Applied Mathematics: Body and Soul

Springer-Verlag Berlin Heidelberg GmbH

K. Eriksson· D. Estep· (.Johnson

Applied Mathematics: Body and Soul [VOLUME 2]

Integrals and Geometry in IR"

Springer

Kenneth Eriksson Claes Johnson

Chalmers University of Technology Department of Mathematics 41296 Goteborg, Sweden e-mail: kennethlclaes@math.chalmers.se

Cataloging-in-Publication Data applied for

Donald Estep

Colorado State University Department of Mathematics Fort Collins, CO 80523-1874 USA e-mail: estep@math.colostate.edu

A catalog record for this book is available from the Library of Congress.

Bibliographie information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>.

Mathematics Subject Classification (2000): 15-01,34-01,35-01,49-01,65-01,70-01,76-01

ISBN 978-3-642-05658-1 ISBN 978-3-662-05798-8 (eBook) DOI 10.1007/978-3-662-05798-8

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on mierofilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH, Violations are liable for prosecution under the German Copyright Law.

springeronline.com III Springer-Verlag Berlin Heidelberg 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 Softcover reprint of the hardcover 1 st edition 2004 The use of general descriptive names, registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cover design: design & production, Heidelberg and Anders Logg, Department of Computational Mathematics, Chalmers University of Technology !ypesetting: Le-TeX Jelonel<, Schiiiidt & Vockler GbR, Leipzig Prmted on acid-free paper SPIN 10999554 46/31l1ck-54321

To the students of Chemical Engineering at Chalmers during 1998-2002, who enthusiastically participated in the development of the reform project behind this book.

Preface

I admit that each and every thing remains in its state until there is reason for change. (Leibniz)

The Need of Reform of Mathematics Education

Mathematics education needs to be reformed as we now pass into the new millennium. We share this conviction with a rapidly increasing number of researchers and teachers of both mathematics and topics of science and engineering based on mathematical modeling. The reason is of course the computer revolution, which has fundamentally changed the possibilities of using mathematical and computational techniques for modeling, simula­tion and control of real phenomena. New products and systems may be developed and tested through computer simulation on time scales and at costs which are orders of magnitude smaller than those using traditional techniques based on extensive laboratory testing, hand calculations and trial and error.

At the heart of the new simulation techniques lie the new fields of Computational Mathematical Modeling (CMM), including Computational Mechanics, Physics, Fluid Dynamics, Electromagnetics and Chemistry, all based on solving systems of differential equations using computers, com­bined with geometric modeling/Computer Aided Design (CAD). Compu­tational modeling is also finding revolutionary new applications in biology, medicine, environmental sciences, economy and financial markets.

VIII Preface

Education in mathematics forms the basis of science and engineering education from undergraduate to graduate level, because engineering and science are largely based on mathematical modeling. The level and the quality of mathematics education sets the level of the education as a whole. The new technology of CMM/CAD crosses borders between traditional engineering disciplines and schools, and drives strong forces to modernize engineering education in both content and form from basic to graduate level.

Our Reform Program

Our own reform work started some 20 years ago in courses in CMM at advanced undergraduate level, and has through the years successively pen­etrated through the system to the basic education in calculus and linear algebra. Our aim has become to develop a complete program for mathe­matics education in science and engineering from basic undergraduate to graduate education. As of now our program contains the series of books:

1. Computational Differential Equations, (CDE)

2. Applied Mathematics: Body & Soul I-III, (AM I-III)

3. Applied Mathematics: Body & Soul IV-, (AM IV-).

AM I-III is the present book in three volumes I-III covering the basics of calculus and linear algebra. AM IV- offers a continuation with a series of volumes dedicated to specific areas of applications such as Dynamical Systems (IV), Fluid Mechanics (V), Solid Mechanics (VI) and Electromag­netics (VII), which will start appearing in 2003. CDE published in 1996 may be be viewed as a first version of the whole Applied Mathematics: Body & Soul project.

