Math seems to me to be the antithesis of that process. I have a background in biology and I enjoy the cautiousness with which biologists and biochemists approach the process of proving or testing a hypothesis. For example, why should 2 x 2 = 4? Why not 5, or 0 or 100? It is similar to a language, the inventors of which came up with grammatical rules according to their whims and fancies, and which new learners are expected to accept as gospel truth. It seems to me that math is filled with assumptions with no rhyme or reason whatsoever. This is part of my fundamental gripe with math and why I never understood anything beyond how to count my change. But how does one come up with a postulate? Are we free to assume whatever we want? What is to prevent me from assuming any kind of nonsense I want and then building a system of proofs from it - something like building lego buildings? Of course the lego structures will be 'internally consistent' in that it forms a complete world by itself, but for practical purposes, it would be totally useless. Not a great proof and written after it was proven that this could not be proved.Īs I understand it, the postulates/axioms are assumptions and they are used to construct theorems. On a side note, in 1890, Charles Dodgson (aka Lewis Carroll author of Alice in Wonderland) published a book with a "proof" of the parallel postulate using the first 4 postulates. We can see different versions of systems where the parallel postulate is false by assuming that either there are no parallel lines, or that for any line and point not on a line there are an infinite number of parallel lines. Later (1868) it was proved that the two systems were equally consistent and as consistent as the real number system. In the late 19th century (approximately 1823), three different mathematicians (Bolyai, Lobachevsky and Gauss) proved independently that there was a different system that could be used that assumed the 5th postulate was incorrect. Some really great proofs were created by mathematicians trying to prove the parallel postulate. Mathematicians kept trying to prove that the 5th postulate (commonly known as the parallel postulate) could be proved from the first four postulates and thus was unnecessary. This should get you started.There was a big debate for hundreds of years about whether you really needed all 5 of Euclid's basic postulates. ScheyĪnd for numerical methods in linear algebra take a look at, Numerical linear algebra, Trefethen. Or Introductory Functional Analysis with Applications, Erwin Kreyszigįor a review of multivariable calculus I liked Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, H.M. Linear Functional Analysis (Springer Undergraduate Mathematics Series) Know you will want to study this further. You should at least read the first part of the book, and possibly the second part. Take a look at something like Royden's Real Analysis book. It seems to me that to understand SIAM publications you will need mathematics at the graduate level. As you pointed out, the constraint on brevity is impossible, unless it is only to review the material. I myself am also an engineer interested in fluid mechanics. If you have mastered the concepts of calculus and are looking for a challenge, this book may be for you. You may also learn some analysis along the way, but since he is dealing only with the real line, he doesn't "weigh you down" with various topological concepts that may or may not interest you. In this textbook, Andreescu exposes you to a lot of techniques (especially in dealing with inequalities) that may help you with the Putnam. I think a great book for doing calculus problems that may prepare you for the Putnam is Titu Andreescu's book- " Problems in Real Analysis: Advanced Calculus on the Real Line".Īndreescu's texts are known to be useful in preparing students for mathematics competitions. Last but not least, I see that you mentioned the Putnam. But more importantly, Courant and John give a lot of motivation for calculus through discussing its relevance to the physical world (there is even discussion of Fourier Series in the study of a vibrating drum).Ĭourant and John also introduce some concepts of modern analysis (calculus done more rigorously) such as monotone sequences, a precise discussion of a limit, etc towards the end of the book, if that interests you. I think the problems are a bit harder than those in Thomas' text. If you want another book that may suit your taste, I recommend " Introduction to Calculus and Analysis" by Richard Courant and Fritz John. I don't think these problems are the routine exercises that you have grown weary of in Thomas' Calculus. Take a look at the calculus test for the years 1998-2011. If you are looking for some challenging calculus problems, I would suggest looking at the HMMT (Harvard-MIT Mathematics Tournament) problems archive.
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