabus: Fall 2008 - Spring 2009

The lectures and homeworks will work through Parts I and II of Lang's text in the fall, all of Part IV in the first half of the spring. The last half of the spring will cover most of Spivak's book and a few topics from Part III of Lang. Except for taking the treatment of multivariate integration from Spivak, the course will follow Lang, but in a rather riverine and nonlinear fashion - we will always make progress, but not always in the same direction, page-wise. This will be especially true of the fall semester; the spring will follow Part IV of Lang more linearly. The structure of the subject, and Lang's well-written book, lend themselves to a variety of tours. I will take you on one that seems compelling to me. It will include a few digressions into topics not covered by either Lang or Spivak - I will distribute supporting material from other sources for those topics.

In the list below, I've identified the sections (sometimes page ranges) in the text which are the source for the lecture material. I use the notation "Chapter.Section [page range]" - for example I.1 [4-5] means Chapter one, section one, pages four through five. You should always read these sections carefully, ideally before the class in which they are covered.

The topics in the list below are not each necessarily limited to one class period - a topic may take up a whole period, or than one, or several.

  1. Basic notions, the integers, induction, countability. 0.1-0.4.
  2. Algebraic properties of the reals (I.1) and of vector spaces (VI.1). R as a vector space over R.
  3. Order properties of R, absolute value and norms (I.2, VI.2).
  4. Integers, rational numbers, and idealized floating point numbers as subsets of R (I.3).
  5. Bases, dimension, linear maps, norms, inner products, norm equivalence theorems. [This material will be supported by readings from other texts, also VI.2.]
  6. Sequences, convergence, Cauchy sequences, complete normed vector spaces, completeness axiom for R, approximation by rationals, floating point numbers (I.4,II.1 (34-39), VI.4 (143-144)).
  7. Bolzano-Weierstrass theorem, (II.1 (39), VIII.1).
  8. Limits (II.2, VII.1).
  9. Continuous functions on subsets of R, intermediate value theorem and its consequences, continuous functions on subsets of normed vector spaces, uniform and Lipshitz continuity, (II.4, VII.2).
  10. Open and closed sets (VI.5), compactness and open covers VIII.4).
  11. Continuous functions on compact sets, proof of the Norm Equivalence Theorem (II.4, VIII.2).
  12. Limits in function spaces, completeness of C0[a,b] (VII.3).
  13. The integral in 1D (X).
  14. The derivative in 1D, relation to integral (III, X).
  15. Integral norms.(VI.3, VII.
  16. Completion of normed vector spaces, Lp spaces. R as completion of Q (VI.4, VII.4).
  17. Compact sets in C0[a,b], Ascoli's theorem [This material will be supported by readings from other texts].
  18. Approximation by polynomials and Weierstrass's theorem.

Topics to be discussed in the spring term (402):

  1. Series (IX).
  2. Calculus in normed vector spaces: the derivative as a linear map (XV, XVII)
  3. The Contraction Mapping Theorem and its applications (XVIII, XIX)
  4. Differentiable Manifolds (Spivak, Calculus on Manifolds)
  5. Integration in Rn(Spivak, Calculus on Manifolds)
  6. Applications of Calculus, drawn from Chapters XI-XIV and other sources

Students with Disabilities: If you have a documented disability that will impact your work in this class, please contact me to discuss your needs. Additionally, you will need to register with the Disability Support Services Office in the Ley Student Center.