Lecture Notes on Undergraduate Math Kevin Zhou [email protected] These notes are a review of the basic undergraduate math curriculum, focusing on the content most relevant for physics. The primary sources were: • Oxford's Mathematics lecture notes, particularly notes on M2 Analysis, M1 Groups, A2 Metric Spaces, A3 Rings and Modules, A5 Topology, and ASO Groups. The notes by Richard Earl are particularly clear and written in a modular form. • Rudin, Principles of Mathematical Analysis. The canonical introduction to real analysis; terse but complete. Presents many results in the general setting of metric spaces rather than R. • Ablowitz and Fokas, Complex Variables. Quickly covers the core material of complex analysis, then introduces many practical tools; indispensable for an applied mathematician. • Artin, Algebra. A good general algebra textbook that interweaves linear algebra and focuses on nontrivial, concrete examples such as crystallography and quadratic number fields. • David Skinner's lecture notes on Methods. Provides a general undergraduate introduction to mathematical methods in physics, a bit more careful with mathematical details than typical. • Munkres, Topology. A clear, if somewhat dry introduction to point-set topology. Also includes a bit of algebraic topology, focusing on the fundamental group. • Renteln, Manifolds, Tensors, and Forms. A textbook on differential geometry and algebraic topology for physicists. Very clean and terse, with many good exercises. Some sections are quite brief, and are intended as a telegraphic review of results rather than a full exposition. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Metric Spaces 4 1.1 Definitions..........................................4 1.2 Compactness........................................6 1.3 Sequences..........................................8 1.4 Series............................................ 10 2 Real Analysis 14 2.1 Continuity.......................................... 14 2.2 Differentiation....................................... 17 2.3 Integration......................................... 19 2.4 Properties of the Integral................................. 21 2.5 Uniform Convergence................................... 25 3 Complex Analysis 28 3.1 Analytic Functions..................................... 28 3.2 Multivalued Functions................................... 30 3.3 Contour Integration.................................... 32 3.4 Laurent Series........................................ 36 3.5 Application to Real Integrals............................... 40 3.6 Conformal Transformations................................ 42 3.7 Additional Topics...................................... 45 4 Linear Algebra 48 4.1 Exact Sequences...................................... 48 4.2 The Dual Space....................................... 50 4.3 Determinants........................................ 51 4.4 Endomorphisms....................................... 52 5 Groups 56 5.1 Fundamentals........................................ 56 5.2 Group Homomorphisms.................................. 60 5.3 Group Actions....................................... 63 5.4 Composition Series..................................... 66 5.5 Semidirect Products.................................... 69 6 Rings 72 6.1 Fundamentals........................................ 72 6.2 Quotient Rings and Field Extensions........................... 73 6.3 Factorization........................................ 73 6.4 Modules........................................... 73 6.5 The Structure Theorem.................................. 73 7 Point-Set Topology 74 7.1 Definitions.......................................... 74 7.2 Closed Sets and Limit Points............................... 77 7.3 Continuous Functions................................... 78 3 Contents 7.4 The Product Topology................................... 79 7.5 The Metric Topology.................................... 80 8 Algebraic Topology 82 8.1 Constructing Spaces.................................... 82 8.2 The Fundamental Group.................................. 82 8.3 Group Presentations.................................... 82 8.4 Covering Spaces...................................... 82 9 Methods for ODEs 83 9.1 Differential Equations................................... 83 9.2 Eigenfunction Methods................................... 85 9.3 Distributions........................................ 90 9.4 Green's Functions..................................... 92 9.5 Variational Principles................................... 94 10 Methods for PDEs 98 10.1 Separation of Variables................................... 98 10.2 The Fourier Transform................................... 102 10.3 The Method of Characteristics.............................. 