Our program also contains a variety of software (collected in the Math­ematics Labomtory) , and complementary material with step-by step in­structions for self-study, problems with solutions, and projects, all freely available on-line from the web site of the book. Our ambition is to offer a "box" containing a set of books, software and additional instructional ma­terial, which can serve as a basis for a full applied mathematics program in science and engineering from basic to graduate level. Of course, we hope this to be an on-going project with new material being added gradually.

We have been running an applied mathematics program based on AM I-III from first year for the students of chemical engineering at Chalmers since the Fall 99, and we have used parts of the material from AM IV- in advanced undergraduate/beginning graduate courses.

Preface IX

Main Features of the Program:

• The program is based on a synthesis of mathematics, computation and application.

• The program is based on new literature, giving a new unified presen­tation from the start based on constructive mathematical methods including a computational methodology for differential equations.

• The program contains, as an integrated part, software at different levels of complexity.

• The student acquires solid skills of implementing computational meth­ods and developing applications and software using Matlab.

• The synthesis of mathematics and computation opens mathematics education to applications, and gives a basis for the effective use of modern mathematical methods in mechanics, physics, chemistry and applied subjects.

• The synthesis building on constructive mathematics gives a synergetic effect allowing the study of complex systems already in the basic ed­ucation, including the basic models of mechanical systems, heat con­duction, wave propagation, elasticity, fluid flow, electro-magnetism, reaction-diffusion, molecular dynamics, as well as corresponding multi­physics problems.

• The program increases the motivation of the student by applying mathematical methods to interesting and important concrete prob­lems already from the start.

• Emphasis may be put on problem solving, project work and presen­tation.

• The program gives theoretical and computational tools and builds confidence.

• The program contains most of the traditional material from basic courses in analysis and linear algebra

• The program includes much material often left out in traditional pro­grams such as constructive proofs of all the basic theorems in analysis and linear algebra and advanced topics such as nonlinear systems of algebraic / differential equations.

• Emphasis is put on giving the student a solid understanding of basic mathematical concepts such as real numbers, Cauchy sequences, Lips­chitz continuity, and constructive tools for solving algebraic/differen­tial equations, together with an ability to utilize these tools in ad­vanced applications such as molecular dynamics.

X Preface

• The program may be run at different levels of ambition concerning both mathematical analysis and computation, while keeping a com­mon basic core.

AM I-III in Brief

Roughly speaking, AM I-III contains a synthesis of calculus and linear algebra including computational methods and a variety of applications. Emphasis is put on constructive/computational methods with the double aim of making the mathematics both understandable and useful. Our am­bition is to introduce the student early (from the perspective of traditional education) to both advanced mathematical concepts (such as Lipschitz continuity, Cauchy sequence, contraction mapping, initial-value problem for systems of differential equations) and advanced applications such as Lagrangian mechanics, n-body systems, population models, elasticity and electrical circuits, with an approach based on constructive/computational methods.

Thus the idea is that making the student comfortable with both ad­vanced mathematical concepts and modern computational techniques, will open a wealth of possibilities of applying mathematics to problems of real interest. This is in contrast to traditional education where the emphasis is usually put on a set of analytical techniques within a conceptual framework of more limited scope. For example: we already lead the student in the sec­ond quarter to write (in Matlab) his/her own solver for general systems of ordinary differential equations based on mathematically sound principles (high conceptual and computational level), while traditional education at the same time often focuses on training the student to master a bag of tricks for symbolic integration. We also teach the student some tricks to that purpose, but our overall goal is different.

