دانلود کتاب Lectures on advanced mathematical methods for physicists

The two-volume textbook Comprehensive Mathematics for the Working Computer Scientist, of which this is the second volume, is a self-contained comprehensive presentation of mathematics including sets, numbers, graphs, algebra, logic, grammars, machines, linear geometry, calculus, ODEs, and special themes such as neural networks, Fourier theory, wavelets, numerical issues, statistics, categories, and manifolds. The concept framework is streamlined but defining and proving virtually everything. The style implicitly follows the spirit of recent topos-oriented theoretical computer science. Despite the theoretical soundness, the material stresses a large number of core computer science subjects, such as, for example, a discussion of floating point arithmetic, Backus-Naur normal forms, L-systems, Chomsky hierarchies, algorithms for data encoding, e.g., the Reed-Solomon code. The numerous course examples are motivated by computer science and bear a generic scientific meaning. This text is complemented by an online university course which covers the same theoretical content, albeit in a totally different presentation. The student or working scientist who gets involved in this text may at any time consult the online interface which comprises applets and other interactive tools This book presents a survey of Topology and Differential Geometry and also, Lie Groups and Algebras, and their Representations. The first topic is indispensable to students of gravitation and related areas of modern physics (including string theory), while the second has applications in gauge theory and particle physics, integrable systems and nuclear physics. Part I provides a simple introduction to basic topology, followed by a survey of homotopy. Calculus of differentiable manifolds is then developed, and a Riemannian metric is introduced along with the key concepts of connections and curvature. The final chapters lay out the basic notions of simplicial homology and de Rham cohomology as well as fibre bundles, particularly tangent and cotangent bundles. Part II starts with a review of group theory, followed by the basics of representation theory. A thorough description of Lie groups and algebras is presented with their structure constants and linear representations. Root systems and their classifications are detailed, and this section of the book concludes with the description of representations of simple Lie algebras, emphasizing spinor representations of orthogonal and pseudo-orthogonal groups. The style of presentation is succinct and precise. Involved mathematical proofs that are not of primary importance to physics student are omitted. The book aims to provide the reader access to a wide variety of sources in the current literature, in addition to being a textbook of advanced mathematical methods for physicists.  Read more... 1. Topology. 1.1. Preliminaries. 1.2. Topological spaces. 1.3. Metric spaces. 1.4. Basis for a topology. 1.5. Closure. 1.6. Connected and compact spaces. 1.7. Continuous functions. 1.8. Homeomorphism. 1.9. Separability -- 2. Homotopy. 2.1. Loops and homotopies. 2.2. The fundamental group. 2.3. Homotopy type and contractibility. 2.4. Higher homotopy groups -- 3. Differentiable manifolds I. 3.1. The definition of a manifold. 3.2. Differentiation of functions. 3.3. Orient ability. 3.4. Calculus on manifolds : vector and tensor fields. 3.5. Calculus on manifolds : differential forms. 3.6. Properties of differential forms. 3.7. More about vectors and forms -- 4. Differentiable manifolds II. 4.1. Riemannian geometry. 4.2. Frames. 4.3. Connections, curvature and torsion. 4.4. The volume form. 4.5. Isometry. 4.6. Integration of differential forms. 4.7. Stokes' theorem. 4.8. The Laplacian on forms -- 5. Homology and cohomology. 5.1. Simplicial homology. 5.2. De Rham cohomology. 5.3. Harmonic forms and de Rham cohomology -- 6. Fibre bundles. 6.1. The concept of a fibre bundle. 6.2. Tangent and cotangent bundles. 6.3. Vector bundles and principal bundles -- 7. Review of groups and related structures. 7.1. Definition of a group. 7.2. Conjugate elements, equivalence classes. 7.3. Subgroups and cosets. 7.4. Invariant (normal) subgroups, the factor group. 7.5. Abelian groups, commutator subgroup. 7.6. Solvable, nilpotent, semisimple and simple groups. 7.7. Relationships among groups. 7.8. Ways to combine groups -- direct and semidirect products. 7.9. Topological groups, Lie groups, compact Lie groups -- 8. Review of group representations. 8.1. Definition of a representation. 8.2. Invariant subspaces, reducibility, decomposability. 8.3. Equivalence of representations, Schur's lemma. 8.4. Unitary and orthogonal representations. 8.5. Contragredient, adjoint and complex conjugate representations. 8.6. Direct products of group representations -- 9. Lie groups and Lie algebras. 9.1. Local coordinates in a Lie group. 9.2. Analysis of associativity. 9.3. One-parameter subgroups and canonical coordinates. 9.4. Integrability conditions and structure constants. 9.5. Definition of a (real) Lie algebra : Lie algebra of a given Lie group. 9.6. Local reconstruction of Lie group from Lie algebra. 9.7. Comments on the G[symbol])[symbol] relationship. 9.8. Various kinds of and operations with Lie algebras -- 10. Linear representations of Lie algebras -- 11. Complexification and classification of Lie algebras. 11.1. Complexification of a real Lie algebra. 11.2. Solvability, Levi's theorem, and Cartan's analysis of complex (semi) simple Lie algebras. 11.3. The real compact simple Lie algebras -- 12. Geometry of roots for compact simple Lie algebras -- 13. Positive roots, simple roots, Dynkin diagrams. 13.1. Positive roots. 13.2. Simple roots and their properties. 13.3. Dynkin diagrams -- 14. Lie algebras and Dynkin diagrams for SO(2l), SO(2l+1), USp(2l), SU(l+1). 14.1. The SO(2l) family -- D[symbol] of Cartan. 14.2. The SO(2l+1) family -- B[symbol] of Cartan. 14.3. The USp(2l) family -- C[symbol] of Cartan. 14.4. The SU(l+1) family -- A[symbol] of Cartan. 14.5. Coincidences for low dimensions and connectedness -- 15. Complete classification of all CSLA simple root systems. 15.1. Series of lemmas. 15.2. The allowed graphs [symbol]. 15.3. The exceptional groups -- 16. Representations of compact simple Lie algebras. 16.1. Weights and multiplicities. 16.2. Actions of E[symbol] and SU(2)[symbol] -- the Weyl group. 16.3. Dominant weights, highest weight of a UIR. 16.4. Fundamental UIR's, survey of all UIR's. 16.5. Fundamental UIR's for A[symbol], B[symbol], C[symbol], D[symbol]. 16.6. The elementary UIR's. 16.7. Structure of states within a UIR -- 17. Spinor representations for real orthogonal groups. 17.1. The Dirac algebra in even dimensions. 17.2. Generators, weights and reducibility of U(S) -- the spinor UIR's of D[symbol]. 17.3. Conjugation properties of spinor UIR's of D[symbol]. 17.4. Remarks on antisymmetric tensors under D[symbol] = SO(2l). 17.5. The spinor UIR's of B[symbol] = SO(2l+[symbol]). 17.6. Antisymmetric tensors under B[symbol] = SO(2l+1) -- 18. Spinor representations for real pseudo orthogonal groups. 18.1. Definition of SO(q, p) and notational matters. 18.2. Spinor representations S([symbol]) of SO(p, q) for p + q [symbol] 2l. 18.3. Representations related to S([symbol]). 18.4. Behaviour of the irreducible spinor representations S[symbol]. 18.5. Spinor representations of SO(p, q) for p+q = 2l+1. 18.6. Dirac, Weyl and Majorana spinors for SO(p, q)