UCSCIMAT11: UCU SCI 1 course

Calculus and linear algebra

Yuri Kuznetsov, Heinz Hanßmann

fall	time	place
lectures	monday 15:45 - 17:30	Newton E
exercises	thursday 11:00 - 12:45	Newton E

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 optimization 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. optimize 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,
6. make basic use of the symbolic manipulation program Mathematica for the purpose of evaluating expressions and creating graphs.

Literature

V. Blåsjö,

Intuitive Infinitesimal Calculus

http://intellectualmathematics.com/calculus/

D.W. Jordan & P. Smith,

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

Oxford University Press (2008)

Contents

week 35: Review of basic differentiation and integration.

week 36: Differential equations.

week 37: separation of variables, partial fractions.

week 38: Integration techniques.

week 39: Second-order differential equations.

week 40: Complex numbers and differential equations.

week 41: Power series.

week 42: Midterm exam.

week 44: Vectors.

week 45: Matrices.

week 46: determinants, eigenvectors and eigenvalues.

week 47: diagonalization. Uses of linear algebra.

week 48: Multivariable functions.

week 49: Gradients.

week 50: Final exam.