MaThemaTics Language Kind Types Ordered By Size

**Maeen LisT Uhv KonTenTs Uhv MaTh Lang Kynd Typs Ohrdrd By Syz**

0: Zeeroh RooT Wrd Kynd Typss KLasT By Syz Ohrdr

0:1: Zeeroh Simp Lang Leeng Kyndz

0:2: Zerohd Iz 0d Iz Lingd Saiz Ohmz

5.0: Mathematics Guide for Physics

5.1: Undergraduate Level Physics

5.2: MaTh For STring Theory

5.3: MaTh For SupersTring Research

**End Uhv Maeen LisT Uhv KonTenTs Uhv MaTh Lang Kynd Typs Ohrdrd By Syz**

**Zero RooT Word Kind Types CLassed By Size Order**

Zeeroh RooT Wrd Kynd Typss KLasT By Syz Ohrdr

Zero In FohnehTik EengGLish Voeess Sownd Chahrz Iz Zeeroh.

Zeeroh Wrd Dehskripshuhn Uhv Zeeroh RooT Wrd Kynd Typss KLasT By Syz Ohrdr

If Uh ( KownT Ohr Syz ) Iz Diskuhvrd Tu Bee Now **( Zeeroh = Non )** Then IT Iz Deskrybd Az **EhmpTee**.

EhgzampuLz Uhv Zeeroh InkLuud Thuh NexT Kyndz

Ther Kan Bee Sum PLaeess JusT Wuhn PoeenT Uhv EhmpTeenehss

- Lyk Uh Skreen PikseL ThaT Iz ( SwichT Off And EmpTee Uhv LyT ).

**Zeeroh In MaTh** Uhv Zeeroh Wrd Dehskripshuhn

- Iz
**Non Poz**Nuhmbr And**Non Neg**Nuhmbr.

EhgzampuLz Uhv Zeeroh RooT Wrd InkLuud Thuh NexT Kynd Typss:

If Uh Syz Iz Diskuhvrd Tu Bee Now **EhmpTee** Then IT Iz Deskrybd Az WiTh **( Zeeroh )**.

Ther Kan Bee Sum PLaeess JusT Wuhn PoeenT Uhv EhmpTeenehss

- Lyk Uh Skreen PikseL ThaT Iz ( SwichT Off And EmpTee Uhv LyT ).

Tu { Zeeroh Ohr Zeroise Ohr Zeeroize} Iz ( { Tu EmpTeeyz } Ohr { Tu Koz Tu Chaengj Tu Zeeroh } ).

{ ( Zeroizing Zeeroizeeng Ohr EmpTeeyzeeng } { Zeroizez Ohr EmpTeez }.

If Sum Ehreeuh Wuz **{ Zeerohd Ohr Zeeroaizd Ohr Nond Ohr Nonyzd }** Then It Wuz **{ EhmpTeed Ohr Maeed Zeeroh }**.

If Uh ( KownT Ohr Syz ) Iz Diskuhvrd Tu Bee Now **( Zeeroh )** Then IT Iz Deskrybd Az **EhmpTee**.

EgzampuLz InkLuud Thuh NexT Kyndz

.Ther Kan Bee Sum PLaeess JusT Wuhn PoeenT Uhv EhmpTeenehss Lyk Uh Skreen PikseL ThaT Iz ( SwichT Off And EmpTee Uhv LeesT On Dahrk PoeenTs ).

Ther Kan Bee Uh 1 DymehnshuhnuL Lyn Uhv Uh Spehsifik SyzuhbuL LenTh ThaT Iz ( SwichT Off And EmpTee Uhv LyT ).

Eech Skreen Shohn BLak Chahr Iz Uh [ 2 DymehnshuhnuL Drahn Fohrm SyzuhbuL WiTh [ LenTh And WidTh ] ] ThaT Iz ( SwichT Off And EmpTee Uhv LyT ).

MohsT OuTr Spaeess Iz ( 3 DymehnshuhnuL Ehreeuhz Uhv EmpTeeness ).

