Vector Analysis
| Structure Type: | Study unit |
|---|---|
| Code: | IXS9108 |
| Curriculum: | TT 2021 |
| Level: | Bachelor of Engineering |
| Year of Study: | 3 (2023-2024) |
| Semester: | Spring |
| Credits: | 3 cr |
| Responsible Teacher: | Mäkelä, Jarmo |
| Language of Instruction: | Finnish |
Learning Outcomes
The basic problem of vector analysis is: How to differentiate and integrate vectors? This question raises in certain engineering applications of mathematics, especially when we consider a flow of given substance, no matter whether that substance is, say, water flowing in a pipeline, or energy carried by a radio wave. During this course a student will learn the basic concepts and theorems of vector analysis, and to apply them in the problems of mechanics, fluid mechanics, and electrical engineering. The student also learns a general way to analyze by means of matrices those kinds of functions which transform a vector to a new vector. Functions of that kind are known as linear maps, or operators.
Student's Workload
81 h, which contains 42 h of contact studies at VAMK and 24 h at University of Vaasa.
The assessment of student’s own learning 1 h is included in contact lessons.
Prerequisites / Recommended Optional Courses
Integral Calculus, Differential Equations and Series, Advanced Analysis.
Contents
1. Summary of the basic vector calculus.
2. Vector fields.
3. Differentiation of a vector with respect to a parameter.
4. Line integrals of vector fields.
5. Gradient, divergence, curl.
6. Potential of the vector field.
7. Conservative vector field.
8. Surface integral.
9. Green’s theorem.
10. Two-dimensional surfaces embedded in three-dimensional space.
11. Curvelinear coordinates on a surface.
12. Coordinate curves and their tangent vectors.
13. Normal of a surface; orientable surfaces.
14. Calculation of the area of a general two-surface.
15. Flux of a vector field through a surface.
16. Stokes’s theorem.
17. Volume integral.
18. Gauss’s theorem.
19. Maxwell’s equations.
20. Continuiyty equation.
21. A summary of basic operations of matrices.
22. Linear vector space.
23. Operators on linear vector spaces.
24. Matrix representation of the operators in the basis ijk of R3.
25. Operator product.
26. Linear dependence. Basis.
27. Orthonormal basis.
28. Matrix representation of operators in the orthonormal basis.
29. Change of basis.
30. Eigenvalues and eigenvectors.
31. Hermitean operators and matrices.
32. Principal axis representation of second order curve.
33. Normal modes of coupled oscillators (example: a double pendulum on a plane).
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 works, an examination.