advanced_tools
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
advanced_tools [2017/11/29 11:24] jakobadmin [Advanced Tools] |
advanced_tools [2018/05/07 06:55] (current) jakobadmin |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| - | {{indexmenu_n>6}} | + | {{indexmenu_n>8}} |
| ~~NOTOC~~ | ~~NOTOC~~ | ||
| Line 8: | Line 8: | ||
| <WRAP group> | <WRAP group> | ||
| <WRAP half column> | <WRAP half column> | ||
| + | |||
| + | <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. | Although this is a travel guide to physics, you'll find here, of course, lots of pages about mathematics. | ||
| Line 15: | Line 21: | ||
| 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. | ||
| - | <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> | + | |
| </WRAP> | </WRAP> | ||
| Line 25: | Line 29: | ||
| </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.//” | ||
| - | ===== 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. | ||
| * 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 | * A wonderful guided tour of the world of math and physics is "Road to Reality" by Penrose | ||
advanced_tools.1511951064.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
