A single-variable
function
eats one number and returns one number. The real world rarely cooperates: the temperature in
a room depends on where you stand — two coordinates — and the pressure of a gas
depends on volume and temperature. 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. Everything in
this branch of calculus is the one-variable story told one variable at a time.
Evaluating, step by step
Evaluation is exactly as dull as you hope: substitute, then arithmetic. Take the paraboloid
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 — f(x, y) need not equal
f(y, x) — but for this symmetric paraboloid it happens to.
The graph is a surface
The graph of z = f(x, y) lives in three dimensions: for every point
(x, y) in the domain you plot the height
z = f(x, y) above it. The result is a surface — a
landscape floating over the plane. The paraboloid x^2 + y^2 is a
bowl with its lip at the origin; -(x^2 + y^2) is the same bowl
flipped into a dome.
Reading a contour map
A surface is hard to draw on flat paper, so cartographers solved the problem centuries ago:
a contour map. A level curve (or level set) is the set of
points where the function takes a fixed value c,
\{(x, y) : f(x, y) = c\}.
It is the shadow cast on the plane by slicing the surface at height
z = c. For f = x^2 + y^2 the level curve
x^2 + y^2 = c is a circle of radius
\sqrt{c} (for c > 0); the family of them
is a set of nested rings, like a target. 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.
The contour map is not a calculus contrivance — it is how the world already draws
three-dimensional data. On a topographic map the level curves are lines of constant
altitude, and a mountaineer reads the bunching of contours as a cliff. On a
weather chart isobars are level curves of constant pressure and
isotherms level curves of constant temperature; the tight gradient between
a high and a low is exactly where the wind howls.
The same picture generalises up a dimension: for
w = f(x, y, z) the level sets
f = c are surfaces in space (the
isothermal surfaces of a heated solid, say). We cannot draw a 4-D graph, but we can always
draw its 3-D level surfaces — which is why level sets, not graphs, are the working tool in
higher dimensions.