Functions of two variables

1. The plane ℝ²

The set ${\mathbb{R}}^2$ consists of all ordered pairs $(x,y)$ of real numbers. Once a Cartesian coordinate system is fixed in the plane, each pair corresponds to exactly one point $P=(x,y)$, and vice versa, so ${\mathbb{R}}^2$ is identified with the plane. The pair is ordered, therefore $(1,3)$ and $(3,1)$ are different points. A function of two variables assigns a number to each point of a region of the plane, and describing these regions requires the notions of distance and neighborhood.

2. Distance between two points

The distance between the points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$ is

The differences $x_2-x_1$ and $y_2-y_1$ are the legs of a right triangle whose hypotenuse is the segment $PQ$, so the formula is the Pythagorean theorem. For $P=(1,1)$ and $Q=(4,5)$ we have $d(P,Q)=\sqrt{3^2+4^2}=5$. The distance of $P=(x,y)$ from the origin $O$ is $\sqrt{x^2+y^2}$, and the points at distance $r$ from a point $(x_0,y_0)$ form the circle with equation $(x-x_0{)}^2+(y-y_0{)}^2=r^2$.

3. Circular neighborhood

Let $P_0=(x_0,y_0)$ be a point and $r>0$ a real number. The circular neighborhood with center $P_0$ and radius $r$ is the set of points whose distance from $P_0$ is less than $r$:

It is the disk with center $P_0$ and radius $r$ without its boundary circle. Neighborhoods play in the plane the role that the intervals $(x_0-r,x_0+r)$ play on the line. For example, $(2,3)$ belongs to $B((1,2),2)$, because its distance from the center is $\sqrt{2}<2$, while $(3,4)$ does not belong to it, because its distance is $\sqrt{8}>2$.

4. Interior and boundary points

Let $A$ be a subset of the plane and $P_0$ a point.

• $P_0$ is interior to $A$ if there is a neighborhood $B(P_0,r)$ contained in $A$.
• $P_0$ is exterior to $A$ if there is a neighborhood $B(P_0,r)$ that contains no points of $A$.
• $P_0$ is a boundary point of $A$ if every neighborhood of $P_0$ contains both points of $A$ and points not in $A$.

The set of boundary points is called the boundary of $A$. A boundary point may or may not belong to $A$. For the disk $A=\{(x,y):x^2+y^2<1\}$ consider a point $P_0$ with $x^2+y^2<1$. Its distance from the origin is $d(P_0,O)<1$, so its distance from the circle is $1-d(P_0,O)$. The disk with center $P_0$ and radius $r=1-d(P_0,O)$ is therefore contained in $A$, and $P_0$ is interior. The points of the circle $x^2+y^2=1$, on the other hand, are boundary points, because every disk centered at one of them contains points closer to the origin and points farther from it.

5. Open, closed and bounded sets

Let $A$ be a subset of the plane.

• $A$ is open if all its points are interior, that is, if it contains none of its boundary points.
• $A$ is closed if it contains all its boundary points.
• $A$ is bounded if it is contained in a large enough disk, that is, if there exists $R>0$ such that $d(P,O)\le R$ for every $P\in A$.

The disk $x^2+y^2<1$ is open and bounded; the disk $x^2+y^2\le 1$ is closed and bounded; the half-plane $y>0$ is open and not bounded. As a rule, strict inequalities between continuous functions give open sets and non-strict ones give closed sets. Closed and bounded sets are those on which the Weierstrass theorem holds, and they will come back in the topic on unconstrained and constrained extrema.

1. Function of two variables

Let $D$ be a subset of ${\mathbb{R}}^2$. A function of two variables $f:D\to \mathbb{R}$ assigns to each point $(x,y)\in D$ one and only one real number $f(x,y)$. The set $D$ is the domain, and the variables $x$ and $y$ are independent, that is, they can be varied separately. For example, $f(x,y)=x^{2}y-3y$ at $(2,1)$ equals $4-3=1$. Many economic quantities depend on more than one variable; for example, the quantity produced depends on the capital and labor employed, and a consumer’s utility on the quantities of the goods purchased.

2. Natural domain

When a function of two variables is given only by its formula, its natural domain is the set of points $(x,y)$ for which the formula makes sense. The conditions are the same as for functions of one variable:

• denominators are nonzero;
• the arguments of even-index roots are greater than or equal to zero;
• the arguments of logarithms are strictly positive.

The difference lies in the solutions, which are regions of the plane and are represented by a drawing. The most frequent boundary curves are lines, parabolas and circles.

• The line $ax+by+c=0$ divides the plane into two half-planes; in one $ax+by+c>0$ holds, in the other $ax+by+c<0$, and to tell which is which one substitutes a point not on the line, usually the origin.
• The inequality $y>g(x)$ identifies the points that lie above the graph of $g$.
• The inequality $(x-x_0{)}^2+(y-y_0{)}^2<r^2$ identifies the points interior to the circular region with center $(x_0,y_0)$ and radius $r$.

In the drawings, the boundary curves that belong to the domain are drawn with a solid line, those that do not belong to it with a dashed line.