Euclidean Spaces

The main concepts of the topology, metric and functions of Euclidean space.

5.8.2026 - 31.7.2027

5 cr

Registration period closed

Intermediate studies

Euclidean spaces and their topology:

The inner product and norm, Cauchy-Schwarz inequality, the triangle inequality
Sequences and their convergence
Open and closed sets, the boundary, the closure, and the limit point
Compactness, Heine-Borel theorem
Paths, pathwise connectedness, domains

Functions in Euclidean spaces:

The limit and different characterisations of continuity
Continuous functions and compactness, existence of extreme values
Partial derivatives, directional derivatives and gradients, the intermediate value theorem
Differentiability and linear approximations, the chain rule
Line integrals along regular curves

Contains

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Additional information and registration

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Additional information:

Fields

Natural sciences

Scope

5 cr

Code

MATH.MA.620

After the completion of the course the students know the main concepts and results of the topology and metric of euclidean spaces. In addition, the students knows concepts of continuity, differentiability and lineintegral of functions and the related basic results in Euclidean spaces. The students are capable of defining the main concepts precisely and writing justifications and applying results in simple proofs and calculations. The students are able to verify the most important results. The students can apply the concepts and results in the later studies. The students are expected to be able to discuss the concepts of the course and to deliver oral and written presentations of their mathematical thinking.

Organiser

Tampere University

Contact details

avoin.tau@tuni.fi

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