Prolog

• 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

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

Homework submission: class@ckw.phys.ncku.edu.tw

• Hamiltonian dynamics

• Electrodynamics

• Thermodynamics

• Statistics

• Fluid Dynamics

• Defects

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

Geometrical Methods of Mathematical Physics

1. Some Basic Mathematics

2. Differentiable Manifolds And Tensors

3. Lie Derivatives And Lie Groups

4. Differential Forms

5. Applications In Physics

6. Connections For Riemannian Manifolds And Gauge Theories

Bernard F. Schutz,

Cambridge University Press (80)

1. Some Basic Mathematics

1.1 The Space n 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 §1.5 for details.

Structures with only internal operations:

• Group ( G, )

• Ring ( R, +, ) : ( no e, or x1 )

• 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:

is a field.

n is a vector space.

1.1. The Space n And Its Topology

• Goal: Extend multi-variable calculus (on n) to curved spaces without metric.– Bonus: vector calculus on 3 in curvilinear coordinates

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

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

A system of subsets Ui of a set X defines a topology on X if

1. , X U

1

2. ii

U

U

1

3.N

ii

U

U

( Closure under arbitrary unions. )

( Closure under finite intersections. )

Elements Ui of are called open sets.

A topological space is the minimal structure on which concepts of

neighborhood, continuity, compactness, connectedness

can be defined.

Trivial topology: = { , X }

→ every function on X is dis-continuous

Discrete topology: = 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, …

= 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 nd R R R

,r d r x y y xN

2, , i i

i

d x y x y x y

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

Real number = complete Archimedian ordered field.

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

Usual topology of n = Topology with open balls as open sets

Metric-free version: Define neighborhoods r(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., n is Hausdorff separated.

1.2. Mappings

:f X Y by x y f x

Map f from set X into set Y, denoted,

associates each xX uniquely with y = f (x) Y.

Domain f XDomain of f =

Range f YRange of f =

Image of M under f = f M f x x M

Inverse image of N under f = 1f N x f x N

f 1 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.

f x f x x x

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 of f & g is the map

g f x g y

Elementary calculus version:

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

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

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

Then f is continuous at x0 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 Ni.e.,

Open set version:

f continuous: Open set in domain (f ) is mapped to open set in codomain (f ).

f discontinuous: Open set in domain (f ) is mapped to set not open in codomain (f ).

Counter-example:

f continuous but

Open M → half-closed f(M)

f is continuous if every open set in domain (f ) is mapped to an open set in codomain (f ) ?

Wrong!

Correct criterion:

f is continuous if every open set in codomain( f) has an open inverse image.

Open N → half-closed f 1(N)

Continuity at a point:

f : X → Y is continuous at x if the inverse image of any open neighborhood of f (x) is open,

i.e., f 1( [f(x)] ) is open.

Continuity in a region:

f is continuous on M X if f is continuous xM,

i.e., the inverse image of every open set in M is open.

Differentiability of f : n →

1, ,exists & is continuousif 1, ,

k n

kkj

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.

f is smooth → k = whatever value necessary for problem at hand.

: 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 :

f is invertible in some neighborhood of x0 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

1.3. Real Analysis

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

f C if f is analytic over Domain( f)

0

00

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

: ng R R is square integrable on S n if 2 n

S

g d x x exists.

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

2 n

S

f g d x x x for any given 0

An operator on functions defined on n maps functions to functions.

E.g., D ff

x

0

,x

G f x f y g x y dy

3

23

ff f

xE

Commutator of operators: ,A B AB BA

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

A & B commute if , 0A B

E.g., , ,d d

A B f x fdx dx

d d f d d fx x

dx dx dx dx

d f

dx

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

, ,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:

(,+) 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.

It’s common practice to refer to group (G, ) simply as group G.

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

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 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.

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

It is an isomorphism if f is 1-1 onto.(+,) & (,+) are isomorphic with f = log so that

log log log ,x y x y x y R

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. x1 R 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,z R,

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., & are fields under algebraic multiplication & addition.

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

( V, + ; R ) is a module if

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

2. R is a ring.

3. The scalar multiplication RV→V by (a,v) a v satisfies

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

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

a b a b v v v

ab a bv v

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

We’ll only use F = = or .

( A, , + ; R ) is an algebra over ring R if

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 simultaneous equations, matrix algebra, & vector spaces.

Mathematical justification:

( M, , + ; ) , where M is 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

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 .

Given a basis { ei }, we have1

ni i

i ii

nv v V

vv e e

vi are called the components of v.

Einstein’s notation

i iV span span e e

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

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

A norm on a linear space V over field or 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

2i

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 V endowed with a norm.

Examples:

max in x x

Euclidean norm

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

| 0v u

1. | | | v u w v w u w

, , &V a u v w K

| : V V K

s.t.

|. |2 a av u v u

*3. | |v u u v

4. | 0v v

or, for physicists,

| |a av u v u

5. | 0 v v v 0

u & v are orthogonal

Sometimes this is called a sesquilinear product and the term inner product is reserved for the case v | u = u | v .

, | v u v u v uby

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 22 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 :

( θ angle between u & v )

1.6 . The Algebra of Square Matrices

A linear transformation T on vector space (V, + ; ) 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 of V, then

iixx e i

iT T xx e iix T e

i jj iT x Tx eSetting j

i j iT Te e we have j

jT x e

j j iiT T xx T j

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

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

( · = matrix multiplication )

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 xe

k ik i

AB xe

k k jj ii

AB A B AB A B~

j ij iA B x e

AB x

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

~

Similarly,

We’ll henceforth drop the symbol

In general: AB BA

Transpose: iT jij

AA

Adjoint: *i jij

A A

Unit matrix:

i ijj

I

Inverse: 1 1 A A AA I A is non-singular if A-1 exists.

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

Determinant:

1

1 1det n

n

iii i nA a a

1

1

1

1 is an even permutation of 1

1 is an odd permutation of 1

0 otherwisen

n

i i n

i i n

if i i n

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

obtained by deleting the i-th row & j-th column of A.

det i ij j

i

A A cof ALaplace expansion: j arbitrary

1

det

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: iiTr AA

Similarity transform of A by non-singular B: 1A B AB 1A B AB~

Det & Tr are invariant under a similarity transform:

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

Miscellaneous formulae

T T TAB B A 1 1 1 AB B A

det detT A A det det detAB A B

λ is an eigenvalue of A if v 0 s.t.

A v v A v v~

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

det 0 A I

v is then called the eigenvector belonging to λ.

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

det ii

A ii

Tr A

Eigenvalues of A & AT are the same.