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In single variable calculus, integral calculus is taught "back to back" with differential calculus. This is generally true, although to a lesser extent, with multivariable calculus.

Yet I don't see the teaching of integral equations accompanying that of (ordinary) differential equations; instead, differential equations are often paired with linear algebra or partial differential equations. For instance, on this site, there was a tag for differential equations but not for integral equations (until I created one).

Why is that?

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    $\begingroup$ Just a wild guess (which doesn't address the issue of tags in Stack Exchange), but I wonder if it's because after the 1930s or so integral equations became subsumed under functional analysis? $\endgroup$
    – Dave L Renfro
    Apr 7, 2018 at 12:11
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    $\begingroup$ We probably should be teaching more integral equations with PDEs. I just have a token week or two in my intro to DEqns class , so getting to Green's functions is a bit much for my course. In retrospect, it ought to have played a larger role in the PDE course I took as an undergrad. I think you have a point here to some extent. But, maturity and difficulty of the topic makes it formidable for many audiences. That explains the direction that Dave Renfro points out. $\endgroup$
    – James S. Cook
    Apr 7, 2018 at 18:08
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    $\begingroup$ I never encountered them in science or engineering, but saw ODEs and PDEs all over the place. [Note for pedants, I am not saying there is not a single journal article in science or engineering that with an integral equation. I am saying it is not a common part of undergrad physics or engineering, including the harder classes junior and senior year.] $\endgroup$
    – guest
    Apr 7, 2018 at 23:24
  • $\begingroup$ @guest at the graduate level, even introductory, it becomes important. Particularly, the interplay between the method of images and Green's functions in the graduate electricity and magnetism course (ala Jackson). For a nice undergrad book which makes a good case for the importance of Green's functions I think Haberman's text is helpful. See books.google.com/books/about/… $\endgroup$
    – James S. Cook
    Apr 11, 2018 at 4:00
  • $\begingroup$ OK, Green's function is part of pde books. what does that have to do with a full topic on integral equations, kernels, etc.? I mean the Laplace transform has an integral in it also and that is part of regular sophomore ODE course. $\endgroup$
    – guest
    Apr 11, 2018 at 11:55

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While there is a logical symmetry between differential equations and integral equations, it seems that (as they say) "laws of nature" are written in differential equations, not integral equations. As far as I can tell, rigorous study of integral equations arose in the late 1800's in part because of the relative difficulty of rigorous study of (especially partial) differential equations. That is, bounded operators (e.g., inverses of differential operators!), especially compact operators, were far easier to prove theorems about than unbounded operators (differential operators).

Nevertheless, people using differential equations throughout the 19th and 20th centuries often were not slowed-down by worries about mathematical niceties (and very often reached correct conclusions with or without "proofs" whose legitimacy could be understood in the context of the times).

That is, integral equations seem mostly to be of interest either as a (temporary?) substitute for differential equations, or as a technical warm-up for rigorous treatment of differential equations.

The relatively more-recent Schwartz' Kernel Theorem, Grothendieck's "nuclear spaces" (and pseudo-differential operators, singular integral operators, etc.) can certainly be viewed as extrapolations of ideas about integral equations, but it is my impression that people applying such ideas are also less finicky about distinguishing "generalized functions" (as in Schwartz' distributions) from classical functions, so may consider Schwartz' Kernel theorem as "obvious".

So, perhaps, we don't teach introductory integral equations along with introductory differential equations because the differential equations have the more direct applications, and because the greater ease of rigorizing integral equations is irrelevant.

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  • $\begingroup$ Just a comment: Maxwell's equations were first formulated in the integral form, as far as I known, and the concepts of flux, circulation, etc. are a little easier to understand and visualize that divergence and curl. And in general I think treating integral equations like some exotic animal is a mistake, they are a very natural way to express the laws of Nature $\endgroup$
    – Yuriy S
    Nov 14, 2018 at 11:28

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