CONTOUR MAPS

(To control animations: place the cursor over the graphic, then use 'right click' on a PC or 'control+click' on a MAC to select/deselect 'play' and 'loop'.)

Imagine you are out hiking in Yosemite, say. To understand the terrain, what do you do? Well, you whip out your iPhone, load a Google map, and view it in Terrain mode. What you might see is shown to the right. It's simply a contour map of the area with the contours being the lines of constant altitude at equally spaced altitudes

9,000ft, 9,200ft, 9,400ft, 9,600ft, ... .

Before iPhones one would probably have used actual hard copy maps. But the principle is the same: a surface in 3-space is represented as a two-dimensional image by combining all the equally spaced horizontal slices in one plane image.

The slope of the surface in a particular direction depends on how close the contour lines are in that direction; whether the slope is positive or negative in a particular direction depends on whether the values of the countours are increasing or decreasing in that direction; the top of a mountain occurs where the contours surround a point; ... . If these sound like differential calculus concepts, they are! And interpreting them via contour maps makes clearer what the concepts really mean.

Replace Yosemite National Park with the graph of $z = 2 \sin x \sin y$, take horizontal slices at equally spaced values of $z$, and plot them on a single $xy$-plane. Just as the contour map gives a 2D-satellite view of Yosemite, so we now get a satellite view of the graph with a 3D-side-view thrown in as well:

But how can we extract mathematical information from a contour map? The location of the (absolute) maxima and minima are just as clear from the contour map above as from the actual graph. The labels on the contours also indicate the maximum and minimum values. As for interpreting steepness (mathematically = slope) in various directions, look at the following animation:

It is a paraboloid, the graph, say, of $z = x^2 + y^2$. Horizontal cross-sections are being taken at equally spaced values of $z$, say $z =k$ with $k = 0,\, 1, \, 2,\, 3, ...\,9 $ and so on. Algebraically, we see that the cross-sections are circles with radius $r = \sqrt{k}$, so as the cross-sections increase with $k$, the radius increases but increases only as fast as $\sqrt{k}$. This means that the circles expand outward, but get successively closer together. The contour map shows this.

If we reverse the argument and assume the contours are labelled, the fact that the contours are expanding outwards as the height increases must mean that the surface is increasing outwards, and, crucially, the fact that the contours are getting successively close together means that surface is increasingly sucessively faster, in particular, vertical slices of the surface are concave upward. And all this is taking place equally in all directions, so the surface must be symmetric about a vertical axis and increasing equally in all radial directions.

But how would all this change if the paraboloid is replaced by the top half of the cone $z^2 = x^2 + y^2$?

Our iconic hyperbolic paraboloid is a good example of where slope depends critically on direction. Study the next pair of realizations of a hyperbolic paraboloid and then answer the questions following:

How would you label each contour? By a positive, negative, or zero value?

At $P$, in which direction will you head as you start to walk along a contour: the $x$-direction, $y$-direction, or neither?

At $Q$, if you walk in the $x$-direction, would you be walking uphill, downhill, or neither?

At $R$, in which direction will you head as you start to walk along a contour: the $x$-direction, $y$-direction, or neither?

Is $f_x$ positive, negative, or zero at each of $P,\, Q$ and $R$. What can be said about $f_y$?