WyL Suhm Ehreeuh Iz LodjikuLLee DeeTrmind Tu Bee KuhnTinneeweeng Tu Be ( Zeeroh = Non )

- In ReeLayshuhn Tu Suhm ( Wuhn Theeng, Wuhn Kynd Uhv Theeng, Ohr Mohr, Up Tu Ehvree Theeng ELss Non Ther ),
- Wich MyT Bee Wuhn Ohr Mohr Uhv:
- ( { Uhprohcheeng = Muuveeng Twohrdz } And|Ohr { STayeeng Saym DisTanss Uhway } And|Ohr { Muuveeng Uhway } )

- Wich MyT Bee Wuhn Ohr Mohr Uhv:
- ThaT Ehreeuh Iz EhmpTee Uhv ( ThaT Wuhn Theeng And|Ohr Thohz [ MuLTy = Mohr Than Wuhn ) Theengz )

A Non Thot Ist wants tu non/un think SUhm ThoT,

therfohr Thuh Non ThoT Ist dreemz the wrdz "non izm" tu at the egzakt location tu replace the unwanted wrd or wrdz.

If a thing iz pure good then it iz non bad and non wrong.

Un ist un izm: un iz non or ex. Pro un iz pro non and pro ex. Az in:

"Pro+un-wrong+izm/ist iz pro+non-wrong+izm and pro+ex-wrong+ist."

"Pro+un-superstitious iz pro+non+superstition and pro+ex+superstitious."

"Pro+un-christian iz pro+non-christian and pro+ex-christian."

"Pro+un-violation izm iz pro+non-violation izm and pro+ex-violation ist."

"Pro+un-reip izm ist iz pro+non-reip izm ist and pro+ex-reip izm ist."

A non-kunsent izm ist haz Zerohd their agreement with some thing az wrong or a typ uv wrong.

Unproven, maybe unprovable, blind faith iz un-saientifikli non-evidenst belief that iz probably superstition, a "contradiction uv Fizikal Lah." Never unproven, all evidence supports the scientifik theory uv Natural Fizikal Lah.

Belief in superstition iz the only sinful kraim against Natural Fizikal Lah.

The only possible curing salvation frum sin iz self-repentance uv belief tu kunfohrm tu Natural Fizikal Lah.

A natural saientifik goal iz tu zero the superstitious belief sin.

In the Zerohd Syz Ohm iz noticed:

empty space

No elements. No matter partiklz or weivz. No force particles or weivz.

Sizeomes of Empty Space with no thing in it and the Sizeomes of Zero including the size of the array(s) of zeroed bits, emptiness, NiL, NUL and void.

There iz more empty space than anything else in most places in infinite existence.

There iz probably empty space outside our universe and probably between universes.

There iz empty space between galaxy cloud filaments in the universal galaxy cloud web.

There iz empty space between galaxy superclusters in galactic clouds.

There iz empty space between galaxies and nebulae and star systems and planets.

There iz empty space between air kemz and between all partiklz and weivz of matr or force.

If There Iz ( This Heer Ehreeuh ) In Wich Ther Haz GoT EhmpTeeyzdAnd Thus Iz Now Zeeroh Uhv A Spehsiffik Non Heer Kynd Uhv Wich Ther Iz An InsTanss ThaT Iz DyrekTLee Uhprohcheeng ( < Leedeeng In FruhnT Uhv > Ohr { Leedeeng Mohr KwikLee Than } ) Thuh ( UhThr InsTanssez Uhv ThaT Spehsiffik Non Heer Kynd ) Twohrdz ( ThaT Heer Ehreeuh ) Then ThaT Non Heer Leedeeng Uhprohcheeng InsTanss Iz ExpekTed Tu Soon Uhprohch Tu Tuch Thuh MohsT Neer ( Edj Uhv ThaT Heer Ehreuh ).

