Calculus Revisited
Code  Completion  Credits  Range  Language 

01MADR  Z  2  0+2  Czech 
 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 semicontinuous 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, 01MAA24, 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 semicontinuous 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 semicontinuous 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, McGrawHill 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 timetable has been prepared for this course
 The course is a part of the following study plans:

 Matematické inženýrství (elective course)