Introduction to Mathematical Physics 1st Edition by Michael T. Vaughn (PDF)

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A comprehensive survey of all the mathematical methods that should be available to graduate students in physics. In addition to the usual topics of analysis, such as infinite series, functions of a complex variable and some differential equations as well as linear vector spaces, this book includes a more extensive discussion of group theory than can be found in other current textbooks. The main feature of this textbook is its extensive treatment of geometrical methods as applied to physics. With its introduction of differentiable manifolds and a discussion of vectors and forms on such manifolds as part of a first-year graduate course in mathematical methods, the text allows students to grasp at an early stage the contemporary literature on dynamical systems, solitons and related topological solutions to field equations, gauge theories, gravitational theory, and even string theory. Free solutions manual available for lecturers at www.wiley-vch.de/supplements/.

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Editorial Reviews: Review “This book can serve not only as a useful reference but also as an excellent text … .Well written and highly recommended.” (Zentralblatt MATH, 1132, 2008) “…an excellent resource for practicing professionals as well as researchers and faculty…” (CHOICE, December 2007) From the Inside Flap This book is intended to be a survey of mathematical methods which should be available to graduate students in physics and related areas in science and engineering at an early stage in their careers. It may also be useful for practicing professionals who need an elementary introduction to the mathematical areas covered in the text. In addition to the usual topics of analysis such as infinite series, functions of a complex variable, differential and partial differential equations, there is an extensive early introduction to the methods of differential geometry with examples cosen from classical mechanics, thermodynamics and electromagnetic theory. Some simle examples of nonlinear partial differential equations are discussed briefly. There is also a more extensive treatment of group theory than is available in current general mathematical physics textbooks. Free solutions manual available for lecturers at www.wiley-vch.de/supplements/. From the Back Cover This book is intended to be a survey of mathematical methods which should be available to graduate students in physics and related areas in science and engineering at an early stage in their careers. It may also be useful for practicing professionals who need an elementary introduction to the mathematical areas covered in the text. In addition to the usual topics of analysis such as infinite series, functions of a complex variable, differential and partial differential equations, there is an extensive early introduction to the methods of differential geometry with examples cosen from classical mechanics, thermodynamics and electromagnetic theory. Some simle examples of nonlinear partial differential equations are discussed briefly. There is also a more extensive treatment of group theory than is available in current general mathematical physics textbooks. Free solutions manual available for lecturers at www.wiley-vch.de/supplements/. About the Author Michael T. Vaughn is Professor of Physics at Northeastern University in Boston and well known in particle theory for his contributions to quantum field theory especially in the derivation of two loop renormalization group equations for the Yukowa and scalar quartic couplings in Yang-Mills gauge theories and in softly broken supersymmetric theories. Professor Vaughn has taught graduate courses in mathematical physics at the University of Pennsylvania, Indiana University and Texas A&M University as well as at Northeastern. Excerpt. © Reprinted by permission. All rights reserved. Introduction to Mathematical PhysicsBy Michael T. VaughnJohn Wiley & SonsCopyright © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimAll right reserved.ISBN: 978-3-527-40627-2Chapter OneInfinite Sequences and Series In experimental science and engineering, as well as in everyday life, we deal with integers, or at most rational numbers. Yet in theoretical analysis, we use real and complex numbers, as well as far more abstract mathematical constructs, fully expecting that this analysis will eventually provide useful models of natural phenomena. Hence we proceed through the construction of the real and complex numbers starting from the positive integers. Understanding this construction will help the reader appreciate many basic ideas of analysis. We start with the positive integers and zero, and introduce negative integers to allow subtraction of integers. Then we introduce rational numbers to permit division by integers. From arithmetic we proceed to analysis, which begins with the concept of convergence of infinite sequences of (rational) numbers, as defined here by the Cauchy criterion. Then we define irrational numbers as limits of convergent (Cauchy) sequences of rational numbers. In order to solve algebraic equations in general, we must introduce complex numbers and the representation of complex numbers as points in the complex plane. The fundamental theorem of algebra states that every polynomial has at least one root in the complex plane, from which it follows that every polynomial of degree n has exactly n roots in the complex plane when these roots are suitably counted. We leave the proof of this theorem until we study functions of a complex variable at length in Chapter 4. Once we understand convergence of infinite sequences, we can deal with infinite series of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the closely related infinite products