# Calculus

Welcome to the University of Leipzig! I wish you success in your studies.
I would like to provide you with some technical information about the lecture courses Calculus I, II and III.

## 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