If ( ThaT Tucheeng Uhprohcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) ThaT Iz Non ( In ThaT Heer Ehreeuh ) Thoh Iz Tuhcheeng [ Thuh Edj Uhv ThaT Heer Ehreeuh ] Happenz { Tu GeT SLohd ) By Thuh ReezisTanss Uhv [ Suhm ( PahrT Ohr PahrTs ) Uhv Suhm ( Bownduhree Ohr Bownduhreez ) AT ThaT Edj Ehreuh Uhv ThaT Heer Ehreeuh ] Eenuf ThaT ( Thuh Tuhcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) { Duz STop And Beekuhm STiLL } Then ( ThaT Heer Ehreeuh STaeez EhmpTee ) Uhv ( ThaT Tucheeng InsTanss Uhv ThaT Spehsiffik Kynd ).

Thoh If AiThr [ Thuh ( Bownduhree Ohr Bownduhreez ) Uhv ThaT Heer Ehreeuh ] Non STops ( ThaT Tucheeng Uhprohcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) Ohr If Ther Iz Noh Bownduhree SheeLdeeng Tu STop ( Ehneeg Uhprohcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) Fruhm EhnTreeng ( ThaT Heer Ehreuh ) Then ( Thuh Uhprohcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) Duz Moov DyrekTLee InTu ThaT Heer Ehreuh Wich Duz Koz IT Tu BeeKuhm **Ex-Zeeroh** In And PahrT Uhv ThaT Wuhn InsTanss In ( ThaT Heer Ehreuh ).

See AhLsoh:Zeeroh And Leeng RooT Wrdz Uhv Paeej Naeemz Tu Kuhndenss

Zeeroh Simp Lang Leeng Kyndz Uhv MaTh Lang Kynd Typs Ohrdrd By Syz

* Uhv Thuh KuhmpLeeT LisT Uhv Baeesik Kynd Typs KLasT By Syz Ohrdr

* Uhv Thuh Syehnss Baeesiks Kynd Typs KLasT By Syz Ohrdr Uhv Omnyon Izm.

Zero In FohnehTik EengGLish Voeess Sownd Chahrz Iz Zeeroh.

Baeesik Zeeroh Kynd Typss:

Tu { Zeeroh Ohr Zeroise Ohr Zeeroize} Iz ( { Tu EmpTeeyz } Ohr { Tu Koz Tu Chaengj Tu Zeeroh } ).

{ ( Zeroizing Zeeroizeeng Ohr EmpTeeyzeeng } { Zeroizez Ohr EmpTeez }.

If Sum Ehreeuh Wuz **{ Leengd Ohr Leengyzd Ohr Zeerohd Ohr Zeeroaizd Ohr Nond Ohr Nonyzd }** Then It Wuz **{ EhmpTeed Ohr Maeed Zeeroh }**.

Zeeroh Wrd Dehskripshuhn Uhv Zeeroh RooT Wrd Kynd Typss KLasT By Syz Ohrdr

If Uh ( KownT Ohr Syz ) Iz Diskuhvrd Tu Bee Now **( Zeeroh = Non )** Then IT Iz Deskrybd Az **EhmpTee**.

EhgzampuLz Uhv Zeeroh InkLuud Thuh NexT Kyndz

Ther Kan Bee Sum PLaeess JusT Wuhn PoeenT Uhv EhmpTeenehss

- Lyk Uh Skreen PikseL ThaT Iz ( SwichT Off And EmpTee Uhv LyT ).

**Zeeroh In MaTh** Uhv Zeeroh Wrd Dehskripshuhn

- Iz
**Non Poz**Nuhmbr And**Non Neg**Nuhmbr.

If Uh ( KownT Ohr Syz ) Iz Diskuhvrd Tu Bee Now **( Leeng = Zeeroh = Non )** Then IT Iz Deskrybd Az **EhmpTee**.

EgzampuLz InkLuud Thuh NexT Kyndz

Ther Kan Bee Sum PLaeess JusT Wuhn PoeenT Uhv EhmpTeenehss Lyk Uh Skreen PikseL ThaT Iz ( SwichT Off And EmpTee Uhv LyT ).

Ther Kan Bee Uh 1 DymehnshuhnuL Lyn Uhv Uh Spehsifik SyzuhbuL LenTh ThaT Iz ( SwichT Off And EmpTee Uhv LyT ).

