The book by K.F. Riley seems to be very good. I must read up about how you go from 'harmonic waves' to expressing complicated functions as superpositions of simple harmonic waves, to Fourier transforms, to the Delta function. It looks like physics is clearer than I thought.
But for now; may I repost the quote about the delta function.
Regarding integrating first:
As I recall: to differentiate say a falling object you obtain the two components of acceleration; you differentiate the speed at one moment from the rate of change of speed between two instants. To integrate, you integrate across the two instants to get the total distance travelled.
Surely even here; you could see this through 'delta-function' eyes:
As you narrow down your two instants- the moment they become just ONE instant (so their distance apart becomes zero); you have an infinitely precise position? (Impromptu physics here)
But when you talk of the object travelling across a FINITE distance; you have an integration of the falling object. Now you have spread the position getting at least two instants, and can obtain an acceleration reading.
Doesn't the delta function describe a way of looking at an instant; compared with another way of looking at it that involves a wider perspective of two instants? And from this 1 to 2 perspective jump; you perhaps can then treat this pair (1, 1+1) as a new 1, to be paired (old 1, old 1 + 1) + (new 1) thus a new pair; and so on building up perspectives on reality that is like nesting Russian dolls in reverse. Already Relativity, QED, E =Mc squared and Gravity appear obtainable?
The quote again:
"I have found a section in a textbook, about the Dirac delta function.
It appears to show that Dr. Stafford is correct.
In the following, I use the word "by" to denote where a math symbol is next to another (like if the integral symbol is next in a line of symbols, I write "by the integral...")
Page 207 leads to two defining relationships for the Fourier transform
f(t) = 1 divided by: (2pi) to power of a half; by the integral from minus infinity to plus infinity g(omega) exp (i omega t) d omega
g(omega) = 1 divided by: (2pi) to power of a half; by the integral from minus infinity to plus infinity f(t) exp (-1 omega t) d t
Now, the book refers to these earlier defining functions which, substituting one for the other gives the equation
f(x) = (2pi) to the power of: minus a half,
by the integral from -infinity to plus infinity,
times d omega exp (i omega t),
multiplied by (2pi) to the power minus a half, by the integral from minus infinity to plus infinity, by d t dash exp (-1 omega t dash) f(t dash)
the integral from minus infinty to plus infinity , by d t dash f( t dash) x (2pi) to power of minus 1, by the integral from minus infinity to plus infinity, by d omega exp
It says: "Here we have written the differentials immediately following the integral signs to which they refer and in obtaining the second line from the first we have assumed that the order of the integrations can be reversed.
Now, if it is recalled that f(t) is an arbitrary function, and also noted that in 8.46 the left-hand side refers to a value of f at a PARTICULAR value of t, whilst the right-hand side contains an integral over ALL values of the argument of f, then it is clear that the expression
(2pi) to power -1, by the integral from minus infinity to plus infinity, by d omega exp , considered as a function of t dash, has some remarkable properties. The expression is known as the DIRAC delta-FUNCTION and is denoted by delta (t- t dash).
Quantitatively speaking, since the left-hand side of (8.46) is independent of the value of f (t dash) for all t dash not equal to t, and f itself is an arbitrary function, the delta-function must have the effect of making the integral over t dash receive zero contribution from all t dash except in the immediate neighbourhood of t dash = t, where the contribution is so large that a finite value for the integral results."
Source: "Mathematical Methods For The Physical Sciences. An Informal treatment for students of physics and engineering." by K.F. Riley (Lecturer in physics, Cavendish Laboratory. fellow of Clare College, cambridge) (copyright Cambridge University Press 1974) (pages 214, 215)
I assume copoyright law allows me to quote this maths book in discussion here.
While the math detail is out of my depth; I can see enough in their explanation to see this whole thing is Dick Stafford's paper through and through.
"left-hand side refers to a value for f at a PARTICULAR value of t, whilst right-hand side contains an integral over ALL values of the argument of f..."
this looks like one perspective on the left that can be seen from many perspectives on the right
or narrowing on the left and broadening on the right
that is, a partial differential equation that defines "definition" itself?
"Now, since the left-hand side of (8.46) is independent of the value of f (t dash) for all t dash not equal to t and f itself is an arbitrary function..."
t dash looks rather like a specific subset of an examined set
"...the delta function must have the effect of making the integral over t dash receive zero contribution from all t dash except in the immediate neighbourhood of t dash = t, where the contribution is so large that a **************** **** finite **************** value for the integral results."
it looks like here t is the examined set of Dick's set of numbers. Only a t perspective on t dash allows a finite value for the integral over t dash? It sems to be about creating a finite structure, a perspective, from two other perspectives that map to each other.
"The delta function can be visualiused as a very sharp narrow pulse (in space, time, density, etc.) producing an integrated effect of definite magnitude."
(re" Kronecker delta: page 178: "delta little i little j, which has the value 1 when i = j and the value 0 when i not equal j, is known as the Kronecker delta.) It seems to me that the Kronecker delta is not what Dick's work is about, but I can see how it might look like it from a certain restricted perspective.