advanced_tools
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advanced_tools [2017/12/12 05:34] jakobadmin [Recommended Literature] |
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| 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. | 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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| + | ----- | ||
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| + | It is important to understand the different types of mathematical tools and their relationships. A broad generalization goes as follows: | ||
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| + | * __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) | ||
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| + | {{ :stuffstructuresandproperties.png?nolink&400|}} | ||
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| + | 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 $. | ||
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| + | * 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.//” | ||
| - | ===== Recommended Literature ===== | + | * 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. | * Great explanations of many advanced math topics, can be found in the "[[http://web.evanchen.cc/napkin.html|Infinite Napkin]]" book of Evan Chen. | ||
advanced_tools.1513053298.txt.gz · Last modified: 2017/12/12 04:34 (external edit)
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