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 Rn. 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
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.
Final exam is divided into two parts: written and oral. It is necessary to pass both parts of the exam. For details, see:
There is no practical placement
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"
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: