Advanced Analysis
| Structure Type: | Study unit |
|---|---|
| Code: | IXS9109 |
| Type: | Optional obligatory / Professional Studies |
| Curriculum: | TT 2017 |
| Level: | Bachelor of Engineering |
| Year of Study: | 3 (2019-2020) |
| Credits: | 4 cr |
| Responsible Teacher: | Mäkelä, Jarmo |
| Language of Instruction: | Finnish |
Still need to take the course? See the courses during the academic year 2020-2021.
Learning Outcomes
During the Advanced Analysis course the student will deepen the knowledge gained during the courses of Differential calculus and Analysis. She will learn to differentiate multivariable functions, and to optimize them. These knowledge has plenty of practical applications. The largest individual part of the course consists of complex analysis. In that part of the course the student will learn to differentiate and integrate functions of a complex variable. Complex analysis is largely based on power series, and therefore, as a preparation for the complex analysis, the student is first familiarized with the power series of functions of a real variable, their divergence and convergence. Complex analysis, in the level considered in this course, is intented to provide the student the necessary knowledge for the Integral Transforms course. Power series may also be used for the solution of differential equations. The course end with a brief review on the calculus of variations, which may be viewed as a one step forward from the usual differential- and integral calculus.
Student's Workload
The total amount of student's work is 108 h, which contains 56 h of contact studies at VAMK and 32 h at University of Vaasa.
The assessment of student’s own learning 1 h is included in contact lessons.
Prerequisites / Recommended Optional Courses
Ingegral Calculus, Differential Equations and Series.
Contents
1. A brief summary of the single variable analysis.
2. Differential calculus of multivariable functions.
3. Optimization of multivariable functions.
4. Optimization of a multivariable function endowed with constraints by means of Lagrange’s method undetermined multipliers.
5. Power series of functions of a real variable.
6. Convergence and divergence of power series, radius of convergence.
7. Functions of a complex variable.
8. Analytic functions, Cauchy-Riemann equations.
9. Line integrals of functions of a complex variable.
10. Cauchy’s integral theorem.
11. Power series of analytic functions; radius of convergence.
12. Pole of a function of a complex variable.
13. Laurent series.
14. Residue.
15. The residue theorem.
16. Integration of functions of a real variable by means of the residue theorem.
17. Solution of linear differential equations by means of the power series method.
18. Elements of the calculus of variations.
19. Constrained variational problems.
Recommended or Required Reading and Other Learning Resources/Tools
Kreyszig, E: "Advanced Engineering Mathematics", John Wiley & Sons. Material prepared by the lecturer.
Mode of Delivery / Planned Learning Activities and Teaching Methods
Theory, examples and exercises during the lectures, homework exercises.
Assessment Criteria
Grade 5: The student is able to creatively solve problems related with almost all the contents of the course.
Grade 3: The student is able to solve problems related with the central contents of the course.
Grade 1: The student is able to solve basic problems related with the central contents of the course.
Assessment Methods
Homework exercises, practical work, an examination.