Constructive Mathematics: Body & Soul

In our work we have been led to the conviction that the constructive as­pects of calculus and linear algebra need to be strengthened. Of course, constructive and computational mathematics are closely related and the development of the computer has boosted computational mathematics in recent years. Mathematical modeling has two basic dual aspects: one sym­bolic and the other constructive-numerical, which reflect the duality be­tween the infinite and the finite, or the continuous and the discrete. The two aspects have been closely intertwined throughout the development of modern science from the development of calculus in the work of Euler, La­grange, Laplace and Gauss into the work of von Neumann in our time. For

Preface XI

example, Laplace's monumental Mecanique Celeste in five volumes presents a symbolic calculus for a mathematical model of gravitation taking the form of Laplace's equation, together with massive numerical computations giv­ing concrete information concerning the motion of the planets in our solar system.

However, beginning with the search for rigor in the foundations of cal­culus in the 19th century, a split between the symbolic and construc­tive aspects gradually developed. The split accelerated with the inven­tion of the electronic computer in the 1940s, after which the construc­tive aspects were pursued in the new fields of numerical analysis and computing sciences, primarily developed outside departments of mathe­matics. The unfortunate result today is that symbolic mathematics and constructive-numerical mathematics by and large are separate disciplines and are rarely taught together. Typically, a student first meets calcu­lus restricted to its symbolic form and then much later, in a different context, is confronted with the computational side. This state of affairs lacks a sound scientific motivation and causes severe difficulties in courses in physics, mechanics and applied sciences which build on mathematical modeling.

New possibilies are opened by creating from the start a synthesis of constructive and symbolic mathematics representing a synthesis of Body & Soul: with computational techniques available the students may become familiar with nonlinear systems of differential equations already in early calculus, with a wealth of applications. Another consequence is that the basics of calculus, including concepts like real number, Cauchy sequence, convergence, fixed point iteration, contraction mapping, is lifted out of the wardrobe of mathematical obscurities into the real world with direct practical importance. In one shot one can make mathematics education both deeper and broader and lift it to a higher level. This idea underlies the present book, which thus in the setting of a standard engineering program, contains all the basic theorems of calculus including the proofs normally taught only in special honors courses, together with advanced applications such as systems of nonlinear differential equations. We have found that this seemingly impossible program indeed works surprisingly well. Admittedly, this is hard to believe without making real life experiments. We hope the reader will feel encouraged to do so.

Lipschitz Continuity and Cauchy Sequences

The usual definition of the basic concepts of continuity and derivative, which is presented in most Calculus text books today, build on the concept of limit: a real valued function f(x) of a real variable x is said to be con­tinuous at x if limx-->x f(x) = f(x), and f(x) is said to be differentiable at

XII Preface

x with derivative f'(x) if

1. f(x) - f(x) 1m _

x-+x X - X

exists and equals f'(x). We use different definitions, where the concept of limit does not intervene: we say that a real-valued function f(x) is Lipschitz continuous with Lipschitz constant L f on an interval [a, b] if for all x, x E

[a, b], we have If(x) - f(x)1 :::; Lflx - xl·

Further, we say that f (x) is differentiable at x with derivative f' (x) if there is a constant K f (x) such that for all x close to x

If(x) - f(x) - f'(x)(x - x)1 :::; Kf(x)lx - x1 2 •

This means that we put somewhat more stringent requirements on the concepts of continuity and differentiability than is done in the usual def­initions; more precisely, we impose quantitative measures in the form of the constants Lf and Kf(x), whereas the usual definitions using limits are purely qualitative.

Using these more stringent definitions we avoid pathological situations, which can only be confusing to the student (in particular in the beginning) and, as indicated, we avoid using the (difficult) concept of limit in a setting where in fact no limit processes are really taking place. Thus, we do not lead the student to definitions of continuity and differentiability suggesting that all the time the variable x is tending to some value x, that is, all the time some kind of (strange?) limit process is taking place. In fact, continuity expresses that the difference f(x) - f(x) is small if x - x is small, and differentiability expresses that f(x) locally is close to a linear function, and to express these facts we do not have to invoke any limit processes.

