I consider myself an applied mathematician, but I am a mathematician, not an engineer (although my undergraduate education was at what I am proud to consider one of the finest engineering schools in the United States). During the writing of the First Edition of the book there was a lot of discussion about including "applications" in the text, either as exercises or as sections of material. I confess I resisted doing too much of that, and I would like to use this essay to explain, philosophically, why I did so.
My goal in writing the text was to make it as broadly useable as possible. An instructor can teach a very unrigorous class by simply skipping the proofs of the theorems, but a different instructor could teach a very mathematically complete course out of this book. The point is that I, as the author, cannot know what sorts of students or instructors are using the book. I could have taken the time (and pages) to develop any one of a number of application illustrations, but I very strongly believe that doing so would be stepping on the prerogatives of the instructor. And that, I think, is the key point.
Because computational issues are becoming more and more important in many areas of science, more and more departments are teaching their own specialized classes in numerical methods. By keeping the text as "application neutral" as possible, I hope to make it as attractive as possible to a wide range of departments and instructors. But I think the instructors are better placed than I am to decide what applications and illustrations serve their needs best. A heavy dose of engineering examples or illustrations would not be of much interest to a business school professor looking to prepare students in the emerging area of computational finance. And, frankly, to do any applications in an authentic way would take pages and pages of background development. I firmly believe that the instructors are better placed than I am to do that.
I am interested in feedback or (polite) commentary. Feel free to email me at firstname.lastname@example.org.