Contents of Calculus I (Fall 2003/4)
I Real and Complex Numbers (Arithmetic and
Ordering
Axioms, Supremum, Infimum, Order Completeness, Powers, Some
Trigonomery,
Complex Numbers, Inequalities)
II Sequences and Series (Convergence, Cauchy Sequences,
Upper
and Lower Limits, Root and Ratio Tests, Power Series, Absolute
Convergence)
III Functions and Continuity (Limits of Functions,
Continuous
Functions, Discontinuities, Monotonic Functions)
IV Differentiation (Derivatives, Mean Value Theorem,
Continuity
of Derivatives, L'Hospital's Rule, Higher Order Derivatives, Taylor's
Theorem,
Differentiation of Vector-Valued Functions)
V Integration (Antiderivatives, Riemann-Stieltjes
Integral, Integration
and Differentiation, Improper Integrals, Vector-Valued Functions,
Rectifiable
Curves)
Contents of Calculus II (Spring 2004)
VI Basic Topology (countable and uncountable sets, Metric
spaces
and Normed Vector Spaces, open, closed sets, neighborhoods, boundary,
closure,
dense set, compactness, convergent sequences, continuity)
VII Sequences and Series of Functions (Uniform Convergence
of
Sequences and Series, Continuity, Integration, Differentiation,
Stone-Weierstrass
Theorem, Fourier Series)
VIII Functions of Several Variables (Continuity, Partial
Derivatives,
Higher Derivatives, Schwarz's Lemma, Differentiation, Chain Rule,
Inverse
Mapping Theorem, Implicit Function Theorem, Taylor Series, Local
Extrema,
Integration)
IX Integration (Riemann integrals in R^n, Surface
integrals,
Line Integrals, Gauss' divergence theorem, Green's theorem, Stokes'
theorem,
path independence, Green's formulas)
Contents of Calculus III (Fall 2004/5)
X Measure Theory and Integration (ring, algebra, content,
measure,
Lebesgue measure, measurable functions, step functions, Lebesgue
integral,
convergence theorems: Lebesgue, Levi, Fatou)
XI Hilbert Space (Geometry: projection, Riesz' lemma,
Bounded
Linear Operators: self-adjoint, normal, isometric, and unitary
operators,
projections, weak convergence, spectrum and resolvent of bounded
self-adjoint
operators, spectral measure, spectral theorem)
XII Complex Functions (Differentiability: Cauchy-Riemann
equations,
line integrals, Cauchy's integral formula, Liouville's theorem,
Analytic
Functions: power series, identity theorem, continuation, Singularities:
Laurent expansion, residues, real integrals)
Course notes
Chapter1.ps , Chapter1.pdf
Chapter2.ps , Chapter2.pdf
Chapter3.ps , Chapter3.pdf
Chapter4.ps , Chapter4.pdf
Chapter5.ps , Chapter5.pdf
Chapter6.ps , Chapter6.pdf
Chapter7.ps , Chapter7.pdf
Chapter8.ps , Chapter8.pdf
New: Chapter9.ps , Chapter9.pdf
New: Chapter10.ps , Chapter10.pdf
New: Chapter11.ps , Chapter11.pdf
Homework
Homework problems are assigned on Friday and due the next Friday,
starting on October 24. The
problems are available on the web page.
Your
homework will be marked either right (2 points), medium
(1
point) or false (0 points). To pass the course, you will need
50% of
all points at the end of the term.
Literature
- [1]
- R. Courant, Differential and integral calculus I-II,
Wiley Classics
Library, John Wiley & Sons, New York etc., 1988.
- [2]
- J. Dieudonné, Treatise on analysis. Volume I - IX,
Pure and
Applied Mathematics, Academic Press, Boston, 1993.
- [3]
- O. Forster, Analysis 1 - 3 (in German), Vieweg Studium:
Grundkurs
Mathematik, Vieweg, Braunschweig, 2001.
- [4]
- K. Königsberger, Analysis 1 (English),
Springer-Verlag, Berlin,
Heidelberg, New York, 1990.
- [5]
- S. Lang, Undergraduate Analysis, second ed.,
Undergraduate texts
in mathematics, Springer, New York-Heidelberg, 1989.
- [6]
- J. Marsden and A. Weinstein, Calculus. I, II, III,
Undergraduate
Texts in Mathematics, Springer-Verlag, New York etc., 1985.
- [7]
- W. Rudin, Principles of mathematical analysis, third
ed., International
Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New
York-Auckland-Düsseldorf,
1976.
- [8]
- M. Spivak, Calculus, Publish or Perish, Inc., Berkeley,
California,
1980.
- [9]
- M. Spivak, Calculus on Manifolds, Benjamin, New York,
1965.
Our main reference will be [7]. This book is available in 4 copies
in the physics library, one copy is in the mathematics library. Also,
[5]
is available; but it is not very good. The textbooks [6] and [8] are
easy
accessable. Both can be recommended for students which have problems in
the foundations. [9] contains the main facts is easy accessable and has
only 144 pages.
For more detailed information about the ``International Physics
Studies
Program" see
http://www.uni-leipzig.de/~intphys/forms/studyreg.pdf