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Real Analysis

Structure Type: Study unit
Code: IX00BE86
Curriculum: SAT 2024V
Level: Bachelor of Engineering
Year of Study: 3 (2026-2027)
Semester: Autumn
Credits: 5 cr
Responsible Teacher: Mäkelä, Jarmo
Language of Instruction: Finnish

Taking the course in advance? See the courses during the academic year 2023-2024.

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.


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,
6) Gradient, divergence and curl,
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

Suggested Reading: E. Kreyszig: Advanced Engineering Mathematics (Wiley)