Functions of Several Variables

Everything you have done in calculus so far ran on one input. A single-variable function eats one number and returns one number — and that was the right place to learn, the way you learn to ride with training wheels. But look at the quantities the world actually cares about:

The temperature on a weather map depends on where you are: T(x, y) — two inputs, one temperature.
Your monthly loan repayment depends on the principal, the interest rate, and the term: M(P, r, t) — three inputs, one payment.
How cold it feels outside — the wind chill — depends on both the air temperature and the wind speed: W(T, v). The same -5\,°\mathrm{C} feels far crueller in a 30 km/h wind.

A function of several variables takes a whole tuple of inputs and returns a single number,

f : D \subseteq \mathbb{R}^n \to \mathbb{R}, \qquad (x_1, \dots, x_n) \mapsto f(x_1, \dots, x_n).

The domain D is now a region of n-dimensional space, not a slice of the line. For n = 2 we write z = f(x, y), with the domain a region of the plane; for n = 3, w = f(x, y, z), with the domain a chunk of space. The output is still one honest number — what has grown is the address you feed in. Everything in this branch of calculus is the one-variable story told one variable at a time, and this page is about the first skill: reading such a function — as a table, as a surface, and as a contour map.

Evaluating, step by step

Evaluation is exactly as dull as you hope: substitute, then arithmetic. Take the function f(x, y) = x^2 + y^2 and find f(3, 4).

Step 1 — substitute the input coordinates. Put x = 3 and y = 4 into the rule:

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

Step 2 — evaluate each term.

= 9 + 16.

Step 3 — combine.

f(3, 4) = 25.

The order of the inputs matters in general — for g(x, y) = x^2 y we get g(2, 3) = 12 but g(3, 2) = 18 — though for this symmetric x^2 + y^2 it happens not to. An input is an ordered tuple, and swapping coordinates is generally a different point of the domain.

Worked example: the natural domain

When no domain is announced, the convention is the natural domain: every point where the formula makes sense. In one variable that gave you intervals; now it gives you regions of the plane, and finding them is a small geometry puzzle.

Example 1. f(x, y) = \sqrt{\,1 - x^2 - y^2\,}. The square root demands

1 - x^2 - y^2 \ge 0 \quad\Longleftrightarrow\quad x^2 + y^2 \le 1,

so the natural domain is the closed unit disc — every point on or inside the circle of radius 1. (The graph, if you are curious, is the upper half of the unit sphere: a perfect dome sitting on that disc.)

Example 2. g(x, y) = \ln(x + y). The logarithm demands x + y > 0, i.e. y > -x: an open half-plane, everything strictly above the line y = -x. The boundary line itself is excluded — on it the log blows up.

Notice the change of scenery: one-variable domains were unions of intervals, but a two-variable domain can be a disc, a half-plane, a plane with a curve deleted (think 1/(y - x^2), the plane minus a parabola), or something far stranger. Sketching the domain is usually the first move in any problem.

Three pictures of one function

A function of two variables can be read three ways, and fluency means moving between them freely.

1. As a table. Fix a grid of inputs and list the outputs. This is how wind chill is actually published: rows are wind speed v, columns are air temperature T, and each cell is W(T, v) in °C:

v \backslash T	0\,°\mathrm{C}	-5\,°\mathrm{C}	-10\,°\mathrm{C}
10 km/h	−3	−9	−15
20 km/h	−5	−12	−18
30 km/h	−6	−13	−20

Read down a column and you see what more wind does at fixed temperature; read along a row and you see what colder air does at fixed wind. (That "hold one input still and vary the other" reflex is exactly the idea that becomes the partial derivative.) A table is honest but blurry — it only samples the function.

2. As a surface. The graph of z = f(x, y) lives in three dimensions: above every point (x, y) of the domain, plot the height z = f(x, y). The result is a surface — a landscape floating over the plane. The paraboloid z = x^2 + y^2 is a bowl with its lowest point at the origin (height 0 there, rising in every direction); z = -(x^2 + y^2) is the same bowl flipped into a dome; z = x^2 - y^2 is a saddle — a mountain pass that climbs along the x-axis and falls along the y-axis.

3. As a contour map — the star of this page, coming next. A surface is hard to draw on flat paper, so cartographers solved the problem centuries ago: flatten the landscape into curves of constant height.

See a surface for real

A function of two variables is a surface — a landscape of heights floating over the x–y plane. Here is z = \sin x \cos y, a rolling egg-carton of hills and valleys. Drag it to rotate and explore the peaks and troughs from any angle.

Reading a contour map

A level curve (or level set) of f is the set of points where the function takes a fixed value c,

\{(x, y) : f(x, y) = c\}.

Picture slicing the surface with the horizontal plane z = c and dropping the cut edge straight down onto the floor: that shadow is the level curve. Draw the curves for several evenly spaced values of c on one plane and you have a contour map — the hiker's-map view of the function.

