What is mathemathical physics: Definition and 9 Discussions

Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.
Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties that are considered true starting points of the theory under consideration.Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science and the social sciences. Although mathematics is extensively used for modeling phenomena, the fundamental truths of mathematics are independent from any scientific experimentation. Some areas of mathematics, such as statistics and game theory, are developed in close correlation with their applications and are often grouped under applied mathematics. Other areas are developed independently from any application (and are therefore called pure mathematics), but often later find practical applications. The problem of integer factorization, for example, which goes back to Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks.
Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics, most notably in Euclid's Elements. Since its beginning, mathematics was essentially divided into geometry and arithmetic (the manipulation of natural numbers and fractions), until the 16th and 17th centuries, when algebra and infinitesimal calculus were introduced as new areas. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid lockstep increase in the development of both. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method, which heralded a dramatic increase in the number of mathematical areas and their fields of application. The contemporary Mathematics Subject Classification lists more than 60 first-level areas of mathematics.

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  1. D

    I Online Christoffel Symbols Calculator

    I would love to hear from you if you have any suggestions, feedback, or criticism. The goal is to build better and more sophisticated software that would push the boundaries of research in astrophysics!
  2. P

    Intro Math Hassani's books on Mathematical Physics/Methods

    By looking around, it seems like Dr. Hassani's books are great for studying "mathematical methods for the physicist/engineer." One is for the beginner physicist [Mathematical Methods: For Students of Physics and Related Fields] and the other is [Mathematical Physics: A Modern Introduction to Its...
  3. G

    Complex contour integral proof

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  4. G

    Put the eigenvalue function in self-adjoint form

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  5. Safinaz

    I Is the Kroneker Delta Identity Used Correctly in this Paper?

    Is there a Kroneker Delta identity: $$ \delta^3 (k) \delta^3 (k) = \frac{2\pi^2}{k^3} ~ \delta (k_1- k_2) $$? Where k is a wave number. In this Paper: Equations 26 and 27, I think this identity is used to make ##k_1=k_2## and there is an extra negative sign.
  6. G

    I Transfer rank2 tensor to a new basis

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  7. G

    A Upper indices and lower indices in Einstein notation

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  8. advhaver

    Admissions PhD Programs for Mathematical Physics (Experimental Condensed Matter Physics)

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