1._SomeBasicMathematics

7/27/2019 1._SomeBasicMathematics

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Supplementary

Y.Choquet-Bruhat et al,

Analysis, Manifolds & Physics, rev. ed., North Holland (82) H.Flanders,

Differential Forms, Academic Press (63)

R.Aldrovandi, J.G.Pereira,

An Introduction to Geometrical Physics, World Scientific (95) T.Frankel,

The Geometry of Physics, 2nd ed., CUP (03)

B.F.Schutz,

Geometrical Methods of Mathematical Physics, CUP (80)

Main Textbook


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Geometrical Methods of Mathematical Physics

1. Some Basic Mathematics

2. Differentiable Manifolds And Tensors3. Lie Derivatives And Lie Groups

4. Differential Forms

5. Applications In Physics

6. Connections For Riemannian ManifoldsAnd Gauge Theories

Bernard F. Schutz,Cambridge University Press (80)


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1. Some Basic Mathematics

1.1 The Space Rn And Its Topology

1.2 Mappings

1.3 Real Analysis

1.4 Group Theory

1.5 Linear Algebra

1.6 The Algebra Of Square Matrices

See: Choquet, Chapter I.


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Basic Algebraic Structures

See 1.5 for details.

Structures with only internal operations: Group ( G, )

Ring (R, +, ) : ( no e, orx1 )

Field (F, +, ) : Ring with e &x1 except for 0.

Structures with external scalar multiplication:

Module (M, +, ; R )

Algebra (A, +, ; R with e )

Vector space ( V, + ;F)

Prototypes:

Ris a field.

Rn is a vector space.


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1.1. The Space Rn And Its Topology

Goal: Extend multi-variable calculus (on En) to curved spaceswithout metric.

Bonus: vector calculus on E3 in curvilinear coordinates

Basic calculus concepts & tools (metric built-in):

Limit, continuity, differentiability, r-ball neighborhood, - formulism,

Integration,

Essential concept in the absence of metric:

Proximity Topology.


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A system U of subsets Uiof a setXdefines a topology onXif

1. , X U

1

2. ii

U

U

1

3.N

i

i

U

U

( Closure under arbitrary unions. )

( Closure under finite intersections. )

Elements UiofU are called open sets.

A topological space is the minimal structure on which concepts of

neighborhood, continuity, compactness, connectedness

can be defined.


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Trivial topology: U= { ,X }

every function onXis dis-continuous

Discrete topology: U= 2X

every function onXis continuous

Exact choice of topology is usually not very important:

2 topologies are equivalent if there exists an homeomorphism

(bi-continuous bijection) between them.

Tools for classification of topologies:

topological invariances, homology, homotopy,


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= Set of all ordered n-tuples of real numbers

1, ,n n ix x x x

R R

~ Prototype of an n-D continuum

Distance function (Euclidean metric):

:n n

d R R R

,

rd r x y y xN

2

, ,i i

i

d x y x y x y

(Open)Neighborhood /ball of radius rat x :A set Sis open if , . .r rS s t N S x x xN

Real numberR = complete Archimedian ordered field.

A set Sis discrete if , . . \r rS s t S x x x xN N


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Usual topology ofRn= Topologywith open balls as open sets

Metric-free version:

Define neighborhoods Nr(x) in terms of open intervals / cubes.

Preview: Continuity of functions will be defined in terms of open sets.

Hausdorff separated: Distinct points possess disjoint neighborhoods.

E.g., Rn is Hausdorff separated.


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1.2. Mappings

:f X Y by x y f x

Mapffrom set Xinto set Y, denoted,

associates each xX uniquely withy= f(x) Y.

Domain f XDomain off =

Range f YRange off =

Image ofMunderf = f M f x x M

Inverse image ofNunderf = 1f N x f x N

f1 exists ifffis 1-1 (injective):

f is onto (surjective) iff(X) = Y.

f is abijection if it is 1-1 onto.

f x f x x x


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x z g f x

:f X Y

:g Y Z

:g f X Z by

f g

g f

X Y Z

Composition

Given by x y f x

by y z g y

The composition off&gis the map

g f x g y


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Elementary calculus version:

Let f: R R. Thenfis continuous atx0

if

0 00 0 . .s t f x f x x x

Open ball version: Let 0 0x x x x N

Thenfis continuous atx0 if

0 00 0 . .s t f x f x x x N N

Continuity

0 00 0 . .s t f x f x

N N

i.e.,


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Open set version:

fcontinuous: Open set in

domain (f) is mapped to openset in codomain (f).

fdiscontinuous: Open set in

domain (f) is mapped to setnot open in codomain (f).

