We introduce functions that take vectors or points as inputs and output a number.

The world is constantly changing. Sometimes this change is very slow, other times it is shockingly fast. Consider Meteor Crater in northern Arizona:
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This area was once grasslands and woodlands inhabited by bison, camels, wooly mammoths, and giant ground sloths. During the Pleistocene epoch, a meteor only meters in diameter collided with the Earth and this changed very quickly. The collision released around joules of energy, comparable to the energy released by a large nuclear weapon. A fireball extended out kilometers from the center of the impact, destroying all life in its wake. It is estimated it took one-hundred years for the local plant and animal life to repopulate the area. Fifty-thousand years later, the remains of the impact crater are still intact on our ever-changing Earth.

To help us understand events like these, we need to precisely describe what we are observing (in this case, the crater). To do this we use a contour map, often called a topographical map:

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In essence we are looking at the crater from directly above, and each curve in the maps above represents a fixed, constant height. Mathematically, a contour map describes a function of two variables. We will now define a more general case of a function of variables. These are often called functions of several variables.

Let’s investigate functions of two variables, :

Consider and compute .
What is the domain of ? The domain is all vectors allowable as inputoutput for . Because of the square-root, we need such that: Write

This inequality describes the interior of an ellipse centered in the -plane.

What is the range of ? The range is the set of all possible inputoutput values. The square-root ensures that all output is . Since the and terms are squared, then subtracted, inside the square-root, the largest output value comes at , : . Thus the range is the interval .

Now let’s ponder functions of three variables, .

Consider and compute .
What is the domain of ? The domain is all vectors allowable as inputoutput for . Because of denominator in the expression representing , we need to find such that We recognize that the set of all points in that are not in form a lineplanecircle in space that passes through the origin, with normal vector
What is the range of ? The range is the set of all possible inputoutput values. It happens to be all ofa proper subset of . There is no set way of establishing this. Rather, to get numbers near we can let and choose . To get numbers of arbitrarily large magnitude, we can let .

Visualizing functions of several variables

The graph of a function of a single variable, is a curve in a two-dimensional plane. The graph of a function of two variables, is a surface in three-dimensional space. The graph of a function of three variables, is a surface in four-dimensional space. How can we visualize such functions? While technology is readily available to help us graph functions of two variables, there is still a paper-and-pencil approach that is useful. This technique is know as sketching level sets. When working with functions , our level sets are level curves, and when working with functions , our level sets are level surfaces.

Level curves

It may be surprising to find that the problem of representing a three dimensional surface on paper is familiar to most people (they just don’t realize it). Topographical maps, like the one shown in Figure represent the surface of Earth by indicating points with the same elevation with contour lines. Another example would be isotherms, we see these in weather maps:

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Given a function , we can draw a ‘‘topographical map’’ of by drawing level curves (or, contour lines). A level curve at is a curve in the -plane such that for all points on the curve, . Below we see a surface with level curves drawn beneath the surface:

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When drawing level curves, it is important that the values are spaced equally apart as that gives the best insight to how quickly the ‘‘elevation’’ is changing. Examples will help one understand this concept.

If one example is good, two is better.

Level surfaces

It is very difficult to produce a meaningful graph of a function of three variables.

  • A function of one variable is a curve drawn in two dimensions.
  • A function of two variables is a surface drawn in three dimensions.
  • A function of three variables is a hypersurface drawn in four dimensions.

There are a few techniques one can employ to try to ‘‘picture’’ a graph of three variables. One is an analogue of level curves: level surfaces. Given , the level surface at is the surface in space formed by all points where . Time for an example.