Level set examples

Example 1

Let f(x,y) = x2 − y2. We look at the level curves c = x2 − y2.

First, look at the case c = 0. The level curve equation x2 − y2 = 0 factors to (x − y)(x + y) = 0. This equation is satisfied if either y = x or y = −x. Both these are equations for lines, so the level curve for c = 0 is two lines.

If c≠0, then we can rewrite the level curve equation c = x2 − y2 as

1 = x2 c −y2 c .

If you remember you conic sections, you’ll recognize this as the equation for a hyperbola. If c is positive, the hyperbolas open to the left and right. If c is negative, the hyperbolas open up and down.

For example, if c = 1, the equation is x2 − y2 = 1. If c = −1, the equation is y2 − x2 = 1. A number of level curves are plotted below.

PIC

We can “stack” these level curves on top of one another to form the graph of the function. Below, the level curves are the thick blue lines. Drag the green point from 0 to 1. When the point is at 1, each level curve given by c = f(x,y) will be at the height z = c.

This is yet another way to visualize the relationship between the level curves and the graph of z = f(x,y) shown below.

Example 2

Let f(x,y,z) = x2 + y2 + z2. Although we cannot plot the graph of this function, we can graph some of its level surfaces. The equation for a level surface, x2 + y2 + z2 = c, is the equation for a sphere of radius c.

Below is the level surface with c = 1 (i.e., sphere with radius 1) drawn in dark red along with the level surface with c = 4 (i.e., sphere with radius 2) drawn in light green.