Counter-example:

fcontinuous but

OpenM half-closedf(M)

fis continuous if every open set in domain (f) is

mapped to an open set in codomain (f) ?Wrong!


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Correct criterion:

fis continuous ifevery open set in codomain(f)has an open inverse image.

OpenN half-closedf1(N)


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Continuity at a point:

f:X Yis continuous atx if the inverse image of any open

neighborhood off(x) is open,

i.e., f1(N[f(x)] ) is open.

Continuity in a region:

f is continuousonMX iff is continuous xM,

i.e., the inverse image of every open set inMis open.

Differentiability off: Rn R

1, ,

exists & is continuousif 1, ,

k n

k

kj

f x x

xf C j n

0

means is continuousf C f

means i anal ts y icf C f

i.e., Taylor expansion exists.

fis smooth k= whatever value necessary for problem at hand.


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: n nf R R by

1 1 1, , , , , ,n n nx x y y f f x x

x y f x

j j iy f x

Inverse function theorem :

fis invertible in some neighborhood ofx0 if

1

1

, ,det 0

, ,

n i

jn

yJ

y y

xx x

( Jacobian )

Let : nh R R then

n n

M f M

h J d H d x x y y where h Hx y x

Let


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1.3. Real Analysis

:f R R is analytic atx0 iff(x) has a Taylor series atx0

f C iffis analytic over Domain(f)

0

0

0

1

!

nn

nn x

d ff x x x

n d x

2

1exp but is not analytic at 0f C x

x

~C C convergence

:n

g R R is square integrable on SRn if 2 n

S

g d x x exists.

A square integrable functiongcan be approximated by an

analytic function f s.t.

2 n

S

f g d

x x x for any given 0


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An operatoron functions defined on Rnmaps functions to functions.

E.g., D ff

x

0

,

x

G f x f y g x y dy

3

2

3

ff f

x

E

Commutatorof operators: ,A B AB BA

s.t. ,A B f AB BA f A B f B A f

A &Bcommute if , 0A B

E.g., , ,d d

A B f x fdx dx

d d f d d f x x

dx dx dx dx

d f

dx

Domain (AB) C2 but Domain ([A , B ]) C1


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1.4. Group Theory

A group (G, ) is a set G with an internal operation : GG G thatis

, ,x y z x y z x y z x y z G 1. Associative:

2. Endowed with an identity element: s.t.e G x e e x x x G

1 1 1s.t.x G x x x x e x G

3. Endowed with an inverse for each element:

A group (G, +) is Abelian if all of its elements commute:

,x y y x x y G ( Identity is denoted by 0 )Examples:

(R,+) is an Abelian group.

The set of all permutations ofn objects form thepermutation groupSn.

All symmetries / transformations are members of some groups.

Its common practice to refer to group (G, ) simply as group G.


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(S, ) is a subgroup of group (G, ) ifSG.

Rough definition:

A Lie group is a group whose elements can be continuously parametrized.

~ continuous symmetries.

E.g., The set of all even permutations is a subgroup ofSn.

But the set of all odd permutations is not a subgroup ofSn (no e).

Groups (G,) is homomorphic to (H,*) if an onto map f: GH s.t.

* ,f x y f x f y x y G

It is an isomorphism iffis 1-1 onto.

(R+,) & (R,+) are isomorphic with f= log so that

log log log ,x y x y x y R

S Ch t Ch 1


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1.5. Linear Algebra

x y z x y z x y z

Ring (R, , +) is a field if

1. eR s.t. ex =xe =x xR.

2. x1R s.t. x1 x =x x1 = e xR except 0.

(R, , +) is a ring if

1. (R, + ) is an Abelian group.

2. is associative & distributive wrt + , i.e., x,y,zR,

x y z x y x z x y z x z y z

E.g., The set of all nn matrices is a ring (no inverse).

The function space is also a ring (no inverse).

E.g., R & C are fields under algebraic multiplication & addition.

See Choquet, Chap 1

or Aldrovandi, Math.1.


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( V, + ;R ) is a module if

1. ( V, + ) is an Abelian group.

2. R is a ring.

3. The scalar multiplicationRVVby (a,v) av satisfies

a a a v u v u , & ,a b R V u v

Module ( V, + ;F) is a linear (vector) space ifFis a field.

a b a b v v v

ab a bv v4. IfR has an identity e, then ev = v vV.

