Level curves

We can visualize a scalar-valued function of two variables, f : R²$\to$R, by graphing its level curves, i.e., the curves where f (x, y) is a constant value. In fact, you've done this already using the Mathematica function ContourPlot. (ContourPlot also includes shading to indicate which values are larger; we won't do that here.)

A level curve is simply a cross section of z = f (x, y) taken at a constant value of z. For example, by now you know that the z cross sections of an elliptic parabaloid, say z = - x² -2y², are ellipses. For some constant c, the graph of c = - x² -2y² is an ellipse as long as c < 0. Hence, each level curve is an ellipse.

We can plot the level curves for a bunch of different constants c together in a level curve plot. For the above example, the level curve plot looks like:

The smallest ellipse in the center is when c = - 1. Working outward, the level curves are for c = - 2, -3,..., - 10.

To illustrate the relationship between the level curves and the graph, the below CVT includes the graph of the function f (x, y) = - x² -2y² along with a level curve plot hovering about the graph. Drag the blue dot up and down to change the value of c. The level curve f (x, y) = c is shown in magenta in the level curve plot. The level curve is where the plane z = c intersects the graph z = f (x, y).