Eech Skreen Shohn BLak Chahr Iz Uh [ 2 DymehnshuhnuL Drahn Fohrm SyzuhbuL WiTh [ LenTh And WidTh ] ] ThaT Iz ( SwichT Off And EmpTee Uhv LyT ).

MohsT OuTr Spaeess Iz ( 3 DymehnshuhnuL Ehreeuhz Uhv EmpTeeness ).

WyL Suhm Ehreeuh Iz LodjikuLLee DeeTrmind Tu Bee KuhnTinneeweeng Tu Be ( Leeng = Zeeroh = Non )

- In ReeLayshuhn Tu Suhm ( Wuhn Theeng, Wuhn Kynd Uhv Theeng, Ohr Mohr, Up Tu Ehvree Theeng ELss Non Ther ),
- Wich MyT Bee Wuhn Ohr Mohr Uhv:
- ( { Uhprohcheeng = Muuveeng Twohrdz } And|Ohr { STayeeng Saym DisTanss Uhway } And|Ohr { Muuveeng Uhway } )

- Wich MyT Bee Wuhn Ohr Mohr Uhv:
- ThaT Ehreeuh Iz EhmpTee Uhv ( ThaT Wuhn Theeng And|Ohr Thohz [ MuLTy = Mohr Than Wuhn ) Theengz )

If There Iz ( This Heer Ehreeuh ) In Wich Ther Iz Zeeroh Uhv A Spehsiffik Non Heer Kynd Uhv Wich Ther Iz An InsTanss ThaT Iz DyrekTLee Uhprohcheeng ( < Leedeeng In FruhnT Uhv > Ohr { Leedeeng Mohr KwikLee Than } ) Thuh ( UhThr InsTanssez Uhv ThaT Spehsiffik Non Heer Kynd ) Twohrdz ( ThaT Heer Ehreeuh ) Then ThaT Non Heer Leedeeng Uhprohcheeng InsTanss Iz ExpekTed Tu Soon Uhprohch Tu Tuch Thuh MohsT Neer ( Edj Uhv ThaT Heer Ehreuh ).

If ( ThaT Tucheeng Uhprohcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) ThaT Iz Non ( In ThaT Heer Ehreeuh ) Thoh Iz Tuhcheeng [ Thuh Edj Uhv ThaT Heer Ehreeuh ] Happenz { Tu GeT SLohd ) By Thuh ReezisTanss Uhv [ Suhm ( PahrT Ohr PahrTs ) Uhv Suhm ( Bownduhree Ohr Bownduhreez ) AT ThaT Edj Ehreuh Uhv ThaT Heer Ehreeuh ] Eenuf ThaT ( Thuh Tuhcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) { Duz STop And Beekuhm STiLL } Then ( ThaT Heer Ehreeuh STaeez EhmpTee ) Uhv ( ThaT Tucheeng InsTanss Uhv ThaT Spehsiffik Kynd ).

Thoh If AiThr [ Thuh ( Bownduhree Ohr Bownduhreez ) Uhv ThaT Heer Ehreeuh ] Non STops ( ThaT Tucheeng Uhprohcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) Ohr If Ther Iz Noh Bownduhree SheeLdeeng Tu STop ( Ehneeg Uhprohcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) Fruhm EhnTreeng ( ThaT Heer Ehreuh ) Then ( Thuh Uhprohcheeng InsTanss Uhv ThaT Spehsiffik Kynd ) Duz Moov DyrekTLee InTu ThaT Heer Ehreuh Wich Duz Koz IT Tu BeeKuhm **Ex-Zeeroh** In And PahrT Uhv ThaT Wuhn InsTanss In ( ThaT Heer Ehreuh ).

- In Simp Lang Yeeng Voiss Sownd Chahrz Iz ( Zeeroh = Leeng )
- Uv Thuh Leeng Saiz Ohmz Uv Thuh ReeL Prwvd TrwTh Syz Ohmz
- Uv Thuh Baisik Kyndz Syz Ohmz Paidj LisT

NexT Zhong Lang Pin Yin KonsepTs Wr Fruhm:

https://chinese.yabla.com/chinese-english-pinyin-dictionary.php?define=Zero

〇

- líng

- zero

Thuh Wrd "Zero" Iz RehpreezehnTed By Thuh Zhong Lang Syn 〇 SpeLd In Pin Yin Az **líng**.

