Mathematics for Economists
by Lund University
This course will cover basic university level mathematics. It is based on the book Essential Mathematics for Economic Analysis by Knut Sydsaeter, Peter Hammond, Arne Strom and Andrés Carvajal.
In this chapter, we look at the foundations of mathematics. In particularly, we will look at logic and sets. You can go quite far in mathematics without the foundations. However, in order to develop a better understanding of mathematics, foundations are very useful as you will know the exact definitions of concepts we use in mathematics.
Algebra is one of the main and most important part of mathematics. Chapter 2 will cover the algebra needed for this course, such as rules of algebra, fractions, powers, inequalities and logarithms. Most of what you will see in this chapter will be known to you. However, unless you had studied mathematics at the University level, it might be a good idea to go through this chapter. First of all, it will give you a chance to refresh and review material that you have done before. Second, the structure of the material in this chapter is more formalized with a clear separation of definitions and results.
Most of chapter 3 is devoted to equations but we have also included a section on the summation sign. We will introduce all the terminology that we need for a fundamental understanding of what it means to solve an equation. We will focus most of our attention on quadratic equations and systems of linear equations.
This chapter is devoted entirely to one of the most important concepts in mathematics namely functions. We begin the chapter by carefully looking at exactly what we mean by a function introducing the domain, codomain and range of a function. The most important class of functions are the linear functions which we will study extensively. Next, we look at some of the most important nonlinear functions. The chapter is concluded with a few slightly more advanced topics related to functions.
This chapter is entirely devoted to the derivative of a function of one variable. The derivative of function is defined as a limit of a specific ratio (the Newton quotient) and we begin the chapter with a brief introduction to limits. In the rest of the chapter we will learn how to differentiate various functions. To our help we will have a bunch of rules such as the chain rule. We will also need higher order derivatives. For example, we can sometimes use the second order derivative to distinguish between a maximum point and a minimum point. This chapter is concluded with a few more advanced topics such as implicit differentiation.
In this chapter we will look at how to maximize or minimize a function of one variable. There is not much theory in this chapter, the best way of learning how to optimize functions is through lots of problems.
This chapter introduces integrals and anti-derivatives, also called primitive functions or indefinite integrals. If you differentiate an anti-derivative you will come back to the function you started with. We will look at the connection between integrals, which is an area under the graph of the function, and the anti-derivative of the function.
This chapter is an introduction to matrix algebra, or more generally, linear algebra. We will look at matrices and the algebra of matrices, that is, how to add and multiply two matrices and the rules that apply when we add and multiply matrices. In this chapter we also look at the transpose of a matrix, the inverse of a matrix and the determinant of a matrix. We also investigated the relationship between matrix algebra and systems of linear equations.
So far, we have only considered functions of one variable. In this chapter we will look at functions of an arbitrary number of variables. An important concept for a function of one variable was the derivative. When we have a function of several variables, we will have several different derivatives which we will call partial derivatives. Fortunately, finding partial derivatives is generally no more difficult than finding ordinary derivatives. This chapter also introduces the Hessian, a matrix where we have collected all the second order partial derivatives.
In the final chapter of this course we look at maximization and minimization of functions of several variables. Optimization can be either unconstrained or constrained. By constrained optimization, we mean that the variables must satisfy one or several constraints, we are not free to pick any values when we optimize the function. Unconstrained optimization is very similar to optimization of a function of a single variable. For constrained optimization, we will study a solution technique called the method of Lagrange.