Math Core

Most quantities you meet outside a textbook depend on more than one input. The temperature in a room depends on where you stand, the cost of a box depends on its length, width and height, and the monthly payment on a loan depends on the principal, the rate and the term. To do calculus with these quantities, you first need a clear picture of what a function of several variables is and how to see one.

Functions of two variables

A function of two variables takes an ordered pair and returns one number.

Definition

Function of two variables

A function of two variables $f$ assigns to each point $(x, y)$ in a set $D \subseteq \mathbb{R}^2$ exactly one real number $f(x, y)$ . The set $D$ is the domain of $f$ , and the set of all output values is the range.

You will often write $z = f(x, y)$ and call $x$ and $y$ the independent variables and $z$ the dependent variable. Evaluating works just like one-variable functions: substitute and simplify. For $f(x, y) = x^2 y - 3y + 2$ ,

f(2, -1) = (2)^2(-1) - 3(-1) + 2 = -4 + 3 + 2 = 1.

The same idea extends to three or more variables. A function $g(x, y, z)$ takes a point in space and returns a number, such as the temperature at that point.

Finding the domain

When a function is given only by a formula, its domain is every point where the formula makes sense as a real number. The usual restrictions carry over from single-variable calculus:

you cannot divide by zero,
the expression under an even root must be $\ge 0$ ,
the input of a logarithm must be $> 0$ .

The difference is that the domain is now a region in the plane, so describe it with an inequality in $x$ and $y$ and, when you can, a sketch.

Worked example: Domain of a square-root function

Find and describe the domain and range of $f(x, y) = \sqrt{9 - x^2 - y^2}$ .

The radicand must be nonnegative: $9 - x^2 - y^2 \ge 0$ , or $x^2 + y^2 \le 9$ . The domain is the closed disk of radius $3$ centered at the origin, including its boundary circle.

For the range, $x^2 + y^2$ runs from $0$ (at the origin) to $9$ (on the circle), so $9 - x^2 - y^2$ runs from $9$ down to $0$ and its square root runs from $3$ down to $0$ . The range is $[0, 3]$ .

Worked example: Domain of a logarithm

Find the domain of $f(x, y) = \ln(x - y)$ .

The logarithm needs $x - y > 0$ , which is the same as $y < x$ . The domain is every point strictly below the line $y = x$ . The line itself is not included, because $\ln 0$ is undefined.

Graphs are surfaces

The graph of $f(x, y)$ is the set of points $(x, y, z)$ in space with $z = f(x, y)$ . Above each input point in the $xy$ -plane you plot a height, so the graph is a surface.

Some graphs you already know from the quadric surfaces in the first unit. The graph of $f(x, y) = \sqrt{9 - x^2 - y^2}$ is the upper half of the sphere $x^2 + y^2 + z^2 = 9$ . The graph of $f(x, y) = x^2 + y^2$ is a paraboloid opening upward, and the graph of $f(x, y) = 6 - 2x - 3y$ is a plane.

Surfaces are hard to draw by hand, and a single 3D picture hides as much as it shows. That is why the next tool is so useful.

Level curves and contour maps

A topographic map shows a mountain on flat paper by drawing curves of constant elevation. You can do the same for any function of two variables.

Definition

Level curve

The level curves of $f(x, y)$ are the curves $f(x, y) = k$ in the $xy$ -plane, one for each constant $k$ in the range of $f$ . A picture of several level curves is a contour map.

Each level curve is what you get by slicing the surface $z = f(x, y)$ with the horizontal plane $z = k$ and dropping the slice straight down into the $xy$ -plane. If you draw level curves for equally spaced values of $k$ , then curves that crowd together mean the surface is steep there, and curves that spread apart mean it is nearly flat.

Worked example: Level curves of an elliptic paraboloid

Sketch level curves of $f(x, y) = x^2 + 4y^2$ for $k = 4$ , $16$ and $36$ .

Set $x^2 + 4y^2 = k$ . For $k > 0$ this is an ellipse. Dividing by $k$ gives $\dfrac{x^2}{k} + \dfrac{y^2}{k/4} = 1$ , so the ellipse crosses the $x$ -axis at $\pm\sqrt{k}$ and the $y$ -axis at $\pm\tfrac{1}{2}\sqrt{k}$ .

$k = 4$ : intercepts $x = \pm 2$ , $y = \pm 1$ .
$k = 16$ : intercepts $x = \pm 4$ , $y = \pm 2$ .
$k = 36$ : intercepts $x = \pm 6$ , $y = \pm 3$ .

Level curves x² + 4y² = 4, 16, 36 of f(x, y) = x² + 4y².Open in grapher →

The heights $4, 16, 36$ are not equally spaced, but notice that the ellipses spread apart at a steady rate while the heights grow faster and faster. That is the signature of a surface that gets steeper as you move away from the origin, which is exactly what an upward paraboloid does. For $k = 0$ the level "curve" is the single point $(0, 0)$ , the bottom of the bowl, and for $k < 0$ there is no level curve at all.

Worked example: Level curves that are parabolas

Describe the level curves of $f(x, y) = y - x^2$ .

Setting $y - x^2 = k$ gives $y = x^2 + k$ : a family of upward parabolas, each a vertical shift of $y = x^2$ .

Level curves y = x² + k for k = −2, 0, 2, 4.Open in grapher →

Because the curves are identical parabolas shifted by equal amounts, the surface rises at a steady rate as you move straight up in the $y$ direction.

Tip

To find the level curve through a particular point, first evaluate $f$ at that point to get $k$ , then write $f(x, y) = k$ .

Three variables: level surfaces

The graph of a function $w = g(x, y, z)$ would live in four dimensions, so you cannot draw it. Instead, you study its level surfaces $g(x, y, z) = k$ . For $g(x, y, z) = x^2 + y^2 + z^2$ , the level surfaces are spheres centered at the origin with radius $\sqrt{k}$ . If $g$ gives the temperature at each point, the level surfaces are surfaces of constant temperature.

Common mistake

Level curves live in the $xy$ -plane, not on the surface. A level curve $f(x, y) = k$ is an equation in two variables only. Writing $z$ into it, or treating it as the surface itself, is a common source of confusion. The value $k$ tells you which height the curve represents.

Practice

Practice 1

Let $f(x, y) = x^2 y - 3y + 2$ . Find $f(2, -1)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Let $g(x, y, z) = x^2 + yz$ . Find $g(1, 2, -3)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Which set is the domain of $f(x, y) = \sqrt{x - y^2}$ ?

Practice 4

The range of $f(x, y) = \sqrt{16 - x^2 - y^2}$ is an interval $[0, M]$ . Find $M$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

The level curve $x^2 + 4y^2 = k$ of $f(x, y) = x^2 + 4y^2$ passes through the point $(2, 1)$ . Find $k$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

The level curve of $f(x, y) = y - x^2$ at height $k = 3$ is the graph of $y = g(x)$ . Enter $g(x)$ .

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

What are the level surfaces $x^2 + y^2 + z^2 = k$ (with $k > 0$ ) of $g(x, y, z) = x^2 + y^2 + z^2$ ?

Practice 8

The level curve of $f(x, y) = \ln(y - x)$ that passes through $(2, 3)$ is a line $y = h(x)$ . Enter $h(x)$ .

Enter an expression, e.g. 3x^2 - 2x + 1