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+ | {{indexmenu_n>8}} | ||
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+ | ~~NOTOC~~ | ||
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====== Advanced Tools ====== | ====== Advanced Tools ====== | ||
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+ | <WRAP group> | ||
+ | <WRAP half column> | ||
<blockquote> | <blockquote> | ||
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</blockquote> | </blockquote> | ||
+ | Although this is a travel guide to physics, you'll find here, of course, lots of pages about mathematics. | ||
+ | Mathematics is the "language of nature" and physics isn't possible without it. However, in contrast to the usual Wikipedia pages about math topics, our approach can be summarized as follows: "Mathematics is a toolbox and we pick only those tools that are useful in physics". | ||
- | <nspages advanced_tools -h1 -textPages=""> | + | Especially, this means that for every mathematical concept explained here, there is a "Why is it useful?" section that explains where and how the concept is useful in physics. |
- | ===== Recommended Literature ===== | ||
+ | |||
+ | |||
+ | |||
+ | </WRAP> | ||
+ | <WRAP half column><nspages advanced_tools -h1 -textPages=""></WRAP> | ||
+ | </WRAP> | ||
+ | |||
+ | |||
+ | ----- | ||
+ | |||
+ | |||
+ | |||
+ | It is important to understand the different types of mathematical tools and their relationships. A broad generalization goes as follows: | ||
+ | |||
+ | * __stuff__ (e.g. a set, or several sets, etc.) | ||
+ | * can be equipped with __structure__ (e.g. functions, elements, relations, collections of subsets) | ||
+ | * that satisfy certain __properties__ (e.g. equations, inequalities, inclusions) | ||
+ | |||
+ | {{ :stuffstructuresandproperties.png?nolink&400|}} | ||
+ | |||
+ | As an example consider a __function__. A function is a pair of sets $X,Y$ equipped with a structure $f \subset X\times Y$ that satisfies $\forall x \in X \ \exists ! \ y \in Y \text{ s.t. } (x,y) \in f $. | ||
+ | |||
+ | * We can __check__ properties: they are true or false. | ||
+ | * We can __choose__ structures from a __set__ of possibilities. | ||
+ | * We can __choose__ stuff from a __category__ of possibilities. | ||
+ | |||
+ | |||
+ | (Adapted from http://math.ucr.edu/home/baez/qg-spring2004/s04week01.pdf) | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | ===== Recommended Resources ===== | ||
+ | |||
+ | * One of the best student-friendly books to get familiar with many of the most important advanced tools used regularly in physics is [[https://www.springer.com/de/book/9780387989310|Advanced mathematical methods for scientists and engineers]] by Bender and Orszag. “//...a sleazy approximation that provides v good physical insight into what’s going on in some system is far more useful than an unintelligible exact result.//” | ||
+ | |||
+ | * To get an overview of all the different math subfields this video: [[https://www.youtube.com/watch?v=OmJ-4B-mS-Y|The Map of Mathematics]] is recommended. | ||
+ | |||
+ | * Great explanations of many advanced math topics, can be found in the "[[http://web.evanchen.cc/napkin.html|Infinite Napkin]]" book of Evan Chen. | ||
* One of the best books to get familiar with many of the most important advanced tools is "Geometrical methods of mathematical physics" by Bernard F. Schutz | * One of the best books to get familiar with many of the most important advanced tools is "Geometrical methods of mathematical physics" by Bernard F. Schutz | ||
+ | * A wonderful guided tour of the world of math and physics is "Road to Reality" by Penrose | ||
* Also: Differential Geometry, Gauge Theories, and Gravity M. Göckeler, T. Schücker is a good read | * Also: Differential Geometry, Gauge Theories, and Gravity M. Göckeler, T. Schücker is a good read | ||
* Further recommended texts to accompany these: | * Further recommended texts to accompany these: | ||
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- Geometry, Topology,& Physics, Nakahara (another useful survey) | - Geometry, Topology,& Physics, Nakahara (another useful survey) | ||
- Spin Geometry, Lawson & Michelson | - Spin Geometry, Lawson & Michelson | ||
+ | |||
+ | * Robert Geroch, Mathematical Physics | ||
+ | * Ivonne Choquet-Bruhat, Cecile DeWitt-Morette, and Margaret Dillard-Bleick, Analysis, Manifolds, and Physics | ||
+ | * Timothy Gowers, June Barrow-Green, Imre Leader "The Princeton companion to mathematics" | ||
+ | |||
+ | ===== Quotes ===== | ||
+ | |||
+ | |||
+ | <blockquote>Mathematics and physics have gone their separate ways for nearly | ||
+ | a century now and it is time for this to end.<cite>Topology, | ||
+ | Geometry and Gauge Fields: Interactions by Naber</cite></blockquote> | ||
+ | |||
+ | |||
+ | |||
+ | |||
+ | <blockquote> | ||
+ | Physics doesn’t simply borrow the language from mathematics, but manipulates it, adapts it, and reinvents it according to the needs for a description of physical reality. The magic about physics is transforming the pure logic of mathematics into a beautiful narrative about nature. A poet couldn’t write poetry without a language, but language is not sufficient to make poetry. The same happens in science: mathematics is the language, but it takes physics to do poetry. | ||
+ | |||
+ | |||
+ | <cite>https://arxiv.org/pdf/1710.07663.pdf#page23</cite> | ||
+ | |||
+ | </blockquote> | ||