Worked example. For f(x, y) = x^2 + y^2 the level curve at value c is

x^2 + y^2 = c,

which for c > 0 is a circle of radius \sqrt{c}; at c = 0 it collapses to the single point at the origin (the bottom of the bowl), and for c < 0 it is empty — the bowl never dips below zero. The family is a set of nested rings, like a target. And notice the spacing: the levels c = 1, 4, 9 are equally spaced in height, but their radii 1, 2, 3 crowd closer together going outward per unit of height gained — the bowl steepens as you climb its wall, and the map shows it by bunching. Reading the map is a skill worth drilling:

Closely spaced curves mean a steep slope — the height changes fast over a short horizontal distance. Widely spaced curves mean gentle ground.
A point's height is the label of the curve it sits on (interpolating between two neighbours if it lands between them).
Concentric closed loops surround a peak or a pit — a local maximum or minimum.

Slide the level c below and watch the single curve x^2 + y^2 = c sweep outward as a growing circle.

A function of several variables is a map f : D \subseteq \mathbb{R}^n \to \mathbb{R} assigning one real number to each point of a domain D. For n = 2:

Its graph \{(x, y, f(x, y)) : (x, y) \in D\} is a surface in \mathbb{R}^3.
Its level curve at value c is \{(x, y) \in D : f(x, y) = c\}, the cross-section of the graph at height z = c projected onto the plane.
Level curves never cross (a point cannot have two heights), and their spacing encodes steepness: tightly packed where the surface is steep, spread out where it is flat.

A gallery of contour shapes

Different functions leave utterly different fingerprints on the plane, and with a little practice you can name the surface from its map alone.

Worked example: f(x, y) = xy. The level curve xy = c is, for c \ne 0, the hyperbola y = c/x: for c > 0 its two branches sit in the first and third quadrants, for c < 0 in the second and fourth. The zero set xy = 0 is the pair of coordinate axes — the crossed lines that the hyperbolas hug ever closer without touching.

Worked example: f(x, y) = x^2 - y^2. Same story rotated by 45°: x^2 - y^2 = c gives hyperbolas opening left–right for c > 0 and up–down for c < 0, while c = 0 factors as (x - y)(x + y) = 0 — the crossed lines y = \pm x. This is the map of the saddle: stand at the origin and the ground rises ahead and behind (along the x-axis) and falls to your left and right (along the y-axis). Hikers know this map instantly — it is a mountain pass.

One more for the collection: a linear function f(x, y) = ax + by has level curves ax + by = c — a family of parallel, evenly spaced straight lines. The surface is a tilted plane: an infinite ramp, equally steep everywhere, which is exactly what evenly spaced contours say.

Level curves never cross. A crossing point would sit on two curves at once — meaning f takes two different values at one point, which no function does. (The crossed lines of the saddle are not a violation: they are branches of one level set, c = 0.) If your sketch shows contours for two different values intersecting, the sketch is wrong.
Closely spaced contours mean steep, not "more important." Bunched curves say only that the height changes fast there. Nothing about the region being special, big, or high — a dense band can sit at any altitude.
The surface and the contour map are the same function, drawn twice. The graph is a 3-D object, \{(x, y, f(x, y))\}; the contour map flattens it onto the 2-D domain. Neither picture is the function — both are portraits of it, and every fact about one translates into a fact about the other.
A level curve need not be a single curve. It can be two branches of a hyperbola, a point (the bowl's bottom at c = 0), or empty (the bowl at c = -1). "Level set" is the safer name.

Because it is one. Atmospheric pressure is a function p(x, y) of position, and the isobars on a weather chart are precisely its level curves; isotherms are level curves of the temperature field T(x, y). All the reading rules apply verbatim: closed loops ring a HIGH (a peak of the pressure surface) or a LOW (a pit), and tightly bunched isobars mean pressure changing fast — which is where the wind howls. Even better, at large scales the wind does something delightful: instead of blowing from high to low pressure it blows along the isobars (the Earth's rotation deflects it), circling lows anticlockwise in the northern hemisphere. A forecaster glancing at contour spacing and loop direction is doing multivariable calculus by eye.

Video games pull the same trick from the other side. A game "terrain" is usually a heightmap: a grayscale image whose pixel brightness at (x, y) is the height z — literally a function z = f(x, y) stored as a table of samples, which the engine skins into the mountains you run over. Every open-world landscape you have explored was a function of two variables wearing a costume.

For w = f(x, y, z) the graph would need four dimensions, so we give up on graphs — but level sets survive beautifully: the set f(x, y, z) = c is a level surface in ordinary 3-D space. For f = x^2 + y^2 + z^2 the level surfaces are concentric spheres of radius \sqrt{c} — nested shells, the 3-D answer to the target of nested rings. The isothermal surfaces inside a heated solid, or surfaces of constant electric potential around a charge, are exactly this.