Cool! I've just contributed several examples. If anyone is interested in the sheer amount of identities that have been discovered, good books are (many of them gigantic references spanning thousands of pages). When bored, try proving some of those facts, examples build on top of each other. These are not the only examples, as there are
many texts like these in other areas of mathematics and engineering, be it numerical analysis, optimization and variational analysis, statistics, abstract algebra, control theory, geometry and so on.
Table of Integrals, Series, and Products, Gradshteyn & Ryzhik.
Special Integrals of Gradshteyn and Ryzhik, Vols. I and II, Moll for some proofs of the above.
Handbook of Integral Equations, Polyanin & Manzhirov.
Scalar, Vector, and Matrix Mathematics, Bernstein.
Handbook of Number Theory I and II, Sandor, Crstici & Mitrinovic.
Wikipedia also has a plethora of pages with mathematical identities. Some of them:
Can't have a list like that without a mention of Abramowitz & Stegun [0], or its successor, the NIST Digital Library of Mathematical Functions [1]. It's about as comprehensive as it gets.
I'm a chilean maths grad student and save for the qualifying exam, it's quite accurate. So much so that I think I made a mistake clicking on this because as it progressed I started feeling dizzy. Other commenters here also have their relatable experiences, which doesn't make me feel so bad.
He was born in the German Empire, fought in both world wars, lived through Weimar, rise of the Soviet Union, rise of the Third Reich, saw his country split in two, saw the Berlin wall fall and the collapse of the USSR. That's a very high bar for me to reach, and I don't know what the modern equivalent would be.
This cookbook was insightful in the sense that I once tried proving several of its identities via Kronecker products but then found that it was much easier using tensor notation, so easy that I almost forgot Kronecker products altogether. I put together some proofs in my blog [0] (note: it's incomplete and the URL might change!).
Two favorite books of mine I've seldom seen mentioned but I think are gems: The three stigmata of Palmer Eldritch and Time out of joint, both by Philip K. Dick. The first plays with several layers of reality along the book. One time I found myself re-reading some passages because there were so many layers it was hard to follow, but it's worth it. The second is (if I recall correctly) the inspiration for a 90s pop movie starring Jim Carrey. I won't mention the name because it might give away some plot points.
I adapted The Three Stigmata of Palmer Eldritch for the stage many years ago, but was refused permission to perform it by P. K. Dick's estate.
It is truly a masterpiece!
We ended up using Flow My Tears, The Policeman Said, since that was an existing adaption by a friend of Philip K. Dick's that they had no control over.
Not sure I still have the stage play to be honest, it was done in the late 90s, I'd have to really dig back into my old backups.
The whole idea was to use multimedia, video projection, computer animation and live video mixed up with live performance to really mess with the audience's sense of what was real or not.
We managed to pull it off quite nicely with Flow My Tears though. A nice bit was cutting from a live feed of an actress apparently cutting her arm, to a pre recorded close up of the arm being cut and lots of blood. Audience always thought they were seeing what was happening on stage at that point :)
Slightly related: Does anyone else can't concentrate for long periods of time while sitting in front of a computer? I have mild scoliosis and a fear of varicocele (if that's relevant), and fortunately my work involves implementation of numerical algorithms/problem solving that I can first sketch on a notebook while laying on my bed and finally be coded sitting -on a rush-. Same with reading books. I'm thinking of purchasing a one-sided divan sofa and put it next to my desktop.
A height adjustable desk is worth looking into if you don't already have one. Alternating between sitting and standing every 30-60 minutes is great for posture fatigue.
I have a similar mild scoliosis/desk avoidance thing. But what I've found is that it's not related to pain(I can game for hours and hours in a chair), it's more of a energy/bloodflow/breathing issue. Lying prone or supine is actually a good posture for relaxing enough to think carefully. Sitting upright induces attention and focus to immediate tasks but it can also produce woozy idis haze right after eating.
My favorite setup, overall, is floor sitting with a low desk. This lets the posture shift around moment-to-moment, including fully upright seiza type positions and supine or prone positions.
I’ve heard that some people even attempt to maximize walking time when important deadlines are close, because putting on serious faces and pretending to be productive at their desk is anecdotally so counterproductive to them. I think there are much to be discovered in where our creativity come from.
oh that's interesting how do posts get queued there
EDIT: i see they are picked manually "HN's second-chance pool is a way to give links a second chance at the front page. Moderators and a small number of reviewers go through old submissions looking for articles that are in the spirit of the site—gratifying intellectual curiosity—and which seem like they might interest the community."
Precisely, it baffled me how people here went on with it seeing only the first order consequences. I'm a grad student in Chile and I get emails from recruiters in the US three or four times a week. I haven't picked up a job and don't plan to until I finish my degree (soon), and I haven't thoroughly evaluated the quality of each offer. Meanwhile recruiters in my country don't look as much, might as well be because American employers have much to gain from a remote worker compared to local employers. Several of my friends are from Argentina, and they all work for US companies either remotely or on-site, something I would have had a hard time believing just five years ago. This decade will be interesting.
I want to start off by saying that I am not a dev, my job is engineering that uses advanced mathematical theory to solve physical problems.
I have only set up a LinkedIn profile. Recruiters have my email from there, the conferences I've attended and contact information I've passed to my friends (being an engineering grad student helps). It varies, but the lowest is 45k/year, which living in the capital (even more if I lived in a village) of Chile means that as a single male I would live comfortably and without worries.
Yes, 45k USD remote for US companies, as opposed to entry level salary for MSc grads of 18k for the field I'm working in, in Chile. As I said, that's the lowest salary offer of remote jobs. The theory is numerical analysis of partial differential equations, specifically, the finite element method (for CFD, fluid-structure interaction, thin shell mechanics, buckling, and so on). To have a basic understanding of how to code custom FEM programs one needs at least an understanding of the fundamentals of functional analysis + analysis of PDEs + tensor calculus + the specific theory of physical application involved. These are non-trivial skills that take years and years to develop, so I would advise to anyone interested that it's best to get a job as a dev instead. I do it out of passion and because my family has a history of civil/mech eng.
The reason for such spread, I suspect, is that there is not much market for FEM professionals outside of the US and Europe, and if it exists, it's very small and unsophisticated.
Table of Integrals, Series, and Products, Gradshteyn & Ryzhik.
Special Integrals of Gradshteyn and Ryzhik, Vols. I and II, Moll for some proofs of the above.
Handbook of Integral Equations, Polyanin & Manzhirov.
Scalar, Vector, and Matrix Mathematics, Bernstein.
Handbook of Number Theory I and II, Sandor, Crstici & Mitrinovic.
Wikipedia also has a plethora of pages with mathematical identities. Some of them:
https://en.wikipedia.org/wiki/Vector_calculus_identities
https://en.wikipedia.org/wiki/Vector_algebra_relations
https://en.wikipedia.org/wiki/Exterior_calculus_identities
https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spheric...
https://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian...
https://en.wikipedia.org/wiki/List_of_set_identities_and_rel...
https://en.wikipedia.org/wiki/List_of_triangle_inequalities
https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...
... and its several lists of integrals (including trigonometric, exponential, rational).
https://en.wikipedia.org/wiki/Lists_of_integrals#Lists_of_in...
More advanced topics:
http://proximity-operator.net/