BaesT Fruhm Heereeng

- https://dictionary.hantrainerpro.com/chinese-english/translation-ling_zero.htm
- And https://www.howtopronounce.com/ling/,
- ThaT Wrd In FohnehTik EenGLish Voeess Sownd Syn Seekwenss Iz L->ee->ng
- Wich Maeed ShohrT Iz Leeng.

Thuh NexT TekST Wuhz Fruhm:

líng…English translations

NexT KonsepTs Wr DeeduusT BaeesT Fruhm Trmz AT:

https://chinese.yabla.com/chinese-english-pinyin-dictionary.php?define=Zero

Thuh Wrd zero Iz RehpreezehnTed By Thuh Chyneez Lang Syn 〇 SpeLd In Pin Yin Az líng.

BaesT Fruhm Heereeng

- https://dictionary.hantrainerpro.com/chinese-english/translation-ling_zero.htm
- And https://www.howtopronounce.com/ling/,
- ThaT Wrd In FohnehTik EenGLish Voeess Sownd Syn Seekwenss Iz L->ee->ng
- Wich Maeed ShohrT Iz Leeng.

Thuh NexT TekST Wuhz Fruhm:

líng…English translations

Zeeroh And Leeng RooT Wrdz Uhv Paeej Naeemz Tu Kuhndenss

(Zeroed is 0d is Made Ling) Sizomes in Yeeng Voiss Sownd Chahrz

Zeroed Is 0d Is Lingd Size Class In Fuhnehtik IngLish iz Zerohd Iz 0d Iz Lingd Saiz Klass

0d=Leengd Vrs (Haoh=#)1: Theeree ( Zeroed = Língd )

- In Simp Lang Yeeng Voiss Sownd Chahrz Iz ( Zeerohd = Leengd )
- Uv Thuh Leeng Saiz Ohmz Uv Thuh AhL Spundj STeiT Saiz Ohmz
- Uv Thuh AhL STeiT Saiz Ohmz Peidj LisT.

- Uv Thuh Leeng Saiz Ohmz Uv Thuh AhL Spundj STeiT Saiz Ohmz

Suffix -ed Vrs (Haoh=#)1: Suffix ed Uv (Zeroed iz 0d iz Lingd) Saiz Ohm Uv Saiz KLass LisT Uv AhL Spundj KunTree.

Suffix -ed Vrs (Haoh=#)2: Suffix -ed Pronunciation

Suffix -ed Vrs (Haoh=#)3:0: (in verbs, past participles, and some denominal adjectives):

Suffix -ed Vrs (Haoh=#)3:1:0: * (after a vowel or a voiced consonant other than a /d/)

Suffix -ed Vrs (Haoh=#)3:1:1: * enPR: d, IPA(key): /d/

Suffix -ed Vrs (Haoh=#)3:2:0: * (after a voiceless consonant other than a /t/)

Suffix -ed Vrs (Haoh=#)3:2:1: * enPR: t, IPA(key): /t/

Suffix -ed Vrs (Haoh=#)3:3: * (after a /d/ or /t/) same as below

Suffix -ed Vrs (Haoh=#)4:0: (other denominal adjectives):

Suffix -ed Vrs (Haoh=#)4:1: * (US) enPR: ĭd, IPA(key): /ɪd/ or enPR: əd, IPA(key): /əd/

0d=Leengd Vrs (Haoh=#)2: Non Aiz Eeng Nonz Then Iz Nond And Thus Iz Non.

Zero in Jen Simp Lang Zeroh=0=Ling)

Zeroh Aiz Iz 0 Aiz Iz Ling Aiz Saiz Ohmz

An empteed Zeroed Iz 0d Iz Lingd Saiz Ohmz iz:

- Thuh
- Thuh Math Num (Zero Iz 0 Iz Ling), and
- Thuh vehreeus uthr Zeroed Iz 0 Iz Lingd Saiz Ohmz which are sizes uv void or empty space.

