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A function of two real variables defines a surface in three
dimensions, the dimensions being the original two and the
function itself. We can produce a three dimensional image
today, but for many years that was quite impractical, and
mathematicians had to satisfy themselves with two dimensional
images of these surfaces in three dimensions.
There are two fundamental and complementary ways to do this.
The first is to plot equi-value contour lines in the xy plane.
This method is used to show equal pressure lines (called isobars)
in weather maps, or to show height of land surfaces in topographical
maps. When such contours are reasonably smooth, the tangent
to a contour line represents a direction that is in the intersection
of a horizontal plane and the tangent plane to the surface
at that point. Its normal in the xy plane points in the direction
of the gradient vector (plus or minus it) and can also be
used to describe the surface. With a little practice you can
get a pretty good idea what the function surface looks like
from such contour lines. Thus the function rises relatively
steeply where such contour lines for different values of the
function are close to one another, and rises relatively gently
when they are far apart.
A second way to describe a function of two variables is by
drawing little arrows in the direction of the gradient vector
at lots of points, and connecting these into "lines of
increase" (not a common term). These lines will be perpendicular
to the equal-value contour lines, and will go from "local
minimum points of the function to local maximum points",
(or to or from the boundary of the region in which you are
examining the function.) You can get a pretty good idea of
the nature of the function from a picture of this kind. When
the function under investigation represents potential energy
in some physical setting, or potential in electrostatics,
then the lines here described are 'lines of force' which show
the direction that the force on an object or tiny charged
particle point.
For physical applications we really want to be able to visualize
functions of three variables, which are unusually difficult
to describe on one flat surface. We will try to find ways
to do this later on.
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