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# Vector Analysis

Rakennetyyppi: Opintojakso IZS9108 IT 2020 Insinööri (AMK) 3 (2022-2023) Kevät 3 op Mäkelä, Jarmo Englanti

## Osaamistavoitteet

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 to differentiate and integrate vectors in general curvelinear coordinates, and not only in the familiar xyz coordinates.

## Opiskelijan työmäärä

The total amount of student's work is 81 h, which contains 42 h of contact studies. The assessment of student’s own learning 1 h is included in contact lessons.

## Edeltävät opinnot / Suositellut valinnaiset opinnot

Analysis: Differential- and Integral Calculus basics and Differential equations and series.

## Sisältö

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.
6. Divergence.
7. Curl.
8. Potential of the vector field.
9. Conservative vector field.
10. Surface integral.
11. Green’s theorem.
12. Two-dimensional surfaces embedded in three-dimensional space.
13. Curvelinear coordinates on a surface.
14. Coordinate curves and their tangent vectors.
15. Normal of a surface; orientable surfaces.
16. Calculation of the area of a general two-surface.
17. Flux of a vector field through a surface.
18. Stokes’s theorem.
19. Volume integral.
20. Gauss’s theorem.
21. Maxwell’s equations.
22. Continuity equation.
23. General curvelinear coordinates in three dimensions.
24. Surface- and volume integrals in the curvelinear coordinates: The Jacobi determinant.
25. Metric tensor.
26. Covariant and contravariant vector fields.
27. Christoffel symbol.
28. Differentation of vector fields in curvelinear coordinates.

## Opiskelumateriaali

Kreyszig, E: "Advanced Engineering Mathematics", John Wiley & Sons; the material prepared by the lecturer.

## Opetusmuoto / Opetusmenetelmät

Theory, examples and exercises during the lectures. Homework exercises.

## Arviointikriteerit

Grade 1: The student knows those subjects of the course, which are necessary for the forthcoming studies and working life.
Grade 3: The student is well-abled to utilize the course contents.
Grade 5: The student is able to apply creatively the contents of the course.

## Arviointimenetelmät

Homework exercises and an examination.