of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Infinite series are central to the study of solutions, both exact and approximate, to the differential equations that arise in every branch of physics. Many functions that arise in physics are defined only through infinite series, and it is important to understand the convergence properties of these series, both for theoretical analysis and for approximate evaluation of the functions. We review some of the standard tests (comparison test, ratio test, root test, integral test) for convergence of infinite series, and give some illustrative examples. We note that absolute convergence of an infinite series is necessary and sufficient to allow the terms of a series to be rearranged arbitrarily without changing the sum of the series. Infinite sequences of functions have more subtle convergence properties. In addition to pointwise convergence of the sequence of values of the functions taken at a single point, there is a concept of uniform convergence on an interval of the real axis, or in a region of the complex plane. Uniform convergence guarantees that properties such as continuity and differentiability of the functions in the sequence are shared by the limit function. There is also a concept of weak convergence, defined in terms of the sequences of numbers generated by integrating each function of the sequence over a region with functions from a class of smooth functions (test functions). For example, the Dirac δ-function and its derivatives are defined in terms of weakly convergent sequences of well-behaved functions. It is a short step from sequences of functions to consider infinite series of functions, especially power series of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in which the an are real or complex numbers and z is a complex variable. These series are central to the theory of functions of a complex variable. We show that a power series converges absolutely and uniformly inside a circle in the complex plane (the circle of convergence), with convergence on the circle of convergence an issue that must be decided separately for each particular series. Even divergent series can be useful. We show some examples that illustrate the idea of a semiconvergent, or asymptotic, series. These can be used to determine the asymptotic behavior and approximate asymptotic values of a function, even though the series is actually divergent. We give a general description of the properties of such series, and explain Laplace’s method for finding an asymptotic expansion of a function defined by an integral representation (Laplace integral) of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Beyond the sequences and series generated by the mathematical functions that occur in solutions to differential equations of physics, there are sequences generated by dynamical systems themselves through the equations of motion of the system. These sequences can be viewed as iterated maps of the coordinate space of the system into itself; they arise in classical mechanics, for example, as successive intersections of a particle orbit with a fixed plane. They also arise naturally in population dynamics as a sequence of population counts at periodic intervals. The asymptotic behavior of these sequences exhibits new phenomena beyond the simple convergence or divergence familiar from previous studies. In particular, there are sequences that converge, not to a single limit, but to a periodic limit cycle, or that diverge in such a way that the points in the sequence are dense in some region in a coordinate space. An elementary prototype of such a sequence is the logistic map defined by Tλ : x -> xλ = λx(1 – x) This map generates a sequence of points {xn} with xn+1 = λxn(1 – xn) (0 <λ <4) starting from a generic point x0 in the interval 0 Exercise 1.1. Let p be a prime number. Then &radius;p is not rational. Note. Here and throughout the book we use the convention that when a proposition is simply stated, the problem is to prove it, or to give a counterexample that shows it is false. 1.1.2 Algebraic Equations The rational numbers are adequate for the usual operations of arithmetic, but to solve algebraic (polynomial) equations, or to carry out the limiting operations of calculus, we need more. For example, the quadratic equation x2 – 2 = 0 (1.8) has no rational solution, yet it makes sense to enlarge the rational number system to include the roots of this equation. The real algebraic numbers are introduced as the real roots of polynomials of any degree with integer coefficients. The algebraic numbers also form a field. -> Exercise 1.2. Show that the roots of a polynomial with rational coefficients can be expressed as roots of a polynomial with integer coefficients. Complex numbers are introduced in order to solve algebraic equations that would otherwise have no real roots. For example, the equation x2 + 1 = 0 (1.9) has no real solutions; it is “solved” by introducing the imaginary unit i [equivalent] &radius;-1 so that the roots are given by x = [+ or -] i. Complex numbers are then introduced as ordered pairs (x, y) ~ x + iy, of real numbers; x, y can be restricted to be rational (algebraic) to define the complex rational (algebraic) numbers. Complex numbers can be represented as points (x, y) in a plane (the complex plane) in a natural way, and the magnitude of the complex number x + iy is defined by |x + iy| [equivalent] &radius;x2 + y2 (1.10) In view of the identity eiθ = cosθ + i sinθ (1.11) we can also write x + iy = reiθ (1.12) with r = |x + iy| and tan θ = y/x. These relations have an obvious interpretation in terms of the polar coordinates of the point (x, y). We also define arg z [equivalent] θ (1.13) for z [not equal to] 0. The angle arg z is the phase of z. Evidently it can only be defined as mod 2π; adding any integer multiple of 2π to arg z does not change the complex number z, since e2πi = 1 (1.14) Equation (1.14) is one of the most remarkable equations of mathematics. 