Mathematics III 1100-2AF11

The course consists of three parts:
(1) Elements of differential geometry,
(2) Complex analysis,
(3) Elements of distribution theory and the Fourier transform.

In the first part of the course, students will learn analysis on surfaces. The following notions will be introduced: surfaces immersed in mathbb{R}^n, vector fields, differential forms, integration of differential forms over surfaces, and Stokes’ theorem. Elements of classical vector analysis will also be discussed, including the concepts of gradient, divergence, curl, and the Laplacian.

The second part of the course is intended to prepare students for courses in quantum mechanics. The topics covered will include complex differentiability, holomorphic functions, Taylor and Laurent series, integration in the complex plane, and the residue method.

In the third part of the course, students will become acquainted with the Fourier transform for a certain class of functions, as well as with elements of distribution theory necessary for the study of quantum mechanics.

Course coordinators

Term 2024Z:

Katarzyna Grabowska

Term 2025Z:

Paweł Kasprzak

Term 2026Z:

Paweł Kasprzak

Mode

Classroom

Prerequisites (description)

A student beginning the course Mathematics III should be familiar with the basic concepts of linear algebra: vector spaces, linear independence, bases, linear mappings, matrices of linear mappings, etc. They should also have a satisfactory command of differential calculus for real functions of one and several variables. Practical skills in integrating functions of one and several variables will also be useful. The above material is covered in the courses Mathematics I and Mathematics II.

Learning outcomes

After completing the course, the student should be familiar with the basic concepts and techniques of differential geometry used in introductory courses on classical electrodynamics and classical mechanics. They should also be prepared to participate in a course on quantum mechanics. The student should know the basic techniques of integration of complex functions, understand the concept of distributions as generalized functions, and be able to compute Fourier transforms of certain functions.

Assessment criteria

The final grade is based on the student’s performance in the exercise classes, the result of the written examination, and the oral examination. During the oral examination, theoretical knowledge is assessed, including understanding of concepts and familiarity with theorems. Detailed assessment rules for each teaching cycle are determined by the instructor.

Practical placement

not applicable

Bibliography

Michel Spivak, "Calculus on manifolds"
Tristan Needham "Visual complex analysis"
Lars Ahlfors "Complex Analysis"
Gerald B. Follan, "Fourier Analysis and Its Applications"

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: