Calculus Revisited

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The term function - development of the term; misleading character of generality of the term; 'a statistical aspect'; discontinuous functions are still 'close' to continuous ones

Limit passage - supremum, limsup, lim have the same scheme; definition of term filter; usage of filter for all limit passages

Problem of definition of the length of curve - classical approach and its problems; term curve in analysis; the necessity of defining new terms: rectifiable curve; Lebesgue's approach (leads to necessity of definition of new integral - Lebesgue's integral); functional approach: curve length as a lower semi-continuous functional in curve space

Integral theory - historical introduction; determination of surface area of complex figure; effort for finding an universal methodology: Cauchy's approach, Riemann's approach; persisting problems lead Lebesgue to a definition of a new integral; the two fundamental Lebesgue's thoughts; Lebesgue's measure and measurability; existence (and construction) of unmeasurable set (in Lebesgue sense) and the axiom of choice; comparison of Riemann's and Lebesgue's integral and finding the essence of difference; weak spots of Lebesgue's integral; the essence of measure theory; new perspectives in integral theory

Requirements:

Basic course of Calculus, Linear Algebra and Functional analysis (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, 01FA held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. The term function - development of the term; misleading character of generality of the term; 'a statistical aspect'; discontinuous functions are still 'close' to continuous ones

2. Limit passage - supremum, limsup, lim have the same scheme; definition of term filter; usage of filter for all limit passages

3. Problem of definition of the length of curve - classical approach and its problems; term curve in analysis; the necessity of defining new terms: rectifiable curve; Lebesgue's approach (leads to necessity of definition of new integral - Lebesgue's integral); functional approach: curve length as a lower semi-continuous functional in curve space

4. Integral theory - historical introduction; determination of surface area of complex figure; effort for finding an universal methodology: Cauchy's approach, Riemann's approach; persisting problems lead Lebesgue to a definition of a new integral; the two fundamental Lebesgue's thoughts; Lebesgue's measure and measurability; existence (and construction) of unmeasurable set (in Lebesgue sense) and the axiom of choice; comparison of Riemann's and Lebesgue's integral and finding the essence of difference; weak spots of Lebesgue's integral; the essence of measure theory; new perspectives in integral theory

5. Introduction to symmetries of differential equations and its usage for solving ordinary differential equations.

Syllabus of tutorials:

The subject is a seminar.

The term function - development of the term; misleading character of generality of the term; 'a statistical aspect'; discontinuous functions are still 'close' to continuous ones.

Limit passage - supremum, limsup, lim have the same scheme; definition of term filter; usage of filter for all limit passages

Problem of definition of the length of curve - classical approach and its problems; term curve in analysis; the necessity of defining new terms: rectifiable curve; Lebesgue's approach (leads to necessity of definition of new integral - Lebesgue's integral); functional approach: curve length as a lower semi-continuous functional in curve space

Integral theory - historical introduction; determination of surface area of complex figure; effort for finding an universal methodology: Cauchy's approach, Riemann's approach; persisting problems lead Lebesgue to a definition of a new integral; the two fundamental Lebesgue's thoughts; Lebesgue's measure and measurability; existence (and construction) of unmeasurable set (in Lebesgue sense) and the axiom of choice; comparison of Riemann's and Lebesgue's integral and finding the essence of difference; weak spots of Lebesgue's integral; the essence of measure theory; new perspectives in integral theory

Study Objective:

Knowledge:

To gain deeper insight into standardly used terms such as function, theory of integration, measure theory, axiom of choice etc. Further to get acquianted with symmetry methods for solving differential equations.

Skills:

Insight into standardly used terms such as function, theory of integration, measure theory, axiom of choice, solving differential equations using their symmetries

Study materials:

Key references:

[1] W. Rudin - Real and complex analysis, 3rd edition, McGraw-Hill Education (India) Pvt Ltd, 2006

[2] J. Marsden, A Weinstein - Calculus 1,2,3, Springer, 1985

Recommended references:

[1] M. Reed, B. Simon - Methods of modern mathematical physics, I. Functional analysis, Academic Press 1980

[2] P. Hydon - Symmetry Methods for Differential Equations, Cambridge university press, 2000

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Matematické inženýrství (elective course)