BSTE:HighPerformanceComputing
High Performance Computing
Course Characteristics
Key concepts of the class
- Multivariate calculus: derivatives, differentials, maxima and minima
- Multivariate integration
- Functional series. Fourier series
- Integrals with parameters
What is the purpose of this course?
The goal of the course is to study basic mathematical concepts that will be required in further studies. The course is based on Mathematical Analysis I, and the concepts studied there are widely used in this course. The course covers differentiation and integration of functions of several variables. Some more advanced concepts, as uniform convergence of series and integrals, are also considered, since they are important for understanding applicability of many theorems of mathematical analysis. In the end of the course some useful applications are covered, such as gamma-function, beta-function, and Fourier transform.
Course objectives based on Bloom’s taxonomy
- What should a student remember at the end of the course?
By the end of the course, the students should be able to:
- find partial and directional derivatives of functions of several variables;
- find maxima and minima for a function of several variables
- use Fubini’s theorem for calculating multiple integrals
- calculate line and path integrals
- distinguish between point wise and uniform convergence of series and improper integrals
- decompose a function into Fourier series
- calculate Fourier transform of a function
- What should a student be able to understand at the end of the course?
By the end of the course, the students should be able to understand:
- how to find minima and maxima of a function subject to a constraint
- how to represent double integrals as iterated integrals and vice versa
- what the length of a curve and the area of a surface is
- properties of uniformly convergent series and improper integrals
- beta-function, gamma-function and their properties
- how to find Fourier transform of a function
- What should a student be able to apply at the end of the course?
By the end of the course, the students should be able to ...
- find multiple, path, surface integrals
- find the range of a function in a given domain
- decompose a function into Fourier series
Resources and reference material
- Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson
- Jerrold Marsden, Alan Weinstein (1985) Calculus (in three volumes; volumes 2 and 3), Springer
Course Sections
The main sections of the course and approximate hour distribution between them is as follows:
| Section | Section Title | Teaching Hours |
|---|---|---|
| 1 | Differential Analysis of Functions of Several Variables | 24 |
| 2 | Integration of Functions of Several Variables | 30 |
| 3 | Uniform Convergence of Functional Series. Fourier Series | 18 |
| 4 | Integrals with Parameter(s) | 18 |
| hline |
Section 1
Section title:
Differential Analysis of Functions of Several Variables
Topics covered in this section:
- Limits of functions of several variables
- Partial and directional derivatives of functions of several variables. Gradient
- Differentials of functions of several variables. Taylor formula
- Maxima and minima for functions of several variables
- Maxima and minima for functions of several variables subject to a constraint
Section 2
Section title:
Differential Analysis of Functions of Several Variables
Topics covered in this section:
- Limits of functions of several variables
- Partial and directional derivatives of functions of several variables. Gradient
- Differentials of functions of several variables. Taylor formula
- Maxima and minima for functions of several variables
- Maxima and minima for functions of several variables subject to a constraint
Section 3
Section title:
Differential Analysis of Functions of Several Variables
Topics covered in this section:
- Limits of functions of several variables
- Partial and directional derivatives of functions of several variables. Gradient
- Differentials of functions of several variables. Taylor formula
- Maxima and minima for functions of several variables
- Maxima and minima for functions of several variables subject to a constraint
Section 4
Section title:
Differential Analysis of Functions of Several Variables
Topics covered in this section:
- Limits of functions of several variables
- Partial and directional derivatives of functions of several variables. Gradient
- Differentials of functions of several variables. Taylor formula
- Maxima and minima for functions of several variables
- Maxima and minima for functions of several variables subject to a constraint