1.1.3 Infinite Sequences; Irrational Numbers To complete the construction of the real and complex numbers, we need to look at some elementary properties of sequences, starting with the formal definitions: Definition 1.1. A sequence of numbers (real or complex) is an ordered set of numbers in one-to-one correspondence with the positive integers; write {zn} [equivalent] {z1, z2, …}.Definition 1.2. The sequence {zn} is bounded if there is some positive number M such that |zn| for all positive integers n. Definition 1.3. The sequence {xn} of real numbers is increasing (decreasing) if xn+1 > xn (xn+1 n) for every n. The sequence is nondecreasing (nonincreasing) if xn+1 ≥ xn (xn+1 ≤ xn) for every n. A sequence belonging to one of these classes is monotone (or monotonic). Remark. The preceding definition is restricted to real numbers because it is only for real numbers that we can define a “natural” ordering that is compatible with the standard measure of the distance between the numbers. Definition 1.4. The sequence {zn} is a Cauchy sequence if for every ε > 0 there is a positive integer N such that |zp – zq| <ε whenever p, q > N. Definition 1.5. The sequence {zn} is convergent to the limit z (write {zn} -> z) if for every ε > 0 there is a positive integer N such that |zn – z| <ε whenever n > N. There is no guarantee that a Cauchy sequence of rational numbers converges to a rational, or even algebraic, limit. For example, the sequence {xn} defined by xn [equivalent] (1 + 1/n)n (1.15) converges to the limit e = 2.71828 …, the base of natural logarithms. It is true, though nontrivial to prove, that e is not an algebraic number. A real number that is not algebraic is transcendental. Another famous transcendental number is π, which is related to e through Eq. (1.14). If we want to insure that every Cauchy sequence of rational numbers converges to a limit, we must include the irrational numbers, which can be defined as limits of Cauchy sequences of rational numbers. As examples of such sequences, imagine the infinite, nonterminating, nonperiodic decimal expansions of transcendental numbers such as e or π, or algebraic numbers such as &radius;2. Countless computer cycles have been used in calculating the digits in these expansions. The set of real numbers, denoted by R, can now be defined as the set containing rational numbers together with the limits of Cauchy sequences of rational numbers. The set of complex numbers, denoted by BLDBLD, is then introduced as the set of all ordered pairs (x, y) ~ x+iy of real numbers. Once we know that every Cauchy sequence of real (or rational) numbers converges to a real number, it is a simple exercise to show that every Cauchy sequence of complex numbers converges to a complex number. Monotonic sequences are especially important, since they appear as partial sums of infinite series of positive terms. The key property is contained in the Theorem 1.1. A monotonic sequence {xn} is convergent if and only if it is bounded. Proof. If the sequence is unbounded, it will diverge to [+ or -]∞, which simply means that for any positive number M, no matter how large, there is an integer N such that xn > M (or xn <-M if the sequence is monotonic nonincreasing) for any n ≥ N. This is true, since for any positive number M, there is at least one member xN of the sequence with xN > M (or xN <-M)—otherwise M would be a bound for the sequence—and hence xn > M (or xn <-M) for any n ≥ N in view of the monotonic nature of the sequence. If the monotonic nondecreasing sequence {xn} is bounded from above, then in order to have a limit, there must be a bound that is smaller than any other bound (such a bound is the least upper bound of the sequence). If the sequence has a limit X, then X is certainly the least upper bound of the sequence, while if a least upper bound [bar.X] exists, then it must be the limit of the sequence. For if there is some ε > 0 such that [bar.X] – xn > ε for all n, then [bar.X] – ε will be an upper bound to the sequence smaller than [bar.X]. The existence of a least upper bound is intuitively plausible, but its existence cannot be proven from the concepts we have introduced so far. There are alternative axiomatic formulations of the real number system that guarantee the existence of the least upper bound; the convergence of any bounded monotonic nondecreasing sequence is then a consequence as just explained. The same argument applies to bounded monotonic nonincreasing sequences, which must then have a greatest lower bound to which the sequence converges. (Continues…) Excerpted from Introduction to Mathematical Physicsby Michael T. Vaughn Copyright © 2007 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Excerpted by permission of John Wiley & Sons. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Absolutely awful formatting. I’m going to try to return the Kindle version.

⭐Remarkably self-contained and written with great care and attention to detail, this manuscript provides the reader with a strong foundation for the study of mathematical physics. Notably, there is a wonderful balance of mathematical rigor to that of the actual physical interpretations of mathematical results. Examples of this include the chapter on the geometry of physics and the final chapters on group theory. The reader is supplied with the necessary tools and concepts to progress further in these subjects which are essential for modern applications. The problem sets in many cases are also rather unique and correlate very well with the presented material. I expect this text to be a valuable resource to a very wide audience of physicists ranging in specializations from condensed matter, to the physics of biological systems, and to the study of particle physics.

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