Đang tải... (xem toàn văn)
Đây là bộ sách tiếng anh về chuyên ngành vật lý gồm các lý thuyết căn bản và lý liên quan đến công nghệ nano ,công nghệ vật liệu ,công nghệ vi điện tử,vật lý bán dẫn. Bộ sách này thích hợp cho những ai đam mê theo đuổi ngành vật lý và muốn tìm hiểu thế giới vũ trụ và hoạt độn ra sao.
[...]... the complex numbers, convergence of a sequence in a metric space can be defined in terms of the tails of the sequence being trapped by the balls surrounding a point That is, z + z means , for all E > 0, there exists no such that Tn0 Bc(z) ,( 1.2.9) c where B,(z) := { y E X I d(z,y) < E } Comments 1 Any given sequence of points in a metric space (X,d) may not converge at all but, if it does, it converges... topology-that satisfy certain axioms (they are listed in Theorem 1.3) This special collection of subsets is then used to give a purely set-theoretic notion of characteristic topological ideas such as ‘nearness , ‘convergence of a sequence , ‘continuity of a function’ etc From a physical perspective, one could say that topology is concerned with the relation between points and ‘regions’: in particular, open sets... a matter of considerable significance that many of these are metric spaces For example, let C ( [ ab],R)denote the space of all , real-valued, continuous functions on the closed interval [a,b] := { T E R 1 a 5 T 5 b } A metric can be defined on C( [a,b],R)by Another metric is (1.2.22) a sketched in Figure 1.2 This is inequivalent t o the metric in Eq s (1.2.21) Another inequivalent metric is (1.2.23)... This will be discussed in detail later 4 The analogous properties for closed sets are: 0 the intersection of an arbitraq collection of closed sets is closed; CHAPTER 1 A N INTRODUCTION T O TOPOLOGY 14 0 the union of any finite collection of closed sets is closed; 0 the empty set 8 and X itself are closed 5 The topology associated with a metric space is determined equally by either the collection of all... theory, and Yang-Mills theory Evidently, no excuse is needed for teaching a course on differential geometry to postgraduate students of theoretical physics However, the impression of the subject gained from, say, an undergraduate course in general relativity can be rather misleading when viewed from the perspective of modern mathematics Such courses usually employ a very coordinate-based approach to... involving Jacobian transformations is really only valid on the intersection of the domains of the coordinate systems concerned What is neglected in such approaches to differential geometry is the fact that the topology of a manifold may be different from that of a vector space, and hence-in particular-it cannot be covered by a single coordinate system The modern approach to differential geometry is... manifold However, this is far from being the only use of differential geometry in physics For example, the Hamiltonian and Lagrangian approaches to classical mechanics are best described in this way; and the use of differential geometry in quantum field theory has increased steadily in recent decades -for example, in canonical quantum gravity, superstring theory, the non-linear a-model, topological quantum... set X may be combined t o form a new metric Some specific examples of such operations are as follows 1 If di, i = 1 , 2 , , n is a finite set of metrics on X then u 2, , a} is any set of real numbers, , defines a metric on X if {al, each of which is greater than or equal t o zero and such that at least one of them is non-zero 2 If dl and d2 are a pair of metrics on X, a new metric, called the join... be a metric space Sometimes this term is applied to the pair ( X , d ) if it is appropriate to make a reference to the specific metric function, d , involved 2 If Eq (1.2.7) is replaced by the weaker condition “d(z,y) 2 0, with d ( z , z ) = 0 for all z E X ” (ie ., there may be z # y such that d ( z , y ) = 0) then X is said to be a pseudo-metric space, and the function d is a pseudo-metric 3 As in... 2 , can be defined by, for all s, y E X , 1.2 METRIC SPACES 11 One might expect t o be able to use this pair of metrics to define another metric as min(dl(z,y),d a ( z , y ) ) but, however, this fails t o satisfy the triangle inequality Eq (1.2.8) This can be remedied by defining instead the meet of dl and d2 to be, for all x,y E X , where the infinurn is taken over all finite subsets {x = x 1, z , . : Modern Differential Geometry for Physicists (2nd edn.) C J lsham World Scientific Lecture Notes in Physics - Vol. 61 Modern Differential Geometry. Geometry for Physicists Second Edition Chris J lsham Theoretical Phvsics Grow Imperial College of Science, fechnolog y and Medicine UK World Scientific