Prolog Website Homework submission http ckw phys ncku
Prolog Website: Homework submission: • • • http: //ckw. phys. ncku. edu. tw 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 • Statistics • Electrodynamics • Fluid Dynamics • Thermodynamics • Defects
Main Textbook B. F. Schutz, “Geometrical Methods of Mathematical Physics”, CUP (80) 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”, 2 nd ed. , CUP (03)
Geometrical Methods of Mathematical Physics Bernard F. Schutz, Cambridge University Press (80) 1. 2. 3. 4. 5. 6. 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 § 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: R is 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 E 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 U of subsets Ui of a set X defines a topology on X if ( Closure under arbitrary unions. ) ( Closure under finite intersections. ) Elements Ui of U are called open sets. A topological space is the minimal structure on which concepts of neighborhood, continuity, compactness, connectedness can be defined.
Trivial topology: U = { , X } → every function on X is dis-continuous Discrete topology: U = 2 X → 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, …
Real number R = 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
Usual topology of Rn = Topology with 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.
1. 2. Mappings Map f from set X into set Y, denoted, by associates each x X uniquely 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.
Composition Given by by The composition of f & g is the map by
Continuity Elementary calculus version: Let f : R → R. Then f is continuous at x 0 if Open ball version: Let Then f is continuous at x 0 if i. 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 ). f is continuous if every open set in domain (f ) is mapped to an open set in codomain (f ) ? Counter-example: f continuous but Open M → half-closed f(M) 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( N[f(x)] ) is open. Continuity in a region: f is continuous on M X if 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.
Let by Inverse function theorem : f is invertible in some neighborhood of x 0 if ( Jacobian ) Let then where
1. 3. Real Analysis is analytic at x 0 if f (x) has a Taylor series at x 0 if f is analytic over Domain( f) is square integrable on S Rn if A square integrable function g can be approximated by an analytic function f s. t. exists.
An operator on functions defined on Rn maps functions to functions. E. g. , Commutator of operators: s. t. A & B commute if E. g. , Domain (AB) C 2 but Domain ([A , B ]) C 1
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 group Sn. All symmetries / transformations are members of some groups.
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
1. 5. Linear Algebra See Choquet, Chap 1 or Aldrovandi, Math. 1. ( R, , + ) is a ring if 1. ( 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.
( V, + ; R ) is a module if 1. ( V, + ) is an Abelian group. 2. R is a ring. 3. The scalar multiplication R V→V by (a, v) a v satisfies 4. If R has an identity e, then ev = v v V. Module ( V, + ; F ) is a linear (vector) space if F is a field. ( A, , + ; R ) is an algebra over ring R if 1. ( A, , + ) is a ring. 2. ( A, + ; R ) is a module s. t. We’ll only use F = K = R or C. 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, , + ; 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. 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. Einstein’s notation
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
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
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 transformation T 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
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:
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 always n complex eigenvalues and m eigenvectors with m n. Eigenvalues of A & AT are the same.
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