# Duhamel's Principle for Integrals

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 $$f\left(t\right)$$ and $$g\left(t\right)$$ be integrable on $$\left[a,b\right]$$ and let $$\phi\left(x,y\right)$$ be everywhere continuous. Then, in the sense of section $$401$$, the limit of the sum $$\sum_{i=1}^{n}\phi\left(f\left(t_{i}\right),g\left(t\acute{_{i}}\right)\right)\triangle t_{i}$$ 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 $$\intop_{a}^{b}\phi\left(f\left(t\right),g\left(t\right)\right)\,dt$$, which also exits.

$$\begin{equation} \lim_{\left|\Re\right|\rightarrow0}\sum_{i=1}^{n}\phi\left(f\left(t_{i}\right),g\left(t\acute{_{i}}\right)\right)\triangle t_{i}=\intop_{a}^{b}\phi\left(f\left(t\right),g\left(t\right)\right)dt \end{equation}$$

The point to note in the definition in Olmsted is that $$t_{i}$$ and $$t\acute{_{i}}$$ 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?

• I don't think it has gone out of style. I used it recently without knowing the name Duhamel. See this answer:math.stackexchange.com/a/3072835/72031 A special case when $\phi(x, y) =xy$ is proved here: math.stackexchange.com/a/2152482/72031 The proof for the continuous $\phi$ is based on uniform continuity of $\phi$ in the bounded rectangle with diagonal points $(-M, - N), (M, N)$ where $M, N$ are upper bounds for $|f|, |g|$ respectively. – Paramanand Singh Jan 20 '19 at 3:56
• +1 for letting me know that such a thing already existed. While working out the proof for arc-length formula I was thinking to prove the general case for continuous $\phi$ but I figured a way to use the special case where $\phi(x, y) =xy$. – Paramanand Singh Jan 20 '19 at 3:59
• Does anyone know of a published text I can quote that has the proof? – Paul Elliott Feb 12 '19 at 7:50

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 $$f:[a, b] \to\mathbb {R}$$ 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 $$\phi:\mathbb{R} ^2\to\mathbb {R}$$ is continuous on $$\mathbb {R} ^2$$. Then using the above criterion or Lebesgue's criterion we can show that the function $$F:[a, b] \to\mathbb{R}$$ 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 $$\left|\sum_{i=1}^{n}\phi(f(t_i),g(t'_i))(x_i-x_{i-1})-\int_{a} ^{b} \phi(f(x), g(x)) \, dx\right|<\epsilon \tag{1}$$ for any two sets of tags $$t_i, t'_i\in[x_{i-1},x_i]$$.

To find such a $$\delta$$ we note that there is a positive $$\delta_1$$ such that $$\left|\sum_{i=1}^{n}\phi(f(t_i),g(t_i))(x_i-x_{i-1})-\int_{a}^{b}\phi(f(x),g(x))\,dx\right|<\frac{\epsilon} {2}\tag{2}$$ for all partitions $$P$$ of $$[a, b]$$ with norm less than $$\delta_1$$ and any set of tags $$t_i\in[x_{i-1},x_i]$$. 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 $$t_i, t'_i\in[x_{i-1},x_i]$$. 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 $$\left|\sum_{i=1}^{n}\{\phi(f(t_i),g(t_i))-\phi(f(t_i),g(t'_i))\}(x_i-x_{i-1})\right|<\epsilon\tag{4}$$ for all partitions $$P$$ with norm less than $$\delta$$ and any sets of tags $$t_i, t'_i\in[x_{i-1},x_i]$$.

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 $$|S|=|S_1+S_2|\leq |S_1|+|S_2|<\epsilon$$