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Mathematics I & II

Course Description

Mathematics I and II is a sequence of two modules that teaches the multivariate calculus and linear algebra concepts required for applications in business studies and in economics. Our approach is neither proof-driven as in a pure mathematics class nor a collection of random algorithms as in an entirely application-based class. We aim to strike a balance between a foundation that allows you to actually understand the required methods and to foreshadow applications where these methods will reemerge throughout your studies.

Course Structure

The mathematics modules consist of three types of sessions:

1. Lectures where new content is presented. These sessions are very interactive, however, and we take time to answer questions throughout the session.
2. Exercises where we discuss problem sets, if needed. You must work on the problem sets before class and group work is encouraged. The solutions to all problem sets are available from the start of the semester so the exercise sessions pursue a flipped classroom approach where we only address questions uhpon your request. We will not recalculate every single solution. Occasionally, we will also show how to solve problems on the computer. This content is optional.
3. Application sessions where we discuss applications of newly acquired methods to examples from business and economics. We cannot discuss the details of every application from later modules in this class but we can motivate select applications and employ our methods to solve them.

Course Content

Calculus and linear algebra are ideally taught in parallel as they draw on each other. As other classes in your curriculum have an immediate need for optimization strategies, we start with calculus in the winter semester and follow up with linear algebra in the summer semester.

Mathematics I in the winter semester requires you to bring working high school level knowledge of algebra, univariate functions, differentiation and ideally also integration. If you are not joining us straight out of high school, you must review fundamental pre-calculus manipulations of equations etc. on your own or in the prep course. We will very briefly recap elementary univariate functions in this class and then move on to the content in the table below. The agenda is only a rough sketch and depends on holidays etc.

Quick summary:

1. We introduce sequences/series.
2. We review differentiation and integration for univariate functions along with some related concepts that you may not know from high school.
3. Multivariate calculus will take up the most room in this class. We extend the concepts from univariate calculus to multivariate calculus with the goal to solve optimization problems with and without constraints. Most applications will arise in multivariate calculus and we will see how questions translate into multivariate objective functions that we can solve to answer our questions.
4. We will discuss (univariate) dynamical systems in a continous and a discrete setting. A solid understanding of the fundamental relationship between differentiation and integration will carry this analysis.

Detailed content of the module is provided in the official module description.

Mathematics II in the summer semester will allow us to deal with the ubiquitous systems of equations that you will encounter in endless settings. Most students will have less intuition to draw on in linear algebra than in calculus, so we have to cover more ground before we can turn our attention to applications. Once we do, however, we will be powerful enough to tackle a large number of them at once.

Quick summary:

1. We will recap Gaussian elimination and taking dot products as essentially the only computational procedures we need in this class.
2. We will introduce vector spaces and endow them with properties. The discussion of linear maps will connect us to our previous semester of calculus and an extensive discussion of matrices as representations of linear maps along with their subspaces will create the larger picture of mapping between vector spaces.
3. We investigate some of the numerous problems that can be framed as systems of equations.
4. We add rectangular matrices to our larger picture and extend known concepts (inverse, decomposition) to this setting.
5. We study more applications of now under-or overdetermined systems of equations.

Detailed content of the module is provided in the official module description.

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