Foundations — Functions of several variables — graphs, level curves, level surfaces

This page assumes you have seen nothing. We will name every squiggle on the parent note and turn it into a picture before it is ever used again.

0. The alphabet of the topic

Below is every symbol the parent page uses. We build them in an order where each one leans only on the ones before it.

1. A number line and the symbol $\mathbb{R}$

The picture: a straight horizontal line stretching forever both ways, with $0$ in the middle. Every point on that line is one real number.

Why the topic needs it: the whole subject is about numbers going in and a number coming out. $\mathbb{R}$ is the bag those numbers are drawn from. When you later see "$f:{\mathbb{R}}^n\to \mathbb{R}$", the plain R on the right is exactly this ruler — the output always lands somewhere on it.

2. Ordered pairs, triples, and the symbols ${\mathbb{R}}^2$, ${\mathbb{R}}^3$, ${\mathbb{R}}^n$

• ${\mathbb{R}}^2$ = "R to the power 2" = the set of all ordered pairs = the whole flat plane.
• ${\mathbb{R}}^3$ = the set of all triples = all of space.
• ${\mathbb{R}}^n$ = all lists of $n$ numbers. The little raised $n$ ("superscript $n$") counts how many numbers you feed in at once.

The picture: to find $(x,y)$, walk $x$ steps East along the horizontal ruler, then $y$ steps North along a second ruler. Where you stop is the point. For $(x,y,z)$, then rise $z$ steps straight up.

Why the topic needs it: a function of several variables is fed an input point. In ${\mathbb{R}}^2$ that input is a pair $(x,y)$; in ${\mathbb{R}}^3$ a triple $(x,y,z)$. This is literally the "several" in "several variables."

3. The arrow $\to$, the letter $f$, and the phrase "function"

The picture: a box labelled $f$. A point $(x,y)$ slides in the left slot; a single number $z$ drops out the bottom. Exactly one number out — never two, never zero — that is the rule that makes $f$ a function.

Why the topic needs it: everything — graphs, level curves, level surfaces — is a way of looking at this one machine. Without the machine there is nothing to draw.

4. The symbols $\subseteq$ and $D$ (domain), and "range"

The picture: shade a region on the input plane. Points inside the shading are legal inputs; points outside are forbidden (they'd force division by zero, or a square root of a negative, etc.).

Why the topic needs it: you can only draw a graph or level set over the domain. Ask "where does the picture even exist?" and the answer is $D$.

5. Squares, $x^2+y^2$, and distance

The picture: the point $(x,y)$, the origin $(0,0)$, and the straight segment between them form a right triangle whose horizontal leg is $x$ and vertical leg is $y$. The Pythagorean theorem says the segment's length $r$ satisfies $r^2=x^2+y^2$. So $x^2+y^2$ is that length squared, and $\sqrt{x^2+y^2}=r$ is the length itself.

Why the topic needs it: the parent's headline examples ($x^2+y^2$ giving circles, $x^2+y^2+z^2$ giving spheres) are pure distance formulas. Once you see "$x^2+y^2=c$" as "squared-distance equals a constant," the fact that it's a circle of radius $\sqrt{c}$ is obvious — same distance in every direction.

6. The output letters $z$, $w$ and the vertical axis

The picture: for $z=f(x,y)$, plant a third ruler standing straight up out of the input plane. The output number $z$ is the height you climb on that vertical ruler above the input spot. Do this for every input point and the heights trace out a surface — the graph.

7. The constant $c$ and the set-builder $\{\cdot :\cdot \}$

The picture: slice the surface with a perfectly horizontal sheet at height $z=c$. The sheet meets the surface along a curve; drop that curve straight down onto the flat input plane. The shadow is the level curve — all the spots that share the output $c$.

Why the topic needs it: level curves and level surfaces are both built with this one template — freeze $c$, collect all inputs giving $c$. Change one word (curve/surface) depending on whether you had 2 or 3 inputs.

8. Dimension bookkeeping ("$n$ minus one")

Why the topic needs it: this is exactly why $f(x,y)=c$ gives curves but $f(x,y,z)=c$ gives surfaces — the parent's "Mistake 3." Keep the arithmetic $n-1$ in your pocket.

How the foundations feed the topic

Equipment checklist

Test yourself — cover the right side and answer out loud.

Connections

• Parent topic — the full picture
• Limits and continuity of multivariable functions — what "approaching a point in ${\mathbb{R}}^n$" means, built on these coordinates
• Partial derivatives — the next tool once the machine and its graph are understood
• Gradient vector — lives perpendicular to the level sets defined here
• Quadric surfaces — the paraboloids and spheres whose distance formulas we unpacked