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====== Advanced Tools ======


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<blockquote>
Physics is not mathematics, just as mathematics is not physics. Somehow nature chooses only a subset of the very beautiful and complex and intricate mathematics that mathematicians develop, and that precise subset is what the theoretical physicist is trying to look for.

<cite>C.N. Yang</cite>
</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". 

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. 





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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
  * 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
  * Further recommended texts to accompany these: 
    - Geometry of Physics, Frankel 
    - Intro to Lie Algebras & Rep. Th., Humphreys 
    - Geometry, Topology,& Physics, Nakahara (another useful survey)
    - 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>