So far a function took one number and returned one number. Many quantities depend on two: the height of land depends on east-west and north-south position; the temperature on a metal plate depends on $x$ and $y$. A function of two variables $z = f(x, y)$ assigns one number to each point of a region of the plane, its domain. Its graph is a surface in space, like the quadric surfaces you have met.
Surfaces are hard to draw by hand, so we borrow an idea from hiking maps. Slice the graph with horizontal planes $z = k$ and drop each slice onto the $xy$-plane. Each curve $f(x, y) = k$ is a level curve (contour), and a family of them is a contour map.
Key idea
Why it works
Worked examples
Worked example: Domain and range
- The square root needs ${}9 - x^2 - y^2 \ge 0$, that is $x^2 + y^2 \le 9$: the closed disk of radius 3.
- The quantity under the root runs from ${}0$ (on the edge) to ${}9$ (at the center), so $f$ runs from ${}0$ to ${}3$.
- Squaring $z = \sqrt{9 - x^2 - y^2}$ gives $x^2 + y^2 + z^2 = 9$ with $z \ge 0$: the upper half of a sphere.
Answer: domain: the disk $x^2 + y^2 \le 9$; range $[0, 3]$; graph: a hemisphere
Worked example: Reading a contour map
- $x^2 + y^2 = k$ is a circle of radius $\sqrt{k}$: radii ${}1$, ${}1.41$, ${}1.73$, ${}2$.
- The gaps between neighbors are ${}0.41$, ${}0.32$, ${}0.27$: they shrink as you move out.
- Same height step, shorter gaps: the bowl is steeper farther from the origin.
Answer: circles that crowd together outward; steeper away from the center
Worked example: The level curve through a point
- Evaluate $f$ there: ${}4 - 3(2)(-1) = 4 + 6 = 10$.
- So the point is on the level curve $x^2 - 3xy = 10$.
Answer: $k = 10$
Common mistake