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# Real Analysis

Structure Type: Study unit IT00BL30 ETE 2024 Bachelor of Engineering 3 (2026) Autumn 5 cr Mäkelä, Jarmo English

## Learning Outcomes

In the Real Analysis course the student learns to differentiate and integrate multivariable functions. The general theory of series is also briefly discussed. The central topic in the course is vector analysis, where vector fields are differentiated and integrated. Vector fields include, for instance, electric-, and magnetic fields, and the velocity field of a fluid. Hence vector analysis has plenty of applications, among other things, in electromagnetism and fluid dynamics.

135 h, including 50 h of scheduled contact studies.

## Prerequisites / Recommended Optional Courses

Follow-up Course in Technical Mathematics.

## Contents

1) A brief summary of the differentiation and integration of single-variable functions,
2) Optimization of multivariable functions,
3) Vectors and vector fields,
4) Differentiation of vector fields with respect to a parameter,
5) Line integrals of vector fields,
7) Potential of the vector field,
8) Surface integral,
9) Green’s theorem,
10) Change of variables in surface integrals: The Jacobi determinant,
11) Flux of the vector field,
12) Stokes’ theorem,
13) Volume integral,
14) Change of variables in volume integrals,
15) Gauss’ theorem,
16) Differentiation of vector fields in curvilinear coordinates,
17) Convergence and divergence of series,
18) Taylor series,
19) Power series solution of differential equations,
20) Calculus of variations (if there is time).

## Recommended or Required Reading and Other Learning Resources/Tools

E. Kreyszig: Advanced Engineering Mathematics (Wiley).

## Mode of Delivery / Planned Learning Activities and Teaching Methods

Lectures and exercises.

## Assessment Criteria

The general criteria of VAMK.

## Assessment Methods

Exercise activity and a course examination.