Functions of Several Variables

Main idea

A function $z = f(x, y)$ gives a height to every point of the plane, so its graph is a landscape. Its level curves $f(x, y) = k$ are the contour lines on a hiking map of that landscape, a flat picture that still shows where it's steep.

z = f(x, y), \qquad \text{level curves: } f(x, y) = {\htmlData{sym=0}{k}}

What the symbols mean

$k$
Fixed height
All the points where the surface is at height k.

Explanation

A function of several variables is simply a rule that takes in more than one number and gives back one number, like the temperature at each spot on a floor, which depends on how far east and how far north you stand. It's what you need any time a quantity depends on more than one thing. Writing $z = f(x, y)$ just means: feed in a location $(x, y)$ (where you are on the map) and get back one number $z$ (the height, or temperature, or whatever you're measuring there). And a level curve $f(x, y) = k$ is just "all the places where the reading is exactly $k$ ." Nothing more to it than that.

Up to now our functions had one input: $f(x)$ , a curve over a number line. Now there are two inputs, so instead of a number line underneath we have a whole plane, and instead of a curve on top we get a surface. That's the reason the last few lessons spent so much time on surfaces.

Domain. It's the same question as before, "where does the formula make sense?", with the same things to avoid: dividing by zero, square roots of negatives and logs of non-positive numbers. The difference is that the answer is now a region of the plane, not an interval.

Graphs and contour maps. The graph is a surface in 3D, which is pretty but hard to draw by hand. A contour map flattens it out. You draw the level curves for equally spaced values of $k$ , exactly like the contour lines on a hiking map. Think about what crowded lines mean: you climb the same height difference in less sideways distance, so where the curves crowd together, the surface is steep. Spread-out curves mean gentle ground.

Three variables. A function $w = f(x, y, z)$ can't be graphed, since that would need four dimensions. But its level surfaces $f(x, y, z) = k$ can be: the surfaces of equal temperature in a room, or of equal pressure in the atmosphere. It's the contour idea, one dimension up.

See it

The surface z = sin x cos y, a landscape of hills and pits laid out like a checkerboard. Every height stays between −1 and 1, and the peaks sit where sin x and cos y are both ±1, such as at (π/2, 0).

A router's signal strength over a floor, S = 100/(1 + 0.1(x² + y²)), drawn as a landscape. The ring at height k is a level curve, and dropping it onto the floor gives the contour line you'd draw on a coverage map. Notice how unevenly the rings spread: from S = 80 to 70 the radius grows only 0.5 m, where the signal falls off fastest, but from 30 to 20 it grows 1.5 m, out where the surface flattens. The usable edge, S = 20, is a circle of radius √40 ≈ 6.3 m.

Worked examples

Example 1

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

1
Setting $x^2 + y^2 = k$ gives a circle of radius $\sqrt k$ for each $k > 0$ .

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Evaluate $f(x, y) = x^2y - 3$ at $(2, -1)$ .

Problem 2+5 XP

What is the domain of $\ln(x - y)$ ?

Problem 3+5 XP

The level curve of $f(x, y) = x^2 + y^2$ at $k = 4$ is a circle. What is its radius?

Problem 4+5 XP

What do the level curves of $z = 2x + 3y$ look like?

Problem 5+5 XP

Evaluate $f(x, y, z) = x + y^2 + z^3$ at $(1, 2, -1)$ .

Problem 6+5 XP

A router's signal strength at floor position $(x, y)$ (meters) is modeled as $S(x, y) = \frac{100}{1 + 0.1(x^2 + y^2)}$ . The level curve $S = 20$ marks the edge of usable coverage. What is its radius?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

Find the area of the domain of $f(x, y) = \dfrac{\ln\left(x^2 + y^2 - 1\right)}{\sqrt{4 - x^2 - y^2}}$ .

Where it's used

Weather maps (isobars and isotherms), topographic maps and the loss landscape a neural network trains on are all functions of several variables drawn as level curves.

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