1. SylabUZ
2. Faculty of Exact and Natural Sciences
3. winter term 2022/2023
4. Mathematics - First-cycle studies leading to Bachelor's degree
5. Mathematical Analysis 2

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Mathematical Analysis 2 - course description

Aim of the course

To acquaint the student with the differential method of examining monotonicity, extrema and convexity of functions, with the concepts of primitive and Riemann integral. The emphasis is on mastering calculus techniques, in particular integration, as well as applications of differential and integral calculus. The next goal is to present the basics of measure theory,, then the theory of Lebesgue integrals, and the connections between Riemann and Lebesgue integrals.

Prerequisites

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Mathematical Analysis 1. Logic and Set Theory. Linear Algebra 1.

Scope

Lecture

I. Elementary integral calculus

1. Riemann integral and area. Basic properties of integral. Mean value theorem for integrals (8 hours)
2. Relationships of differentiation to integration. Newton-Leibniz fundamental theorem of calculus and its consequences (3 hours)
3. Relationships of uniform convergence to integration. Integrating power series (2 hours)

II. Techniques of integration

1. Trigonometric substitutions (2 hours)
2. Euler’s substitutions (3 hours)
3. Numerical integration: trapezoidal rule, Simpson’s rule (the material should be prepared in student’s own right basing on a literature indicated by the lecturer)

III. Applications of integral calculus

1. Exemplary applications of integration in geometry: areas of regions in the plane, volumes of solids, area of surfaces (3 hours)
2. Center of mass and moments. Theorems of Pappus (the material should be prepared in student’s own right basing on a literature indicated by the lecturer)
3. Work and pressure (the material should be prepared in student’s own right basing on a literature indicated by the lecturer)

IV. Cartesian spaces and different ways of describing their subsets. Limit and continuity of functions of many variables

1. Scalars and vectors (1 hour)
2. Polar coordinate system. Curves in polar coordinates. Area of a region bounded by a curve. Length of a curve (2 hours)
3. Parametric equations of a curve on the plane. Tangent line to a curve. Length of a curve (2 hours)
4. Cylindrical coordinates and spherical coordinates (1 hour)
5. Level sets of functions of two or three variables (1 hour)
6. Limit and continuity of functions of several variables (6 hours)

V. Elements of measure theory

1. σ-algebra and measure. Simple examples (2 hours)
2. Complete  measure (2 hours)
3. Outer measure and Carathéodory's theorem (3 hours)
4. Lebesgue measure (4 hours)
5. Measurable functions. Simple functions. Principle of induction for measurable functions (2 hours)
6. Sequences of measurable functions (1 hour) 

VI. Lebesgue integral

1. Definition and basic properties of the integral. Integrability (2 hours)
2. Theorems about calculating limits under the integral sign (2 hours)
3. Integral with respect to the Lebesgue measure and Riemann integral. Characterization of integrability in the Riemann sense (2 hours)
4. Integral as a function of a set (1 hour)
5. Change of variable in the integral (1 hour)
6. Product measure and Fubini's theorem. Iterated integrals (3 hours)

Classes

I. Elementary differential calculus II

1.  Determination of local and global extrema. Proving inequalities by finding extrema. Function analysis (5 hours)
2. Examining the uniform convergence of sequences of functions and series of functions (2 hours)

II. Elementary integral calculus. Techniques of integration

1. Calculating integrals by using definition (2 hours)
2. Integrating by parts and by substitution. Algorithm of integrating rational functions. Making use of Newton-Leibniz fundamental theorem of calculus (9 hours)

Colloquium (2 hours)

III. Applications of integral calculus

1. Convergence and integration. Integrating series of functions (2 hours)
2. Calculating areas of regions in the plane and volumes of solids (2 hours)
3. Determination of the center of mass and calculating the quantity of work (1 hour)

IV. Cartesian spaces and different ways of describing their subsets. Limit and continuity of functions of many variables

1. Changing Cartesian coordinates into polar ones and conversely (1 hour)
2. Calculating areas of regions and length of curves described by polar equations (2 hours)
3. Determination of lines tangent to a curve described parametrically. Calculating areas of regions and length of curves described parametrically (3 hours)
4. Describing surfaces in spherical and cylindrical coordinates (2 hours)
5. Limit and iterated limits (3 hours)
6. Continuity and continuity in separated variables (2 hours)

Colloquium (2 hours)

V. Elements of measure theory

1. Examples of measures and σ-algebras  (2 hours)
2. Definition of volume  (1 hour)
3. Lebesgue outer measure (1 hour)
4. Calculating the Lebesgue measure of certain sets (2 hours)
5. The existence of  non-measurable sets in Lebesgue sens (1 hour)
6. Types of convergence of sequences of measurable functions (2 hours)

VI. Lebesgue integral I

1. Calculating integrals using definitions. Integrability (3 hours)
2. Examples of applications of the theorems about calculating limits under the integral sign  (1 hour)
3. Examples of integrals with respect to the Lebesgue measure of non-integrable functions in the Riemann sense. Comparison of applicability between Riemann and Lebesgue integrals. Improper Riemann integral (4 hours)
4. Integral as a function of a set (1 hour)

Colloquium (2 hours)

 

Teaching methods

Traditional lecture; exercises in which students solve problems and discuss, as well as prepare biographies of mathematicians whose names appear at the lecture; group work completed with a written study; work with a book and with the help of the Internet. If necessary (determined by the order of the Rector of the University of Zielona Góra), classes can be in online form.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

1. Three colloquia with problems of various degree of difficulties, allowing to verify if students attained learning outcomes at the very least.
2. Exam with indicated point ranges.
The final grade is the arithmetic mean of those of the class and exam. A necessary condition to enter the exam is a positive grade of the classes. A necessary condition to pass the course is a positive grade of the exam.

Recommended reading

1. J. Douglas Faires, Barbara T. Faires, Calculus, Random House, New York, 1988.

2.  J. Douglas Faires, Barbara T. Faires, Calculus of one variable, Random House, New York, 1988.

3. Sheldon Axler, Measure, integration and real analysis, Springer, San Francisco, 2020. 

Further reading

1. William F. Trench, Introduction to real analysis, Pearson Education, 2013.

Notes

Modified by dr hab. Bogdan Szal, prof. UZ (last modification: 13-02-2024 22:34)