UCSCIMAT11: UCU SCI 1 course

Calculus and Linear Algebra

Heinz Hanßmann, Stein Meereboer

fall	time	place
lectures and exercises	monday 17:05 - 18:50 thursday 12:05 - 13:50	Newton N-D

ECTS : 7.5 credit points

This introductory course focuses on basic concepts of calculus starting with functions of a single variable. First you explore scalar linear and nonlinear differential equations. Such equations are key in explaining the dynamic behavior of many different systems in a wide variety of fields. This serves as a motivation to learn about techniques, such as differentiation, integration, expansion in a small variable and complex numbers. Next you learn to use tools to study systems of several variables: vectors, matrices and diagonalization of matrices. You also extend techniques such as differentiation to functions of two variables and learn about their geometrical representation. The course concludes with various approaches to optimisation of functions of two variables. The techniques you learn in this course have proven to be highly effective in a wealth of areas as will be illustrated by examples in various fields. Some attention is paid to underlying mathematical foundations, but the focus is on getting a working knowledge of the methods and on learning to apply the techniques.

Course goals

After completing this course students are able to:

1. apply basic techniques of single-variable calculus, such as differentiation, integration and Taylor expansions,
2. find solutions to simple ordinary differential equations,
3. apply basic techniques of linear algebra, including matrix diagonalization,
4. optimise functions of two variables, with and without constraint,
5. understand and extend these techniques beyond the level of recipes so they can be applied to new problems in their future field of study,

Literature

V. Blåsjö

Intuitive Infinitesimal Calculus

http://intellectualmathematics.com/calculus/

E.J. Herman & G. Strang

Calculus 1-3

OpenStax, Rice University (2016)

D.W. Jordan & P. Smith

Mathematical Techniques,
An introduction for the engineering, physical and mathematical sciences

Oxford University Press (2008)

D. Margalit & J. Rabinoff

Interactive Linear Algebra

Georgia Institute of Technology (2017)

J. Stewart

Single Variable Calculus

Brooks/Cole, Cengage Learning (2010)

Contents

31.8. Review of differentiation.

3.9. Review of basic integration.

7.9. Differential equations.

10.9. Direction fields, Euler approximation.

14.9. Separation of variables.

17.9. Partial fractions.

21.9. Substitution.

24.9. Integration by parts.

28.9. Homogeneous linear second-order differential equations.

1.10. Non-homogeneous linear second-order differential equations.

5.10. Complex numbers.

8.10. Linear second-order differential equations: complex case.

12.10. Questions.

15.10. Midterm exam.

26.10. Vectors.

29.10. Scalar product.

2.11. Linear mappings and matrices.

5.11. Gaussian elimination, inverse matrices, data.

9.11. Determinants.

12.11. Eigenvectors and eigenvalues.

16.11. Diagonalization.

19.11. Power series and geometric series

23.11. Binomial series and divergence.

26.11. Multivariable functions.

30.11. Gradients.

3.12. Unconstrained optimisation.

7.12. Constrained optimisation.

10.12. Questions.

14.12. Final exam.

17.12. Discussion.