Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I

This mathematical course is for beginners at the University of Osnabrück in the subjects Computer sciences, Cognitive sciences, Physics and System sciences. It is an introduction to advanced mathematics on the academic level. It covers logical foundation, sets, mappings, algebraic structures like fields and polynomials, the basics of analysis like sequences, continuity, differentiability, primitive functions and the basics of linear algebra like vector spaces, bases, linear maps, eigenvalues. Enjoy!

Vorli

This is the English version of the material (lecture notes, exercises) of a mathematical course taught at the University of Osnabrück. So this is not an online-course, only the material for a real university course. The material is organized in a modular way (e.g. facts, proofs, examples, exercises are in separate files) so that it can be used also in other courses. The mathematical material is quite standard for a university beginners class, and the course has been taught several times. The English version is a direct translation. Suggestions are welcome.

Information

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Lectures

• 1 - Logic and argumentation

• 2 - Quantifiers and induction

• 3 - Sets and mappings

• 4 - Fields

• 5 - Complex numbers

• 6 - Polynomials

• 7 - Approximation and convergence

• 8 - Completeness

• 9 - Series

• 10 - Continuity

• 11 - Intermediate value theorem

• 12 - Exponential function

• 13 - Trigonometry

• 14 - Differentiability

• 15 - Mean value theorem

• 16 - The number ${}\pi$

• 17 - Taylor series

• 18 - Integration

• 19 - Fundamental theorem of calculus

• 20 - Rules for integration

• 21 - Linear systems

• 22 - Matrices and vector spaces

• 23 - Bases and dimension

• 24 - Linear mappings

• 25 - Invertible matrices

• 26 - Determinants

• 27 - Eigenspaces

• 28 - Diagonalizability

Exercise sheets

• Exercise sheet 1

• Exercise sheet 2

• Exercise sheet 3

• Exercise sheet 4

• Exercise sheet 5

• Exercise sheet 6

• Exercise sheet 7

• Exercise sheet 8

• Exercise sheet 9

• Exercise sheet 10

• Exercise sheet 11

• Exercise sheet 12

• Exercise sheet 13

• Exercise sheet 14

• Exercise sheet 15

• Exercise sheet 16

• Exercise sheet 17

• Exercise sheet 18

• Exercise sheet 19

• Exercise sheet 20

• Exercise sheet 21

• Exercise sheet 22

• Exercise sheet 23

• Exercise sheet 24

• Exercise sheet 25

• Exercise sheet 26

• Exercise sheet 27

• Exercise sheet 28

Further material

• List of definitions
• Important theorems

Forum

Literature

Links