The basic idea that a PDE describes the trajectory of a vector through a vector space is extremely powerful, an overarching idea that is very useful for organizing thoughts about mathematical models. When you actually go to implement it using say Spectral Collocation methods, it really falls into place.

I really think everyone in Engineering and Applied Math should look at John Boyd's book on spectral methods, which happens to be downloadable in PDF form:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.3791&rep=rep1&type=pdf

It really hammers home the idea of representing a function as a linear combination of other functions. Very relevant to my recent post on Fourier methods. The NSA approach of breaking up the domain of the function into a hyperfine grid and then calling the value of the function at each grid point the the coordinate of the function in the nth dimension is a very literal interpretation.

Often expositions aimed at mathematicians elide or fail to really understand the value of the connection between the application and the mathematical theory. So much mathematics has been invented by applied people, and then formalized later in a way that's devoid of connection to the application. I'm looking at you "distribution" theory ðŸ™‚

]]>Apparently it has some significant differences to non-standard analysis eg no transfer principle. Interesting.

https://en.m.wikipedia.org/wiki/Smooth_infinitesimal_analysis

]]>RE programming, I think this is another one with a division. I only really appreciated the beauty of analysis after studying and using numerical methods. Here and functional analysis are (to me) where analysis and algebra really start to come together. Perhaps NSA helps one get there earlier.

]]>http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eating.html

I eat corn sometimes in rows across, and sometimes in random chunks... which means I'm a stochastic mix of algebraist and analyst. I think that maps to nonstandard analysis pretty well.

Seriously, though, the better you are at computer programming, the more advantages you can get by manipulating symbolic things, which is more or less algebra and if you've done a lot of computing, I think you're probably more likely to be interested in doing analysis via algebra.

Also, epsilon-delta is a good explanation for a mathematician interested in connections between mathematical objects, but it's a terrible description if you want to preserve the mapping between objects in the world and mathematical objects. There's a good reason why Newton and Leibniz used infinitesimal reasoning, because they were working with physical ideas.

I'm not that interested in proving abstract mathematical facts about mathematical objects, though I was good enough at it as an undergrad (math major). I like building descriptions of physical processes though, and to me the mapping makes better sense. Also powers of infinitesimal numbers give a good description of what asymptotic analysis is all about in applied mathematics...

I guess it's just preferences really.

]]>Have you always been a fan? How do feel about standard epsilon-delta stuff? I had a mature (returning after many years) student come to me the other day after a lecture on PDEs in which I gave a few different ways of deriving equations.

They said that limits had always been difficult for them and part of the reason they originally dropped out. I found that the ' epsilon delta game' was the explanation that worked best for them ultimately. I'm not sure if they were really comfortable with 'infinitely small' etc on the other hand.

Seems to be (very anecdotally) a difference between those who like a 'process' based (eps delta) description and those who like an 'object' based description (infinitesimals)?

Also reminds me of the frequentist (eg 'putting bounds on a stochastic process') vs Bayesian (eg 'manipulation of probability distribution objects') division somewhat.

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