These are examples of our leading philosophy of giving Calculus a quan­titative form, instead of the usual purely qualitative form, which we believe helps both understanding and precision. We believe the price to pay for these advantages is usually well worth paying, and the loss in generality are only some pathological cases of little interest. We can in a natural way relax our definitions, for example to Holder continuity, while still keeping the quantitative aspect, and thereby increase the pathology of the excep­tional cases.

The usual definitions of continuity and differentiability strive for maximal generality, typically considered to be a virtue by a pure mathematician, which however has pathological side effects. With a constructive point of view the interesting world is the constructible world and maximality is not an important issue in itself.

Of course, we do not stay away from limit processes, but we concen­trate on issues where the concept of limit really is central, most notably in

Preface XIII

defining the concept of a real number as the limit of a Cauchy sequence of rational numbers, and a solution of an algebraic or differential equation as the limit of a Cauchy sequence of approximate solutions. Thus, we give the concept of Cauchy sequence a central role, while maintaining a constructive approach seeking constructive processes for generating Cauchy sequences.

In standard Calculus texts, the concepts of Cauchy sequence and Lip­schitz continuity are not used, believing them to be too difficult to be presented to freshmen, while the concept of real number is left undefined (seemingly believing that a freshman is so familiar with this concept from early life that no further discussion is needed). In contrast, in our construc­tive approach these concepts playa central role already from start, and in particular we give a good deal of attention to the fundamental aspect of the constructibility of real numbers (viewed as possibly never-ending decimal expansions) .

We emphasize that taking a constructive approach does not make math­ematicallife more difficult in any important way, as is often claimed by the ruling mathematical school of formalists/logicists: All theorems of interest in Calculus and Linear Algebra survive, with possibly some small unessen­tial modifications to keep the quantitative aspect and make the proofs more precise. As a result we are able to present basic theorems such as Con­traction Mapping Principle, Implicit Function theorem, Inverse Function theorem, Convergence of Newton's Method, in a setting of several variables with complete proofs as a part of our basic Calculus, while these results in the standard curriculum are considered to be much too difficult for this level.

Proofs and Theorems

Most mathematics books including Calculus texts follow a theorem-proof style, where first a theorem is presented and then a corresponding proof is given. This is seldom appreciated very much by the students, who often have difficulties with the role and nature of the proof concept.

We usually turn this around and first present a line of thought leading to some result, and then we state a corresponding theorem as a summary of the hypothesis and the main result obtained. We thus rather use a proof­theorem format. We believe this is in fact often more natural than the theorem-proof style, since by first presenting the line of thought the differ­ent ingredients, like hypotheses, may be introduced in a logical order. The proof will then be just like any other line of thought, where one successively derives consequences from some starting point using different hypothesis as one goes along. We hope this will help to eliminate the often perceived mystery of proofs, simply because the student will not be aware of the fact that a proof is being presented; it will just be a logical line of thought, like

XIV Preface

any logical line of thought in everyday life. Only when the line of thought is finished, one may go back and call it a proof, and in a theorem collect the main result arrived at, including the required hypotheses. As a conse­quence, in the Latex version of the book we do use a theorem-environment, but not any proof-environment; the proof is just a logical line of thought preceding a theorem collecting the hypothesis and the main result.

The Mathematics Laboratory

We have developed various pieces of software to support our program into what we refer to as the Mathematics Laboratory. Some of the software serves the purpose of illustrating mathematical concepts such as roots of equations, Lipschitz continuity, fixed point iteration, differentiability, the definition of the integral and basic calculus for functions of several vari­ables; other pieces are supposed to be used as models for the students own computer realizations; finally some pieces are aimed at applications such as solvers for differential equations. New pieces are being added continuously. Our ambition is to also add different multi-media realizations of various parts of the material.

