Contour Diagrams

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Section 10.6 Contour Diagrams

Figure 10.6.1 shows the relationship between tables of data, contour lines, and contour diagrams. Figure 10.6.2 then shows the relationship of contour diagrams to the underlying graph.

Figure 10.6.1. The relationship between tables of data and contour diagrams. Click on the check boxes to examine each of these features separately.

Figure 10.6.2. The relationship between contour diagrams and graphs. Click and drag the figure!

Activity 10.6.1.

Can you determine the function whose data is shown in Figures 10.6.1–10.6.2?

The level sets are circles, each with an equation of the form \(x^2+y^2=\hbox{constant}\text{.}\) So the underlying function is \(f(x,y)=x^2+y^2\text{.}\) The three-dimensional graph (click and drag!) in Figure 10.6.2 shows the paraboloid \(z=x^2+y^2\text{,}\) that is, \(z=f(x,y)\text{.}\)

Activity 10.6.2. Using technology to visualize level sets.

After you have thought about these questions yourself, you can use the Sage code below to explore several different mechanisms for visualizing level sets in two dimensions. The code in the first box defines and plots a function of two variables.

Now we can plot a contour diagram of the chosen function \(f\text{.}\)

Now try other functions by plugging something else in for \(f(x,y)\) in the first box and then redoing the other steps. A particularly nice choice is \(f=e^{y^2-x^2}\text{.}\)