107 10.4 Green's Functions for PDEs................................ 110 11 Approximation Methods 114 11.1 Asymptotic Series..................................... 114 11.2 Asymptotic Evaluation of Integrals............................ 118 11.3 Matched Asymptotics................................... 124 11.4 Multiple Scales....................................... 124 11.5 WKB Theory........................................ 124 4 1. Metric Spaces 1 Metric Spaces 1.1 Definitions We begin with some basic definitions. Throughout, we let E be a subset of a fixed set X. • A set X is a metric space if it is has a distance function d(p; q) which is positive definite (except for d(p; p) = 0), symmetric, and satisfies the triangle inequality. • A neighborhood of p is the set Nr(p) of all q with d(p; q) < r for some radius r > 0. Others define a neighborhood as any set that contains one of these neighborhoods, which are instead called \the open ball of radius r about p". This is equivalent for proofs; the important part is that neighborhoods always contain points \arbitrarily close" to p. • A point p is a limit point of E if every neighborhood of p contains a point q 6= p in E. If p is not a limit point but is in E, then p is an isolated point. • E is closed if every limit point of E is in E. Intuitively, this means E \contains all its edges". The closure E of E is the union of E and the set of its limit points. • A point p is an interior point of E if there is a neighborhood N of p such that N ⊂ E. Note that interior points must be in E itself, while limit points need not be. • E is open if every point of E is an interior point of E. Intuitively, E \doesn't have edges". • E is bounded if there exists M and q so that d(p; q) < M for all p 2 E. • E is dense in X if every point of X is a limit point of E or a point of E, or both. • The interior E0 of E is the set of all interior points of E, or equivalently the union of all open sets contained in E. Example. We give some simple examples in R with the usual metric. • Finite subsets of R cannot have any limit points or interior points, so they are trivially closed and not open. • The set (0; 1]. The limit points are [0; 1], so the set is not closed. The interior points are (0; 1), so the set is not open. • The set of points 1=n for n 2 Z. The single limit point is 0, so the set is not closed. • All points. This set is trivially open and closed. • The interval [1; 2] in the restricted space [1; 2] [ [3; 4]. This is both open and closed. Generally, this happens when a set contains \all of a connected component". As seen from the last example above, whether a set is closed or open depends on the space, so if we wanted to be precise, we would say \closed in X" rather than just \closed". Example. There are many examples of metrics besides the usual one. 5 1. Metric Spaces • For any set S, we may define the discrete metric ( 0 x = y; d(x; y) = 1 x 6= y: Note that in this case, the closed ball of radius 1 about p is not the closure of the open ball of radius 1 about p. • A metric on a vector space can be defined from an inner product, which can in turn be defined from a norm. (However, a norm does not necessarily give a valid inner product.) For example, for continuous functions f :[a; b] ! R we have the inner product Z b hf; gi = f(t)g(t) dt a which gives the norm kfk = phf; fi and the metric s Z b 2 d2(f; g) = kf − gk = (f(t) − g(t)) dt: a • Alternatively, we could use the metric d1(f; g) = sup jf(x) − g(x)j: x2[a;b] These are both special cases of a range of metrics. We now consider some fundamental properties of open and closed sets. • E is open if and only if its complement Ec is closed. Heuristically, this proof works because open and closed are `for all' and `there exists' properties, and taking the complement swaps them. Specifically, if q is an interior point of E, then E contains all points arbitrarily close to q. But if q is a limit point of Ec, there exist points arbitrarily close to q that are in Ec. Only one of these can be true, giving the result. • Arbitrary unions of open sets are open, because interior points stay interior points when we add more points. By taking the complement, arbitrary intersections of closed sets are closed. • Finite intersections of open sets are open, because we can take intersections of the relevant neighborhoods. This breaks down for infinite intersections because the neighborhoods can shrink down to nothing, e.g. let En = (−1=n; 1=n)). By taking the complement, finite unions of closed sets are closed. Infinite unions don't work because they can create new limit points. Prop. The closure E is the smallest closed set containing E. Proof. The idea behind the proof of closure is that all limit points of E must be