In our program the students get a training from start in using Matlab as a tool for computation. The development of the constructive mathe­matical aspects of the basic topics of real numbers, functions, equations, derivatives and integrals, goes hand in hand with experience of solving equations with fixed point iteration or Newton's method, quadrature, and numerical methods or differential equations. The students see from their own experience that abstract symbolic concepts have roots deep down into constructive computation, which also gives a direct coupling to applications and physical reality.

Go to http://www.phLchalmers.se/bodysoul/

The Applied Mathematics: Body & Soul project has a web site contain­ing additional instructional material and the Mathematics Laboratory. We hope that the web site for the student will be a good friend helping to (independently) digest and progress through the material, and that for the teacher it may offer inspiration. We also hope the web site may serve as a forum for exchange of ideas and experience related the project, and we therefore invite both students and teachers to submit material.

Preface XV

Acknowledgment

The authors of this book want to thank sincerely the following colleagues and graduate students for contributing valuable material, corrections and suggestions for improvement: Rickard Bergstrom, Niklas Eriksson, Johan Hoffman, Mats Larson, Stig Larsson, Marten Levenstam, Anders Logg, Klas Samuelsson and Nils Svanstedt, all actively participating in the devel­opment of our reform project. And again, sincere thanks to all the students of chemical engineering at Chalmers who carried the burden of being ex­posed to new material often in incomplete form, and who have given much enthusiastic criticism and feed-back.

The source of mathematicians pictures is the MacTutor History of Math­ematics archive, and some images are copied from old volumes of Deadalus, the yearly report from The Swedish Museum of Technology.

My heart is sad and lonely for you I sigh, dear, only Why haven't you seen it

I'm all for you body and soul (Green, Body and Soul)

Contents Volume 2

Integrals and Geometry in ~n 427

27 The Integral 429 27.1 Primitive Functions and Integrals. . . . . . . . . . . 429 27.2 Primitive Function of f(x) = xm for m = 0,1,2,... . 433 27.3 Primitive Function of f(x) = xm for m = -2, -3,... 434 27.4 Primitive Function of f(x) = xT for r =f -1 ..... 434 27.5 A Quick Overview of the Progress So Far ...... 435 27.6 A "Very Quick Proof" of the Fundamental Theorem 435 27.7 A "Quick Proof" of the Fundamental Theorem . . 437 27.8 A Proof of the Fundamental Theorem of Calculus. 438 27.9 Comments on the Notation ..... 27.10 Alternative Computational Methods .. . 27.11 The Cyclist's Speedometer ........ . 27.12 Geometrical Interpretation of the Integral 27.13 The Integral as a Limit of Riemann Sums 27.14 An Analog Integrator ........... .

444 445 445 446 448 449

28 Properties of the Integral 453 28.1 Introduction..................... 453 28.2 Reversing the Order of Upper and Lower Limits. 454 28.3 The Whole Is Equal to the Sum of the Parts. . . 454

XVIII

28.4

28.5 28.6 28.7 28.8

28.9 28.10 28.11 28.12 28.13 28.14 28.15 28.16 28.17

Contents Volume 2

Integrating Piecewise Lipschitz Continuous Functions Linearity ............ . Monotonicity . . . . . . . . . . The Triangle Inequality for Integrals Differentiation and Integration are Inverse Operations . . . . . . . . Change of Variables or Substitution. Integration by Parts . . . . . . . . . The Mean Value Theorem ..... . Monotone Functions and the Sign of the Derivative . A Function with Zero Derivative is Constant. . . . . A Bounded Derivative Implies Lipschitz Continuity . Taylor's Theorem . October 29, 1675 The Hodometer .

