A function of three variables, $w = f(x, y, z)$, gives a number at every point of space: the temperature in a room, the density inside a planet, the pressure in the air. Its graph would need four dimensions, so we use the idea that worked for contour maps one dimension up.
Key idea
Why it works
| $f(x, y, z)$ | Level surfaces $f = k$ |
|---|---|
| $x^2 + y^2 + z^2$ | spheres of radius $\sqrt{k}$ around the origin ($k > 0$) |
| $ax + by + cz$ | parallel planes with normal $\langle a, b, c \rangle$ |
| $z - x^2 - y^2$ | copies of one paraboloid, lifted by $k$ |
| $x^2 + y^2 - z^2$ | one-sheet hyperboloids ($k > 0$), a cone ($k = 0$), two-sheet hyperboloids ($k < 0$) |
Level surfaces have physical names: in a temperature field they are isothermal surfaces, and for an electric potential they are equipotential surfaces. How closely the layers are packed shows how fast $f$ changes: tightly packed layers mean a steep change. Every graph is also a level surface: $z = g(x, y)$ is the level surface $F = 0$ of $F(x, y, z) = z - g(x, y)$. This point of view pays off with gradients and normal vectors, where the gradient of $f$ turns out to be perpendicular to the level surface through each point.
Worked examples
Worked example: A sphere
- $f(1, 2, 2) = 1 + 4 + 4 = 9$.
- The level surface is $x^2 + y^2 + z^2 = 9$: the sphere of radius 3 around the origin.
Answer: the sphere $x^2 + y^2 + z^2 = 9$
Worked example: Three kinds of levels
- $k = 1$: $x^2 + y^2 - z^2 = 1$, a hyperboloid of one sheet around the $z$-axis.
- $k = 0$: $z^2 = x^2 + y^2$, a double cone.
- $k = -1$: $z^2 - x^2 - y^2 = 1$, a hyperboloid of two sheets.
Answer: one sheet, cone, two sheets
Worked example: A plane
- $k = f(1, 1, 1) = 2 - 1 + 3 = 4$.
- The level surface is the plane ${}2x - y + 3z = 4$.
Answer: ${}2x - y + 3z = 4$
Common mistake