*Conducted in terms:*2021Z, 2022Z

*Erasmus code:*11.1

*ISCED code:*0541

*ECTS credits:*9

*Language:*Polish

*Organized by:*Faculty of Physics

*Related to study programmes:*

# Analysis I E 1100-1Ind01

1. Elements of logic and set theory

- Equivalence relations

2. Real numbers

- Existence of suprema and infima

3. Sequences of real numbers

- Cauchy sequences

4. Metric spaces

- Balls, open sets

- Closed sets

- Sequences in metric spaces

5. Elements of general topology

- Continuous mappings

- Compactness

- Connectedness

6. Differential calculus

- Basic theorems

- Bernoulli-de l'Hospital rules

- Taylor's formula

- Maxima and minima

7. The Riemann integral

- Fundamental theorem of differential calculus

8. The functions log and exp

9. Series

- Series with positive terms

- Series with arbitrary terms

10. Sequences and series of functions

- Types of convergence, elementary theorems

- Power series

11. Elementary functions

12. Methods of integration

13. Stone-Weierstrass theorem

14. Additional topics

- Banach principle

- Convex functions

- Integrals with parameter

## Mode

## Prerequisites (description)

## Course coordinators

Term 2022Z: | Term 2021Z: |

## Learning outcomes

1. Mastering elementary mathematical analysis.

2. Gaining basic competence in reading and understanding mathematical texts.

3. Obtaining elementary techniques of analysis of real functions.

4. Obtaining experience and skill of identifying crucial mathematical properties of objects under study and application thereof.

## Assessment criteria

Two mid-terms, final written exam, final oral exam. Grading citeria: mastering the subject, solving problems

## Practical placement

None

## Additional information

Information on *level* of this course, *year of study* and semester when the course
unit is delivered, types and amount of *class hours* - can be found in course structure
diagrams of apropriate study programmes. This course is related to
the following study programmes:

Additional information (*registration* calendar, class conductors,
*localization and schedules* of classes), might be available in the USOSweb system: