Calculus - Information for Freshmen

Contents of Calculus I (Fall and Winter 2004/5)

I Real  and Complex Numbers (Arithmetic  and Ordering Axioms, Supremum, Infimum, Order Completeness, Complex Numbers, Trigonometry, some 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, 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)
 

Contents of Calculus II (Spring and Summer 2005)

VI Sequences of Functions and Basic Topology (Uniform Convergence of Sequences and Series, Continuity, Integration, Differentiation, Fourier Series, countable and uncountable sets, Metric spaces and Normed Vector Spaces, open, closed sets, neighborhoods, boundary, closure, dense set, compactness, convergent sequences, continuity)

VII 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)

VIII Curves and Line Integrals (Curves, Rectifiable Curves, Line Integrals, Potentials, Path Independence, Connectedness, Simply Connectedness)

IX Integration (Riemann integrals in R^n)

Contents of Calculus III (Fall and Winter 2005/6)

9. Integration of Functions of Several Variables ( Differential Forms)

10. Surface Integrals (Surface Area, Gauss' divergence theorem, Green's theorem, Stokes' theorem, Green's formulas)

11. Differential Forms

12. Measure Theory and Integration (ring, algebra, content, measure, Lebesgue measure, measurable functions, step functions, Lebesgue integral, convergence theorems: Lebesgue, Levi)

13. Hilbert Space (Geometry: Projections, Riesz' lemma, Bounded Linear Operators: self-adjoint, normal, isometric, and unitary operators, projections, spectrum and resolvent of bounded self-adjoint operators )

14. Complex Analysis (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)

To the Literature .

Contents of Calculus IV (Spring and Summer 2006)

14. Complex Analysis (Cauchy's integral formula, Liouville's theorem, Singularities: Laurent expansion, residues, real integrals)

15. Partial Differential Equations I (Classification, Characteristics, Well-Posed Problems)

16. Distributions (Test Functions, Support, Differentiation, Tensor Product, Convolution Product, Fourier Transform, Weak Solutions)

17. Partial Differential Equations II (Fundamental Solutions of Laplace, Wave and Heat Equations, Fourier Method)

To the Contents and the Literature .

Homework

You have to solve these questions until next Monday and to hand in the solutions before the class starts.

To get credit, you will need 50% of all points at the end of the term. Exercises are marked by Slava Matveev, Mathematical Institute.

Instruction class

Special Seminars