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I am not talking about Duhamel's Integral. That is something different.

There is a lot of coverage of Duhamel's Principle in the context Diff Eq, but I have a copy of "Advanced Calculus" by John M. H. Olmsted copyright 1951. In Chapter 8 page 238 there is a section on Duhamel's Principle for Integrals. It is a weird integral of a function of two variables where two different system of tags of a net are used for the two different arguments to the function.

Duhamel's Principle for Integrals. Let $fleft(tright)$ and $gleft(tright)$
be integrable on $left[a,bright]$ and let $phileft(x,yright)$
be everywhere continuous. Then, in the sense of section $401$, the limit of the sum $sum_{i=1}^{n}phileft(fleft(t_{i}right),gleft(tacute{_{i}}right)right)triangle t_{i}$ where $a_{i-1}leq t_{i}leq a_{i}$ and $a_{i-1}leq t_{i}^{acute{}}leq a_{i}$ as the norm of the net $Re:a=a_{0},a_{1},cdots a_{n}=b$ tends toward
zero, exists and is equal to the definite integral $intop_{a}^{b}phileft(fleft(tright),gleft(tright)right),dt$,
which also exits.

begin{equation}
lim_{left|Reright|rightarrow0}sum_{i=1}^{n}phileft(fleft(t_{i}right),gleft(tacute{_{i}}right)right)triangle t_{i}=intop_{a}^{b}phileft(fleft(tright),gleft(tright)right)dt
end{equation}

The point to note in the definition in Olmsted is that $t_{i}$ and $tacute{_{i}}$ do not have to be equal, just in the same sub-interval.

Olmsted proves Duhamel's principle for the continuous case. Olmstead uses Duhamel's Principle to prove the general formula for the length of an arc.

I have a copy of Baby Rudin, and he proves the same formula for the length of an arc without mentioning Duhamel's Principle for Integrals. When I search the Internet for Duhamel's Principle, all the references to Duhamel's Principle are to the Diff Eq version.

Questions:

1) Has Duhamel's Principle for Integrals gone out of style?

2) Where can I find a proof of the general case of Duhamel's Principle for Integrals?

3) What is the relationship between Duhamel's Principle for Integrals and the Diff EQ version of Duhamel's Principle?

The proof of Duhamel Principle is based on the following criterion of Riemann integrability given by Riemann himself:

Criterion for Riemann Integrability: Let the function $f:[a, b] tomathbb {R} $ be bounded on $[a, b] $. Then $f$ is Riemann integrable on $[a, b] $ if and only if for any given positive real numbers $epsilon, sigma$ we can find a positive real number $delta$ such that for any partition (net) $P$ of $[a, b] $ with norm (mesh) $||P||<delta$ the subintervals of $P$ on which the oscillation of $f$ is at least $sigma$ have a total combined length less than $epsilon$.

The above theorem is a precursor to the famous criterion given by Lebesgue which says that a bounded function is Riemann integrable on a closed interval if and only if the set of its discontinuities on this interval is of measure zero.

Let's assume that functions $f, g$ from $[a, b] $ to $mathbb {R} $ are Riemann integrable on $[a, b] $ and the function $phi:mathbb{R} ^2tomathbb {R} $ is continuous on $mathbb {R} ^2$. Then using the above criterion or Lebesgue's criterion we can show that the function $F:[a, b] tomathbb{R} $ defined by $F(x) =phi(f(x), g(x)) $ is Riemann integrable on $[a, b] $.

Let's start with an arbitrary $epsilon >0$. Let $A, B$ be positive upper bounds for $|f|, |g|$ on $[a, b] $ respectively and let $M$ be a positive upper bound for $|phi|$ on the rectanglular region $mathcal{R} $ with diagonal points $(-A, - B), (A, B) $. We are supposed to find a $delta>0$ such that for all partitions $P={x_0,x_1,x_2,dots ,x_n} $ of $[a, b] $ with norm $||P||<delta $ we have $$left|sum_{i=1}^{n}phi(f(t_i),g(t'_i))(x_i-x_{i-1})-int_{a} ^{b} phi(f(x), g(x)) , dxright|<epsilon tag{1}$$ for any two sets of tags $t_i, t'_iin[x_{i-1},x_i]$.

To find such a $delta$ we note that there is a positive $delta_1$ such that $$left|sum_{i=1}^{n}phi(f(t_i),g(t_i))(x_i-x_{i-1})-int_{a}^{b}phi(f(x),g(x)),dxright|<frac{epsilon} {2}tag{2}$$ for all partitions $P$ of $[a, b] $ with norm less than $delta_1$ and any set of tags $t_iin[x_{i-1},x_i]$. Our job is done if we can find a $delta_2>0$ such that $$left|sum_{i=1}^{n}{phi(f(t_i),g(t_i))-phi(f(t_i),g(t'_i))}(x_i-x_{i-1})right|<frac{epsilon} {2}tag{3}$$ for all partitions $P$ with norm less than $delta_2$ and any sets of tags $t_i, t'_iin[x_{i-1},x_i]$. Clearly we can choose $delta=min(delta_1,delta_2)$ and if $P$ is any partition with norm less than $delta $ then both inequalities $(2),(3)$ hold simultaneously and imply $(1)$.

Thus our desired goal is to ensure the inequality $(3)$ and changing symbols a bit it is sufficient to prove that for any given $epsilon>0$ there is a $delta>0$ such that $$left|sum_{i=1}^{n}{phi(f(t_i),g(t_i))-phi(f(t_i),g(t'_i))}(x_i-x_{i-1})right|<epsilontag{4}$$ for all partitions $P$ with norm less than $delta$ and any sets of tags $t_i, t'_iin[x_{i-1},x_i]$.

To find such a $delta$ we note that $phi$ is uniformly continuous on the rectanglular region $mathcal{R} $ and hence there is a $delta'>0$ such that $$|phi(x, y) - phi(x', y') |<frac{epsilon} {2(b-a)} $$ for all points $(x, y) $ and $(x', y')$ in $mathcal{R} $ with $sqrt{(x-x') ^2+(y-y')^2}<delta'$. Since $g$ is Riemann integrable on $[a, b] $ there is a positive $delta$ such that for all partitions $P$ of $[a, b] $ with norm less than $delta$ the combined length of subintervals of $P$ where the oscillation of $g$ is at least $delta'$ is less than $epsilon/4M $. Next consider the sum, henceforth denoted by $S$, on left in the inequality $(4)$ which can be split into two parts based on the subintervals created by partition $P$. The first part, say $S_1$, is based on those sub-intervals $[x_{i-1},x_i]$ where the oscillation of $g$ is less than $delta'$. For these intervals the difference $$|phi (f(t_i), g(t_i)) - phi(f(t_i), g(t'_i)) |<frac{epsilon} {2(b-a)}$$ and hence $|S_1|<epsilon/2$. The second part, say $S_2$, is based on subintervals where the oscillation of $g$ is at least $delta'$ and the combined length of such subintervals is less than $epsilon /4M $ so that $|S_2|<epsilon /2$ and then $$|S|=|S_1+S_2|leq |S_1|+|S_2|<epsilon $$