Prolog

1. Prolog Website: http://ckw.phys.ncku.edu.tw Homework submission: class@ckw.phys.ncku.edu.tw • 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

2. Main Textbook 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)

3. 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

4. 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.

5. Basic Algebraic Structures See §1.5 for details. Structures with only internal operations: • Group ( G,  ) • Ring ( R, +,  ) : ( no e, or x1 ) • Field ( F, +,  ) : Ring with e & x1 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.

6. 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, … • Integration, … • Essential concept in the absence of metric: Proximity → Topology.

7. 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.

8. 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, …

9. Real numberR = complete Archimedian ordered field. = Set of all ordered n-tuples of real numbers ~ Prototype of an n-D continuum Distance function (Euclidean metric): (Open) Neighborhood / ball of radius r at x : A set S is open if A set S is discrete if

10. Usual topology of Rn= Topologywith open balls as open sets Metric-free version: Define neighborhoods Nr(x) in terms of open intervals / cubes. Hausdorff separated: Distinct points possess disjoint neighborhoods. E.g., Rn is Hausdorff separated. Preview: Continuity of functions will be defined in terms of open sets.

11. 1.2. Mappings Map f from set Xinto set Y, denoted, by associates each xXuniquely with y= f (x) Y. Domain of f = Range of f = Image of M under f = Inverse image of N under f = 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.

12. Composition Given by by The composition of f & g is the map by

13. 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.,

14. 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 ). f is continuous if every open set in domain (f ) is mapped to an open set in codomain (f ) ? Wrong! Counter-example: f continuous but Open M → half-closed f(M)

15. Correct criterion: f is continuous ifevery open set in codomain( f) has an open inverse image. Open N→ half-closed f 1(N)

16. 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( N[f(x)] ) is open. Continuity in a region: f is continuouson M Xif f is continuous  xM, i.e., the inverse image of every open set in M is open. Differentiability of f : Rn → R f is smooth → k = whatever value necessary for problem at hand. i.e., Taylor expansion exists.

17. by Let Inverse function theorem : f is invertible in some neighborhood of x0 if ( Jacobian ) Let then where

18. 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.

19. 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

20. 1.4. Group Theory A group (G,  ) is a set G with an internal operation  : GG → G that is 1. Associative: 2. Endowed with an identity element: 3. Endowed with an inverse for each element: It’s common practice to refer to group (G,  ) simply as group G. 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 groupSn. All symmetries / transformations are members of some groups.

21. Rough definition: 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

22. See Choquet, Chap 1 or Aldrovandi, Math.1. 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 nn matrices is a ring (no inverse). The function space is also a ring (no inverse). Ring ( R,  , +) is a field if 1.  eR s.t. ex = xe = x  xR. 2.  x1 R s.t. x1  x = x  x1 = e  xR except 0. E.g., R & C are fields under algebraic multiplication & addition.

23. ( V, + ; R ) is a module if • ( V, + ) is an Abelian group. • 2. R is a ring. • 3. The scalar multiplicationRV→V by (a,v)  av satisfies 4. If R has an identity e, then ev = v vV. 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. Examples will be given in Chap 3

24. 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 nn matrices, is an algebra . Elements of vector space V are denoted either by bold faced or over-barred letters. 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.

25. A norm on a linear space V over field KR or C is a mapping s.t. ( Triangular inequality ) ( Linearity ) ( 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

26. An inner product on a linear space (V, + ; K) is a mapping by s.t. or, for physicists, Sometimes this is called a sesquilinear product and the term inner product is reserved for the case  v | u  =  u | v . u & v are orthogonal

27. 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 )

28. 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 )

29. 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 

30. In general: Transpose: Adjoint: Unit matrix: Inverse: A is non-singular if A-1 exists. The set of all non-singular nn matrices forms the group GL(n,K). Determinant:

31. 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:

32. 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.