Fields, an Heuristic Approach
Imagine
a mathematical cube. Imagine also that the cube is empty. Now place
a single mathematical point inside the cube. (The location of the point
is arbitrary, so long as the point is inside.) Place a second point
somewhere else inside the cube, and allow this second point to approach
the first until the spacing between the two is arbitrarily small. Repeat
this procedure over and over until the cube is densely filled with mathematical
points; i.e., either (a) so that there is no volume inside the cube,
regardless of how small, in which there is not at least one mathematical
point, or, equivalently, (b) so that if, at any point P inside the cube,
you were to construct a sphere with center at P, then allow the sphere
to become arbitrarily small, there would always be at least one other
point inside the sphere besides P. Call this space of mathematical points
a continuum. In itself, the continuum has no other function than to
serve as a substratum for constructing various types of mathematical
spaces and fields. Let us now use this continuum of points to illustrate
the concept of the field.
First,
we must orient the cube. Choose an orientation such that you can distinguish
a left face, a right face, and a face nearest you along whose normal
line you are observing the cube. We shall use the left-right axis of
the cube as a reference direction. Now, to every point inside the cube
let us assign a color from the visible spectrum according to some agreed-upon
rule. (Mathematically, this operation may be done by assigning a unique
frequency to every point.) Call the color at any given point a field
quantity, and call the resultant whole a color field (. . . so far,
so good). A field is a continuum of points with a mathematical quantity
assigned to every point according to some kind of rule. We will now
consider three different types of color field, each constructed according
to a different rule.
Time-Invariant
(Static) Continuous Color Field:
Imagine a color field in which the color changes 'gradually' from left
to right, say starting with red at the left face of the cube and progressing
smoothly through the rainbow to indigo at the right face. Let the colors
remain non-varying with time. Such a field is called a time-invariant
(static) continuous field. Now pick any two points along the left-right
axis and compare their colors. There is a measurable difference (arithmetic
difference in frequency). This difference becomes arbitrarily small
as the points become arbitrarily close. When the distance between the
points approaches zero, the ratio of the difference between the colors
(frequencies) and the distance between the points approaches a finite
value. This ratio is called the first derivative at the point, and,
with x representing distance and
representing frequency, is written
d/dx
= the limit, as x
approaches zero, of the ratio /x
where
the symbol
is shorthand for "the difference in ..." The first derivative tells
us how quickly the color changes with distance as we move along the
left-right axis of the cube. (Actually, we were not limited to the left-right
axis; any direction might have been chosen along which to evaluate the
derivative; the left-right axis was merely chosen here for conceptual
convenience).
If
first derivatives are evaluated at two different points along the left-right
axis and compared the same way, then it is also possible to show that
the change in first derivative also becomes arbitrarily small as the
points become arbitrarily close. Again, the ratio of the differences
approaches a finite value. This ratio is called the second derivative,
and is written
d
2/dx2.
The
second derivative tells us how quickly the first derivative changes
as we move along the left-right axis. Since the first and second derivatives
are finite at every point inside the cube, they are said to exist everywhere
within the field, and to be continuous functions of location. Such a
field is said to be well behaved.
Discontinuous
color field:
Retain the previous orientation of the cube, 'erase' the continuous
field (in your imagination), and divide the cube with a plane so that
it is possible to speak of a right half cube and a left half cube. Now
imagine a color field in which all the points in the left half cube
are a uniform and vivid green, and all the points in the right half
field are a uniform and vivid red. Each half field is well behaved,
and, in fact, exhibits no change in color (frequency) from point to
point whether the points are arbitrarily close or not (the first and
second derivatives are all equal to zero). But conditions in the field
near the plane dividing the two half cubes are very different: Consider
one point PL in the left half cube, and a second point PR in the right
half cube. The difference in the color (frequency) associated with these
two points remains finite and constant, regardless of how arbitrarily
close we allow the points to become. When the distance between the points
approaches zero, the ratio of differences approaches infinity. Therefore,
the derivatives along the plane separating the right and left half cubes
do not exist (i.e., do not have finite values), and the field is said
to be discontinuous everywhere along the plane. At such a field discontinuity,
the field is not well behaved. (In some physical or engineering applications,
such a discontinuity is called a shock boundary.)
Time-Varying
Continuous Field:
Now let us return to the continuous color field. We initially assumed
that the colors remained constant with time; i.e., we assumed that the
cube would appear always the same regardless of when we chose to look
at it. We said that such a field was called a time invariant (static)
field. But the assumption of time invariance is not essential to our
example of a continuous field (or a discontinuous one for that matter).
So now, beginning with the color field of example (1), imagine that,
at every point in the field, the color is gradually (continuously) changing.
In fact, let us specify a rule for the change to help the imagination
along. First, imagine a line. Now, place the rainbow colors along the
line with red at the left and indigo at the right. Place a movable pointer
on the line. Now pick a point in the field, and place the pointer at
the corresponding color on the line. Let the pointer gradually move
to the right until it reaches the end of the line. Then let the pointer
move to the left, and so on, back and forth. Do this same operation
at every point in the field letting the pointer move at the same rate
(speed) each time. The overall effect is that, everywhere in the field,
the full spectrum will sooner or later be exhibited. If I stand at a
single point, the color at that point will change gradually through
the rainbow. If I stand outside the cube and observe the field as a
whole, waves of color will appear to be traveling across the cube along
the left-right axis. This field is of particular interest because most
fields dealt with in physics or engineering exhibit wave phenomena;
e.g., the transmission field of a radio or television station, the wave
field on the surface of a lake of ocean. The field we have been describing
is called a time varying (dynamic) field; and, since the overall color
pattern repeats from time to time, the field is further called a periodic
time varying field, with the period equal to the difference between
any two successive times in which the cube appears the same. Points
separated along a left-right axis will still exhibit different colors
at any given time (as when a snapshot of the cube is taken), and this
difference may be thought of in terms of a phase difference throughout
the field. Thus, a point near the left face might be going from red
to orange at the same time that a point near the right face is going
from green to blue.
This
heuristic model is offered to facilitate visualization of the field
concept, a formidable concept that can be daunting when first encountered
by students of mathematics or physics.