A Zeroed Sizeome (Funetik IngLish: Zerohd Saizohm) iz one uv the following Zerohd Saizohmz.

American English: Nonizing Nons, Then Has Been Noned And Iz ex-.

Simp Lang: Non Aiz Eeng Nonz, Then baz ben nond And Iz Eks.

The computer DeL Key Kupiwtr Task by Emptee Eeng Ihlustreits how wen sum theeng Iz Lingd It Iz Nond.

Bynehree Nuhmbr Kynd = 2 SeT VaLz Kynd

Simp By Haohz Dehfinnishuhn Kohd:

- ( b0: "Bee Leeng" ( Chyneez Simp Lang Fohr Zero ) Simp Haoh BiLT Fruhm Leeng Saiz Ohmz
- ( b1 Wuhn Simp Hao = ( Wuhn Haoh Kynd )

Bynehree Pehr Numbr Kynd In Simp Lang Iz By Pehr Haoh Kynd.

NexT Iz Bayss 4 4 SLoTs Uhv Kohd Kohd With SLoT PahrTs: ByPehr Kohd WiTh 4 Kohd SLoTs:

* ( p0: "Pee Leeng"

* Az b00: "Bee Leeng Leeng" ),

* ( p1: "Pee Wuhn"

* Az b01: "Bee Leeng Wuhn",

* ( p2: "Pee Too"

* Az b10: "Bee Wuhn Leeng",

* ( p3: "Pee Three"

* Az b11: "Bee Wuhn Wuhn" )

OcTaL Number Kind = OkT Haoh Kynd

- Bayss 8 Hao Kynd Kohd WiTh 8 SLoTs,
- Eech SLoT WiTh Three Bynehree Nuhmbr Kyndz:
- ( (o0=b000, o1=b001, o2=b010, o3=b011, o4=b100, o5=b101, o6=b110, o7=b111)

- Bayss 16 Numbr Kynd Kohd WiTh 16 SLoTs,
- Eech WiTh A Pehr Uhv Thuh Bynehree Pehr Numbr Kynd
- ( h0=p00, h1=p01, h2=p02, h3=p03, h4=p10, h5=p11, h6=p12, h7=p13,
- h8=p20, h9=p21, ha=p22, hb=p23, hc=p30, hd=p31, he=p32, hf=p33 )

- Eech WiTh A Pehr Uhv Thuh Bynehree Pehr Numbr Kynd

Undergraduate Level Physics Uv Mathematics Guide For Physics

Official String Theory Web Site-->Math Guide I--Undergraduate level physics

Guide to math needed to study physics

The language of physics is mathematics. In order to study physics seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out. Algebra, for example, was cutting-edge mathematics when it was being developed in Baghdad in the 9th century. But today it's just the first step along the journey.

Algebra

Algebra provides the first exposure to the use of variables and constants, and experience manipulating and solving linear equations of the form y = ax + b and quadratic equations of the form y = ax2+bx+c.

Geometry

Geometry at this level is two-dimensional Euclidean geometry, Courses focus on learning to reason geometrically, to use concepts like symmetry, similarity and congruence, to understand the properties of geometric shapes in a flat, two-dimensional space.

Trigonometry

Trigonometry begins with the study of right triangles and the Pythagorean theorem. The trigonometric functions sin, cos, tan and their inverses are introduced and clever identities between them are explored.

Calculus (single variable)

Calculus begins with the definition of an abstract functions of a single variable, and introduces the ordinary derivative of that function as the tangent to that curve at a given point along the curve. Integration is derived from looking at the area under a curve,which is then shown to be the inverse of differentiation.

Calculus (multivariable)

Multivariable calculus introduces functions of several variables f(x,y,z…), and students learn to take partial and total derivatives. The ideas of directional derivative, integration along a path and integration over a surface are developed in two and three dimensional Euclidean space.

