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Prolog

Website: http://ckw.phys.ncku.edu.tw

Homework submission: [email protected]

- Algebra / Analysis vs Geometry
- Relativity → Riemannian Geometry
- Symmetry → Lie Derivatives → Lie Group → Lie Algebra
- Integration → Differential forms → Homotopy, Cohomology
- Tensor / Gauge Fields → Fibre Bundles
- Topology

- Hamiltonian dynamics
- Electrodynamics
- Thermodynamics

- Statistics
- Fluid Dynamics
- Defects

Supplementary

B.F.Schutz,

“Geometrical Methods of Mathematical Physics”, CUP (80)

- 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)

Geometrical Methods of Mathematical Physics

Bernard F. Schutz,

Cambridge University Press (80)

- Some Basic Mathematics
- Differentiable Manifolds And Tensors
- Lie Derivatives And Lie Groups
- Differential Forms
- Applications In Physics
- Connections For Riemannian Manifolds And Gauge Theories

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.

Basic Algebraic Structures

See Choquet, Chap 1 or Aldrovandi, Math.1.

- Group ( G, )
- Ring ( R, +, ) : ( no e, x -1 )
- Field ( F, +, ) : Ring with e & x-1 except for 0.
- Module ( M, +, ; R )
- Algebra ( A, +, ; R with e )
- Vector space ( V, + ; F )

Prototypes:

Ris a field.

Rn is a vector space.

1.1. The Space Rn And Its Topology

- Goal: Extend multi-variable calculus (on En) to curved spaces without metric.
- Bonus: vector calculus on E3 in curvilinear coordinates

- Basic calculus concepts & tools (metric built-in):
- Limit, continuity, differentiability, …
- r-ball neighborhood, δ-ε formulism, …

- Essential concept in the absence of metric:
Proximity → Topology.

Real number R = complete Archimedian ordered field.

= Set of all n-tuples of real numbers

~ Prototype of an n-D continuum

Distance function (Euclidean metric):

(Open) Neighborhood of radius r at x :

A set S is open if

A set S is discrete if

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

Hausdorff separated: Distinct points possess disjoint neighborhoods.

A system U of subsets Uiof a set X defines a topology on X if

( Closure under arbitrary unions. )

( Closure under finite intersections. )

Elements Uiof U are called open sets.

A topological space is the minimal structure on which concepts of

neighborhood, continuity, compactness, connectedness

can be defined.

Usual topology of Rn= Topologywith open balls as open sets

Metric-free version: Define Nr(x) in terms of open intervals.

Trivial topology: U= { , X }

→ every function on X is dis-continuous

Discrete topology: U= 2X

→ every function on X is 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

1.2. Mappings

Map f from set Xinto set Y, denoted,

by

associates each xXuniquely with y= f (x) Y.

Domain of f =

Range of f =

Image of M under f =

Inverse image of N under f =

f1 exists iff f is 1-1 (injective):

f is onto (surjective) if f (X) = Y.f is a bijection if it is 1-1 onto.

Continuity

Elementary calculus version:

Let f : R → R. Then f is continuous at x0 if

Open ball version: Let

Then f is continuous at x0 if

i.e.,

- Two possible interpretations in terms of open sets:
- Every open set in Domain( f) is mapped into an open set in Range(f).
- Every open set in Range( f) has an open inverse image.

Every open set in Domain( f) is mapped into an open set in Range(f)? No!

Open M → half-closed f(M)

Every open set in Range( f) has an open inverse image? Yes.

Open N→ half-closed f -1(N)

f : X → Y is continuous at x if the inverse image of any open neighborhood of f (x) is open, i.e., f -1(N[f(x)]) is open.

Continuity in a region:

f is continuouson M Xif f is continuous xM, i.e.,

the inverse image of every open set in M is open.

Differentiability of f : Rn → R

Inverse function theorem :

f is invertible in some neighborhood of x0 if

( Jacobian )

Let

then

where

1.3. Real Analysis

is analytic at x0 if f (x) has a Taylor series at x0

if f is analytic over Domain( f)

is square integrable on S Rn if

exists.

A square integrable function g can be approximated by an analytic function f s.t.

An operator on functions defined on Rnmaps functions to functions.

E.g.,

Commutator of operators:

s.t.

A & Bcommute if

E.g.,

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

1.4. Group Theory

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

1. Associative:

2. Endowed with an identity element:

3. Endowed with an inverse for each element:

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

( Identity is denoted by 0 )

Examples:

(R,+) is an Abelian group.

The set of all permutations of n objects form the permutation group Sn.

All symmetries / transformations are members of some groups.

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

~ continuous symmetries.

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

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

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

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

It is an isomorphism if f is 1-1 onto.

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

1.5. Linear Algebra

- ( R, , +) is a ring if
- ( R, + ) is an Abelian group.
- 2. is associative & distributive wrt + , i.e., x,y,z R,

E.g., The set of all nn matrices is a ring under matrix multiplication & addition.

Ring ( R, , +) is a field if

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

2. x-1 R s.t. x-1 x = x x-1 = e xR except 0.

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

- ( V, + ; R ) is a module if
- ( V, + ) is an Abelian group.
- 2. R is a ring.
- 3. The scalar multiplicationRV→V by (a,v) av satisfies

4. If R has an identity e, then ev = v vV.

We’ll only use F = K = R or C.

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

( A, , + ; R ) is an algebraover ring Rif

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

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

For historical reasons, the term “linear algebra” denotes the study of linear simultaneous equations, matrix algebra, & vector spaces.

Mathematical justification:

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

Linear combination:

{ vi} is linearly independent if

A basis for V is a maximal linearly independent set of vectors in V.

The dimension of V is the number of elements in its basis.

An n-D space V is sometimes denoted by V n .

Einstein’s notation

Given a basis { ei }, we have

vi are called the components of v.

A subspace of V is a subset of V that is also a vector space.

A norm on a linear space V over field KR or C is a mapping

s.t.

( Triangular inequality )

( Linear )

( Positive semi-definite )

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

A normed vector space is a linear space V endowed with a norm.

Examples:

Euclidean norm

An inner product on a linear space (V, + ; K) is a mapping

s.t.

or, for physicists,

( Sesquilinear )

u & v are orthogonal

Inner Product Spaces

Inner product space linear space endowed with an inner product.

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

Properties of an inner product space:

( Cauchy-Schwarz inequality )

( Triangular inequality )

( Parallelogram rule )

The parallelogram rule can be derived from the cosine rule :

( θ angle between u & v )

1.6 . The Algebra of Square Matrices

A linear transformationT on vector space (V, + ; K) is a map

s.t.

If { ei} is a basis of V, then

Setting

we have

T ji= (j,i)-element of matrix T

Writing vectors as a column matrix, we have

( =matrix multiplication )

In linear algebra, linear operators are associative, then

~

Similarly,

~

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

We’ll henceforth drop the symbol

Transpose:

Adjoint:

Unit matrix:

Inverse:

A is non-singular if A-1 exists.

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

Determinant:

Cofactor: cof(Aij) = (-)i+j determinant of submatrix obtained by deleting the i-th row & j-th column of A.

Laplace expansion:

j arbitrary

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

Trace:

Similarity transform of A by non-singular B:

~

Det & Tr are invariant under a similarity transform:

Miscellaneous formulae

λ is an eigenvalue of A if v 0 s.t.

~

v is then called the eigenvector belonging to λ.

For an n-D space, λ satifies the secular equation:

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

Eigenvalues of A & AT are the same.

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