455 456 457 457

458 459 461 462 464 464 465 465 468 469

29 The Logarithm log( x) 473 29.1 The Definition oflog(x) ... . . . 473 29.2 The Importance of the Logarithm. 474 29.3 Important Properties oflog(x) 475

30 Numerical Quadrature 479 30.1 Computing Integrals . . . . . . . . . . . . 479 30.2 The Integral as a Limit of Riemann Sums 483 30.3 The Midpoint Rule. . 484 30.4 Adaptive Quadrature . . . . . . . . . 485

31 The Exponential Function exp(x) = eX 491 31.1 Introduction..................... 491 31.2 Construction of the Exponential exp(x) for x ~ 0 493 31.3 Extension of the Exponential exp(x) to x < 0 498 31.4 The Exponential Function exp(x) for x E lR. . . . 498 31.5 An Important Property of exp(x) . . . . . . . . . 499 31.6 The Inverse of the Exponential is the Logarithm 500 31. 7 The Function aX with a > 0 and x E lR. . . . . . • 501

32 Trigonometric Functions 505 32.1 The Defining Differential Equation . . . . . . . . . . . 505 32.2 Trigonometric Identities . . . . . . . . . . . . . . . . . 509 32.3 The Functions tan(x) and cot (x) and Their Derivatives 510 32.4 Inverses of Trigonometric Functions. 511 32.5 The Functions sinh(x) and cosh(x) . . . . . . . . 513 32.6 The Hanging Chain. . . . . . . . . . . . . . . . . 514 32.7 Comparing u" + k2u(x) = 0 and u" - k2u(x) = 0 515

Contents Volume 2 XIX

33 The Functions exp(z), log(z), sin(z) and cos(z) for z E C 517 33.1 Introduction.......... 517 33.2 Definition of exp(z) . . . . . . 517 33.3 Definition of sin(z) and cos(z) 518 33.4 de Moivres Formula. 518 33.5 Definition of log(z) . . 519

34 Techniques of Integration 34.1 Introduction ..... .

521 521

34.2 Rational Functions: The Simple Cases 522 34.3 Rational Functions: Partial Fractions . 523 34.4 Products of Polynomial and Trigonometric

or Exponential Functions .......... 528 34.5 Combinations of Trigonometric and Root Functions. 528 34.6 Products of Exponential and Trigonometric Functions 529 34.7 Products of Polynomials and Logarithm Functions .. 529

35 Solving Differential Equations Using the Exponential 531 35.1 Introduction.................. 531 35.2 Generalization to u'(x) = A(X)U(X) + f(x) . . . 532 35.3 The Differential Equation u"(x) - u(x) = 0 . . 536 35.4 The Differential Equation 2:~=o akDku(x) = 0 537 35.5 The Differential Equation 2:~=o akDku(x) = f(x) . 538 35.6 Euler's Differential Equation. . . . . . . . . . . . . 539

36 Improper Integrals 541 36.1 Introduction.............. 541 36.2 Integrals Over Unbounded Intervals. 541 36.3 Integrals of Unbounded Functions. 543

37 Series 37.1 Introduction. 37.2 Definition of Convergent Infinite Series. 37.3 Positive Series ........ . 37.4 Absolutely Convergent Series ..... . 37.5 Alternating Series ............ . 37.6 The Series 2::1 t Theoretically Diverges! 37.7 Abel. 37.8 Galois ................... .

38 Scalar Autonomous Initial Value Problems 38.1 Introduction ........... . 38.2 An Analytical Solution Formula . 38.3 Construction of the Solution . . .

547 547 548 549 552 552 553 555 556

559 559 560 563

XX Contents Volume 2

39 Separable Scalar Initial Value Problems 39.1 Introduction .............. . 39.2 An Analytical Solution Formula . . . . 39.3 Volterra-Lotka's Predator-Prey Model 39.4 A Generalization ........ .

40 The General Initial Value Problem 40.1 Introduction ........... . 40.2 Determinism and Materialism .. 40.3 Predictability and Computability 40.4 Construction of the Solution ... 40.5 Computational Work . . . . . . . 40.6 Extension to Second Order Initial Value Problems 40.7 Numerical Methods . . . . . . . . . . . . . . . . . .