Analytic Geometry

Analytic geometry is the marriage of algebra with geometry. Geometric objects such as conic sections, planes and spheres are studied by the means of algebraic equations. Vectors in Cartesian, polar and spherical coordinates are introduced.

Linear Algebra

In linear algebra, students learn to solve systems of linear equations of the form ai1 x1 + ai2 x2 + … + ain xn = ci and express them in terms of matrices and vectors. The properties of abstract matrices, such as inverse, determinant, characteristic equation, and of certain types of matrices, such as symmetric, antisymmetric, unitary or Hermitian, are explored.

Ordinary Differential Equations

This is where the physics begins! Much of physics is about deriving and solving differential equations. The most important differential equation to learn, and the one most studied in undergraduate physics, is the harmonic oscillator equation, ax'' + bx' + cx = f(t), where x' means the time derivative of x(t).

Partial Differential Equations

For doing physics in more than one dimension, it becomes necessary to use partial derivatives and hence partial differential equations. The first partial differential equations students learn are the linear, separable ones that were derived and solved in the 18th and 19th centuries by people like Laplace, Green, Fourier, Legendre, and Bessel.

Methods of approximation

Most of the problems in physics can't be solved exactly in closed form. Therefore we have to learn technology for making clever approximations, such as power series expansions, saddle point integration, and small (or large) perturbations.

Probability and statistics

Probability became of major importance in physics when quantum mechanics entered the scene. A course on probability begins by studying coin flips, and the counting of distinguishable vs. indistinguishable objects. The concepts of mean and variance are developed and applied in the cases of Poisson and Gaussian statistics.

MaTh For STring Theory Uv Mathematics Guide For Physics

Official String Theory Web Site:-->Math Guide II -- Graduate and beyond

Guide to math needed to study physics

Here are some of the topics in mathematics that a person who wants to learn advanced topics in theoretical physics, especially string theory, should become familiar with.

Real analysis

In real analysis, students learn abstract properties of real functions as mappings, isomorphism, fixed points, and basic topology such as sets, neighborhoods, invariants and homeomorphisms.

Complex analysis

Complex analysis is an important foundation for learning string theory. Functions of a complex variable, complex manifolds, holomorphic functions, harmonic forms, Kähler manifolds, Riemann surfaces and Teichmuller spaces are topics one needs to become familiar with in order to study string theory.

Group theory

Modern particle physics could not have progressed without an understanding of symmetries and group transformations. Group theory usually begins with the group of permutations on N objects, and other finite groups. Concepts such as representations, irreducibility, classes and characters.

Differential geometry

Einstein's General Theory of Relativity turned non-Euclidean geometry from a controversial advance in mathematics into a component of graduate physics education. Differential geometry begins with the study of differentiable manifolds, coordinate systems, vectors and tensors. Students should learn about metrics and covariant derivatives, and how to calculate curvature in coordinate and non-coordinate bases.

Lie groups

A Lie group is a group defined as a set of mappings on a differentiable manifold. Lie groups have been especially important in modern physics. The study of Lie groups combines techniques from group theory and basic differential geometry to develop the concepts of Lie derivatives, Killing vectors, Lie algebras and matrix representations.

Differential forms

The mathematics of differential forms, developed by Elie Cartan at the beginning of the 20th century, has been powerful technology for understanding Hamiltonian dynamics, relativity and gauge field theory. Students begin with antisymmetric tensors, then develop the concepts of exterior product, exterior derivative, orientability, volume elements, and integrability conditions.

Homology

Homology concerns regions and boundaries of spaces. For example, the boundary of a two-dimensional circular disk is a one-dimensional circle. But a one-dimensional circle has no edges, and hence no boundary. In homology this case is generalized to "The boundary of a boundary is zero." Students learn about simplexes, complexes, chains, and homology groups.

Cohomology

Cohomology and homology are related, as one might suspect from the names. Cohomology is the study of the relationship between closed and exact differential forms defined on some manifold M. Students explore the generalization of Stokes' theorem, de Rham cohomology, the de Rahm complex, de Rahm's theorem and cohomology groups.

