*Conducted in terms:*2020Z, 2021Z

*Erasmus code:*11.1

*ISCED code:*0541

*ECTS credits:*9

*Language:*Polish

*Organized by:*Faculty of Physics

*Related to study programmes:*

# Calculus III 1100-2AF10

The lecture "Calculus III" is the continuation of the previous lectures "Calculus I" and "calculus II". It covers three fields in mathematics that are needed to complete the mathematical education of future physicists. We start with the introduction to differential geometry dealing with surfaces that are subsets of R^{n}. The second part is calculus on complex plane and in the third part of the lecture we discuss generalized functions and Fourier transform. The plan of the lecture is the following:

I. Introduction to differential geometry

a. The definition of the surface

b. The tangent space

c. The cotangent space, the differential of the function

d. Differential forms, integration, Stokes theorem

e. Curves in R3

f. Vector fields, vector analysis

II. Calculus on complex plane

a. Complex numbers

b. Holomorphic functions, Cauchy-Riemann equations

c. Integration along curves

d. Taylor and Laurent series

e. Multivalued functions, logarithm

f. Residuum and its applications in integration

III. Introduction to the theory of generalized functions and Fourier transform

a. Fourier series

b. Fourier transform

c. Generalized functions, Dirac delta.

Description by Katarzyna Grabowska, September 2009

## Type of course

## Course coordinators

Term 2021Z: | Term 2020Z: |

## Learning outcomes

Student who has passed the exam should

- know and understand basic definitions and theorems of differential geometry and its applications in theoretical physics

- be able to (practically) calculate integral of differentoal forms on surfaces in order to determine the area of the surface or the flux of a vector filed through the surface

- know what does it mean to deffine a geometrical object in the way independent of the set of coordinates on the surface

- know basic definitions and theorem concernig the theory of generalized finctions and Fourier transform

- be able to calculate Fourier transforms of certain functions

- be prepared to start the course on Quantum Mechanics.

## Assessment criteria

Final exam is divided into two parts: written and oral. It is necessary to pass both parts of the exam. For details, see:

https://kampus-student2.ckc.uw.edu.pl/mod/forum/discuss.php?d=50104#p100494

## Practical placement

There is no practical placement

## Bibliography

1. Paweł Urbanski "Analiza III" (Skrypt KMMF)

2. Michael Spivak "Analiza na rozmaitościach"

3. Tristan Needham "Visual complex analysis"

4. Franciszek Leja "Funkcje Zespolone"

5. Vasilij Sergiejewicz Władimirow "Urawnienia matematiczeskoj fiziki"

## 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: