Further clarification was done, by using Maple to integrate this expression for the Dirac Delta Function multiplied by a few sample functions. Here is the Maple code which was used:
deltafn:=1/Pi*b/((x-x0)^2 + b^2);
b
deltafn := -------------------
2 2
Pi ((x - x0) + b )
We now let b have a small value of 1/1000 and we let
x0 be 1:
deltafn1:=subs({x0=1,b=1/1000},deltafn);
1
deltafn1 := 1/1000 -------------------------
2
Pi ((x - 1) + 1/1000000)
Let us try three test functions, x, sin(Pi*x/2),
and exp(-(x-1)^2). These all have a value of 1
when x = 1, and so the integral of these times
the delta function should be 1. We let the integration
range be from -100 to 100:
evalf(int(deltafn1*x,x=-100..100));
.9999872666
evalf(int(deltafn1*sin(Pi*x/2),x=-100..100));
.9984304363
evalf(int(deltafn1*exp(-(x-1)^2),x=-100..100));
.9988726201
As you can see, these values are all pretty close to 1 and
confirm that our approximation for the delta function is pretty
good.
Next we try to illustrate the process of taking the limit as b goes to zero, by plotting the above expression for the delta function, with x0 = 1 and letting b take 10 steps from 1 to 1/10. This is done in a Maple loop, as follows:
plotlist:=[];
plotlist := []
for i from 1 to 10 do
plotlist:=[op(plotlist),plot(subs({x0=1,b=1/i},deltafn),
x=-1..3,0..3.2,
color=COLOR(RGB,evalf(cos(Pi/2*(i-1)/9)),0,evalf(sin(Pi/2*(i-1)/9))))]:
od:
with(plots):
display(plotlist);
| The plot shown to the left illustrates the process of making
b smaller. The red curve is for b = 1 and
the blue curve is for b = 1/10. The curves with
a colour inbetween red and blue are for intermediate b
values. As b becomes smaller and smaller, the peak at x = 1 becomes taller and narrower, wheras the values for x away from 1 become smaller and smaller. In the limit as b goes to 0, the function becomes zero everywhere except for x = 1, where there is an infinitely tall, infinitesimally skinny peak, whose integral is 1.
|
Also using this relation (Lemma 5.4), we showed how to find a solenoidal vector field F having a specified curl [Transparancy 4]. However, there was a technical hitch [Transparancy 5], in that vector field must vanish on the surface of the region under consideration. This can usually be circumvented by considering a surface so large that the vector field tapers off to zero magnitude.
This all leads up to the Fundamental Theorem of Vector Analysis, which states that any vector field can be written as the sum of a gradient and a curl [Transparancy 5] (subject to certain continuity conditions). The theorem was proved quite easily [Transparancy 6] again using the very handy Lemma 5.4.
(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 70 and 95 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 240 K, which takes a bit longer to download.)