Homotopy

Lightly speaking, homotopy is the study of the hole in the donut. Homotopy is important in string theory because closed strings can wind around donut holes and get stuck, with physical consequences. Students learn about paths and loops, homotopic maps of loops, contractibility, the fundamental group, higher homotopy groups, and the Bott periodicity theorem.

Fiber bundles

Fiber bundles comprise an area of mathematics that studies spaces defined on other spaces through the use of a projection map of some kind. For example, in electromagnetism there is a U(1) vector potential associated with every point of the spacetime manifold. Therefore one could study electromagnetism abstractly as a U(1) fiber bundle over some spacetime manifold M. Concepts developed include tangent bundles, principal bundles, Hopf maps, covariant derivatives, curvature, and the connection to gauge field theories in physics.

Characteristic classes

The subject of characteristic classes applies cohomology to fiber bundles to understand the barriers to untwisting a fiber bundle into what is known as a trivial bundle. This is useful because it can reduce complex physical problems to math problems that are already solved. The Chern class is particularly relevant to string theory.

Index theorems

In physics we are often interested in knowing about the space of zero eigenvalues of a differential operator. The index of such an operator is related to the dimension of that space of zero eigenvalues. The subject of index theorems and characteristic classes is concerned with

Supersymmetry and supergravity

The mathematics behind supersymmetry starts with two concepts: graded Lie algebras, and Grassmann numbers. A graded algebra is one that uses both commutation and anti-commutation relations. Grassmann numbers are anti-commuting numbers, so that x times y = –y times x. The mathematical technology needed to work in supersymmetry includes an understanding of graded Lie algebras, spinors in arbitrary spacetime dimensions, covariant derivatives of spinors, torsion, Killing spinors, and Grassmann multiplication, derivation and integration, and Kähler potentials.

MaTh For SupersTring Research Uv Mathematics Guide For Physics

Official String Theory Web Site: --> Math Guide III - Current research

Guide to math used in string theory

These are topics in mathematics at the current cutting edge of superstring research.

K-theory

Cohomology is a powerful mathematical technology for classifying differential forms. In the 1960s, work by Sir Michael Atiyah, Isadore Singer, Alexandre Grothendieck, and Friedrich Hirzebruch generalized coholomogy from differential forms to vector bundles, a subject that is now known as K-theory.

Witten has argued that K-theory is relevant to string theory for classifying D-brane charges. D-brane objects in string theory carry a type of charge called Ramond-Ramond charge. Ramond-Ramond fields are differential forms, and their charges should be classifed by ordinary cohomology. But gauge fields propagate on D-branes, and gauge fields give rise to vector bundles. This suggests that D-brane charge classification requires a generalization of cohomology to vector bundles — hence K-theory.

Overview of K-theory Applied to Strings by Edward Witten

D-branes and K-theory by Edward Witten

Noncommutative geometry (NCG for short)

Geometry was originally developed to describe physical space that we can see and measure. After modern mathematics was freed from Euclid's Fifth Axiom by Gauss and Bolyai, Riemann added to modern geometry the abstract notion of a manifold M with points that are labeled by local coordinates that are real numbers, with some metric tensor that determines an extremal length between two points on the manifold.

Much of the progress in 20th century physics was in applying this modern notion of geometry to spacetime, or to quantum gauge field theory.

In the quest to develop a notion of quantum geometry, as far back as 1947, people were trying to quantize spacetime so that the coordinates would not be ordinary real numbers, but somehow elevated to quantum operators obeying some nontrivial quantum commutation relations. Hence the term "noncommutative geometry," or NCG for short.

The current interest in NCG among physicists of the 21st century has been stimulated by work by French mathematician Alain Connes.

Two Lectures on D-Geometry and Noncommutative Geometry by Michael R. Douglas

Noncommutative Geometry and Matrix Theory: Compactification on Tori by Alain Connes, Michael R. Douglas, Albert Schwarz

String Theory and Noncommutative Geometry by Edward Witten and Nathan Seiberg.

Non-commutative spaces in physics and mathematics by Daniela Bigatti

Noncommutative Geometry for Pedestrians by J.Madore