Functions of Several Variables

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.

Idea clave

The level curve $f(x, y) = k$ collects every point where $f$ has the value $k$. If the levels are equally spaced in height, then level curves that are close together mean the surface is steep there, and curves far apart mean it is gentle.

Por qué funciona

Neighboring contours differ by the same height step $\Delta z$. If you must walk a horizontal distance $d$ to get from one to the next, the average slope along that walk is $\frac{\Delta z}{d}$. Same rise, shorter run: a bigger slope. This is the same rise-over-run idea as the slope of a line, measured on the map.

The contour map of $f(x, y) = x^2 + y^2$. The circles crowd together farther out, because the bowl gets steeper there. Switch to the 3-D view to see the same curves drawn on the surface.

Worked examples

Ejemplo resuelto: Domain and range

Find the domain and range of $f(x, y) = \sqrt{9 - x^2 - y^2}$, and describe its graph.

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.

Respuesta: domain: the disk $x^2 + y^2 \le 9$; range $[0, 3]$; graph: a hemisphere

Ejemplo resuelto: Reading a contour map

Describe the level curves of $f(x, y) = x^2 + y^2$ for $k = 1, 2, 3, 4$ and say where the surface is steepest.

$x^2 + y^2 = k$ 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.

Respuesta: circles that crowd together outward; steeper away from the center

Ejemplo resuelto: The level curve through a point

Which level curve of $f(x, y) = x^2 - 3xy$ passes through $(2, -1)$?

Evaluate $f$ there: ${}4 - 3(2)(-1) = 4 + 6 = 10$.
So the point is on the level curve $x^2 - 3xy = 10$.

Respuesta: $k = 10$

Error común

Reading close level curves as a gentle slope. On a contour map, crowded lines can look "busy but flat". They mean the opposite: the height changes by the same step over a shorter distance, so the ground is steep. Check: estimate rise over run between two neighboring curves.

Worked examples

Práctica

Se apoya en

Lleva a

Fuentes y lecturas