Functions of Multiple Variables

A function of two variables $f(x, y)$ assigns a number to each point $(x, y)$ of its domain: the temperature at each spot on a map, the height of terrain, the profit from selling $x$ of one product and $y$ of another.

Domain and range

The domain is every $(x, y)$ where the formula makes sense:

• $\sqrt{9 - x^2 - y^2}$ needs $x^2 + y^2 \le 9$: the disk of radius $3$.
• $\ln(x - y)$ needs $x > y$: the half-plane below the line $y = x$.
• $\frac{1}{x^2 + y^2}$ needs $(x, y) \ne (0, 0)$.

The graph

The graph of $f$ is the surface $z = f(x, y)$ in 3D. $z = x^2 + y^2$ is a bowl (a paraboloid, 13.05); $z = \sqrt{9 - x^2 - y^2}$ is the upper half of a sphere.

Level curves and contour maps

The level curve $f(x, y) = k$ is the set of points where $f$ equals $k$: slice the surface at height $k$ and drop the slice onto the $xy$-plane. Drawing several gives a contour map, like a hiking map's elevation lines.

For $f = x^2 + y^2$, the level curves are circles, and they get closer together as you move out: equal steps in height take less and less horizontal distance, because the bowl gets steeper. Close contour lines mean steep; widely spaced lines mean gentle.

Three variables

A function $f(x, y, z)$ can't be graphed (that would need 4D), but its level surfaces $f(x, y, z) = k$ can be. For $f = x^2 + y^2 + z^2$ they're spheres; for $f = x + 2y + 3z$ they're parallel planes.