Well only use

F= K= R orC.

(A, , + ;R ) is an algebraover ringRif

1. (A, , + ) is a ring.

2. (A, + ;R ) is a module s.t.

a a a v u v u v u & ,a R A u v

Examples will be

given in Chap 3

For historical reasons the term linear algebra denotes the study of linear


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For historical reasons, the term linear algebra denotes the study of linear

simultaneous equations, matrix algebra, & vector spaces.

Mathematical justification:

(M, , + ; K) , whereMis the set of all nn matrices, is an algebra .

Linear combination: where &i ii ii

a a V v vK

{ vi} is linearly independent if 0i i ii a a i v 0

Abasis forVis a maximal linearly independent set of vectors in V.

The dimension ofVis the number of elements in its basis.

An n-D space Vis sometimes denoted by Vn .

Given a basis { ei }, we have

1

ni i

i i

i

nv v V

vv e e

vi are called the components ofv.

Einsteins

notation

i iV span span e eA subspace ofVis a subset ofVthat is also a vector space.

Elements of vector space Vare denoted either by bold faced or over-barred letters.


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A norm on a linear space Vover field KR orC is a mapping

: byn V n v v vR s.t. , &V a u v K

3. 0n v

1. n n n v u v u

2

i

i

n x x

2. n a a nv v

4. 0n v v 0

( Triangular inequality )

( Positive semi-definite )

( Linearity )

n is a semi- (pseudo-) norm if only 1 & 2 hold.

A normed vector space is a linear space Vendowed with a norm.

Examples:

max in x x

Euclidean norm


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Inner Product Spaces

Inner product space linear space endowed with an inner product.

An inner product | induces a norm || || by

2 2 22 cos v u v u v u

1. | v u v u

2 2 2

2 cos v u v u v u

|v v v

Properties of an inner product space:

( Cauchy-Schwarz inequality )

2. v u v u ( Triangular inequality )

2 2 2 23. 2 v u v u v u ( Parallelogram rule )

The parallelogram rule can be derived from the cosine rule :

( angle between u & v )


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1.6 . The Algebra of Square Matrices

A linear transformationTon vector space (V, + ; K) is a map :T V V

T a b a T b T v u v u , & ,a b V v uKs.t.

If { ei} is a basis ofV, then

i

ixx e i iT T xx e i

ix T e

i jj iT x Tx eSetting j

i j iT Te e we have j

jT x e

j j i

iT T xx Tji= (j,i)-element of matrix T

Writing vectors as a column matrix, we have T x T x

( = matrix multiplication )


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In linear algebra, linear operators are associative, then

A B C AB Cx x A BC x

A B C x A B C x A B C x

AB A Bx x k j ik j iA B x e

k ik iAB x e

k k j

j iiAB A B AB A B~

j ij iA B x e

AB x

i.e., linear associative operators can be represented by matrices.

~

Similarly,

Well henceforth drop the symbol


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In general: AB BA

Transpose: i

T j

ijAA

Adjoint: *i

j

ijA

A

Unit matrix: i i

jjI

Inverse: 1 1 A A AA I A is non-singularifA-1 exists.

The set of all non-singularnn matrices forms the group GL(n,K).

Determinant: 11 1

det nn

iii i nA a a

1

1

1

1 is an even permutation of 1

1 is an odd permutation of 1

0 otherwise

n

n

i i n

i i n

if i i n


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Cofactor: cof(Aij) = (-)i+j determinant of submatrix obtained

by deleting the i-th row &j-th column ofA.

deti i

j ji

A A cof A Laplace expansion: j arbitrary

1det

ji i

j

cof A

A

A deti i

j k j ki

A cof A A

See T.M.Apostol, Linear Algebra , Chap 5, for proof.

Trace:i

iTr AA

Similarity transform ofA by non-singularB:1

A B AB 1A B AB~

Det & Tr are invariant under a similarity transform:

1det det B AB A 1Tr Tr B AB A


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Miscellaneous formulae

T T TAB B A

1 1 1 AB B A

det detT A A det det detAB A B

is an eigenvalue ofA ifv0 s.t. A v v A v v~

For an n-D space, satifies the secular equation: det 0 A I

v is then called the eigenvectorbelonging to .

There are always n complex eigenvalues and m eigenvectors with mn.

det ii

A ii

Tr A

Eigenvalues ofA & AT are the same.

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1._SomeBasicMathematics