Part 1 - Functions of several variables

Analysis of one variable is very powerful, but most of the time, functions will have several variables.

This series will cover analysis in several variables.

Let’s properly define what a function of several variables is

Definition 1 (Function of two variables)

A function of two variables is a rule that associates to each pair, $(x, y)$ of real numbers, in a set, $D$ , a unique number denoted as, $f(x, y)$ .

$D$ is the so-called domain. The domain is the allowed values of our variable parameters.

The range of $f$ , is the set of all values that $f$ can take.

\{ f(x, y) \mid (x, y) \in D \}.

Example 1

f(x, y) = \dfrac{x^3 y}{y - 1}.

Range:

\{ f(x, y) \mid y - 1 \neq 0 \}.

Domain:

\{ (x, y) \mid y \neq 1 \}.

Let’s define graphs in several variables.

Definition 2 (Graph)

The graph of a function of two variables is the set of all points, $(x, y)$ , which, $(x, y) \in D$ .

In other words, it’s a surface with equation:

z = f(x, y).

Example 2

Evaluate $f(0, 4)$ and graph the surface.

f(x, y) = 8 - 4x - 2y.

f(0, 4) = 8 - 0 - 8 = 0.

Level curves are a very important topic when dealing with several variables.

Definition 3 (Level curves)

The level curves of a function of two variables are the curves with the equation:

f(x, y) = k, \quad \text{where } k \text{ is constant}.

Example 3

Sketch the level curves for $f(x, y) = xy$

Case $k = 0$ :

xy = 0 \quad \| \quad x = 0 \text{ or } y = 0.

Case $k \neq 0$ :

xy = k \quad \| \quad x = \dfrac{k}{y} \text{ or } y = \dfrac{k}{x}.

If we plot for different k values:

Let’s compare limits in one variable to several.

\lim_{x \to a} f(x) = L,

\lim_{(x, y) \to (a, b)} f(x, y) = L.

They work quite similar, let’s see the limit laws:

\lim_{(x, y) \to (a, b)} f(x, y) \pm h(x, y) = L \pm M,

\lim_{(x, y) \to (a, b)} f(x, y) \cdot h(x, y) = L \cdot M,

\lim_{(x, y) \to (a, b)} \dfrac{f(x, y)}{h(x, y)} = \dfrac{L}{M}, \quad M \neq 0.