Bliss's Theorem and/or Duhamel's Principle

[bigbadwolf]

Out if curiosity, is anyone here acquainted with Bliss's Theorem and/or Duhamel's Principle? None of the modern texts seem to mention them.

 

[Daniel Duffy]

I don't know Bliss but Duhamel's principle/integral is for the solution of inhomogeneous PDEs? If yes, then I can give some feedback.

I see there is also a Duhamel's theorem.

 

[bigbadwolf]

I see there is also a Duhamel's theorem.

Probably the same thing as the principle. You find it discussed in the old advanced calculus texts but never modern. You need it to justify, among other things, the formula for arc length, which these days is just defined as an integral (without giving any idea of where it came from). It's similar to the situation for the complex integral, which is just defined in a certain way while the old authors such as Titchmarsh and Copson would actually explain where it came from.

I started a thread perhaps a year back asking what advanced calculus texts people were using -- but no-one came up with titles. The modern books are mostly no good.

 

[Daniel Duffy]

Duhamel's principle is for finding the solution of PDEs. Different stew altogether.

Duhamel's principle - Wikipedia

en.wikipedia.org

I looked Duhamel theorem in Widder. Seems like the limit of a Stieltjes integral with special case Bliss. I couid find no other source. Is is not subsumed under Stieltjes integration?

Books on differential geometry (Schaum,..) motivate arc length.

 

[bigbadwolf]

Duhamel's principle is for finding the solution of PDEs. Different stew altogether.

Duhamel's principle - Wikipedia

en.wikipedia.org

I looked Duhamel theorem in Widder. Seems like the limit of a Stieltjes integral with special case Bliss. I couid find no other source. Is is not subsumed under Stieltjes integration?

Here is someone asking about Bliss's theorem. The person responding is clueless:

Bliss's Theorem

sci.math.narkive.com

Here someone has excerpted the theorem and outline proof of Bliss's theorem from some old book:

What is the theorem of Bliss?

I just saw this from a book. What is the real life application of it?

math.stackexchange.com

Bliss's theorem is a special case of Duhamel's principle. The most recent book that has a formulation and proof of it is Taylor and Mann's Advanced Calculus (3rd edition, 1983).

Books on differential geometry (Schaum,..) motivate arc length.

But don't provide any kind of proof. The problem is that you don't quite have a Riemann sum. If you had one, then as the norm partition goes to zero, you get a Riemann integral, which is the definition of arc length. You don't quite have a Riemann sum because for a curve in the plane you have two functions x(t) and y(t), and you are using the mean value theorem on both. The points t' and t" inside an arc segment may not (and usually will not) coincide, so you can't get a Riemann sum for the formula for arc length.[/quote]