## Mathematics for Economists

## Chapter 9 : Functions of several variables

### By Lund University

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.

## Functions of several variables

We begin this section by looking at functions of two variables. The graph of a function of two variables will be a three dimensional figure. We extend this idea to functions with an arbitrary number of variables.

#### Functions of two variables

#### Graph of a function of two variables

#### Functions of several variables

## Functions of several variables: Problems

Exercises on functions of several variables

#### Problem: Evaluate a function of two variables

#### Problem: Evaluate a function of two variables

#### Problem: Find the natural domain of a function of two variables

#### Problem: Level curves

#### Problem: Intersection of level curves

## Partial derivatives

For a function of two variables, we can define two derivatives, one for each variable. Such derivatives are called partial derivatives. Similarly, we get 4 second order partial derivatives that we collect in a matrix called the Hessian. We can generalize this to a function of an arbitrary number of variables. In the final lecture, we will look at the chain rule for functions of several variables.

#### Partial derivatives, rough introduction

#### Partial derivatives

#### Hessian

#### Chain rule for functions of two variables

## Partial derivatives: Problems

Exercises on partial derivatives and Hessian