41 Calculus Tool Bag I 41.1 Introduction ... 41.2 Rational Numbers .......... . 41.3 Real Numbers. Sequences and Limits . 41.4 Polynomials and Rational Functions 41.5 Lipschitz Continuity 41.6 Derivatives ........... . 41. 7 Differentiation Rules . . . . . . . 41.8 Solving f(x) = 0 with f : lR ---t lR 41.9 Integrals .... . 41.10 The Logarithm ....... . 41.11 The Exponential ...... . 41.12 The Trigonometric Functions 41.13 List of Primitive Functions .. 41.14 Series ............ . 41.15 The Differential Equation U + A(X)U(X) = f(x) 41.16 Separable Scalar Initial Value Problems

42 Analytic Geometry in lRn

42.1 Introduction and Survey of Basic Objectives. 42.2 Body jSoul and Artificial Intelligence . 42.3 The Vector Space Structure of lRn. . . 42.4 The Scalar Product and Orthogonality 42.5 Cauchy'S Inequality. . . . . . . . . . . 42.6 The Linear Combinations of a Set of Vectors 42.7 The Standard Basis ............ . 42.8 Linear Independence . . . . . . . . . . . . . . 42.9 Reducing a Set of Vectors to Get a Basis ... 42.10 Using Column Echelon Form to Obtain a Basis 42.11 Using Column Echelon Form to Obtain R(A) .

567 567 568 570 571

575 575 577 577 579 580 581 582

585 585 585 586 586 587 587 587 588 589 590 591 591 594 594 595 595

597 597 600 600 601 602 603 604 605 606 607 608

Contents Volume 2 XXI

42.12 Using Row Echelon Form to Obtain N(A) 610 42.13 Gaussian Elimination. . . . . . . . 612 42.14 A Basis for ~n Contains n Vectors 612 42.15 Coordinates in Different Bases. . . 614 42.16 Linear Functions f : ~n -+ ~ ... 615 42.17 Linear Transformations f : ~n -+ ~m 615 42.18 Matrices . . . . . . . . . . . . . . . . 616 42.19 Matrix Calculus. . . . . . . . . . . . 617 42.20 The Transpose of a Linear Transformation. 619 42.21 Matrix Norms . . . . . . . . . . . . . . . . . 620 42.22 The Lipschitz Constant of a Linear Transformation 621 42.23 Volume in ~n: Determinants and Permutations 621 42.24 Definition of the Volume V(al, ... , an) 623 42.25 The Volume V(al, a2) in ~2 . . • . 624 42.26 The Volume V(al, a2, a3) in ~3 • • 624 42.27 The Volume V(al,a2,a3,a4) in ~4 625 42.28 The Volume V(al' ... ' an) in ~n . 625 42.29 The Determinant of a Triangular Matrix 625 42.30 Using the Column Echelon Form to Compute det A . 625 42.31 The Magic Formula det AB = det A det B . . 626 42.32 Test of Linear Independence . . . . . . . . . . 626 42.33 Cramer's Solution for Non-Singular Systems. 628 42.34 The Inverse Matrix. . . . . . . . . . . . . . . 629 42.35 Projection onto a Subspace ... . . . . . . . 630 42.36 An Equivalent Characterization of the Projection 631 42.37 Orthogonal Decomposition: Pythagoras Theorem 632 42.38 Properties of Projections. . . . . . . . . . . . . . 633 42.39 Orthogonalization: The Gram-Schmidt Procedure. 633 42.40 Orthogonal Matrices . . . . . . . . 634 42.41 Invariance of the Scalar Product

Under Orthogonal Transformations 634 42.42 The QR-Decomposition ...... 635 42.43 The Fundamental Theorem of Linear Algebra 635 42.44 Change of Basis: Coordinates and Matrices 637 42.45 Least Squares Methods. . . . . . . . . . . . . 638

43 The Spectral Theorem 641 43.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . 641 43.2 Basis of Eigenvectors . . . . . . . . . . . . . . . . . . 643 43.3 An Easy Spectral Theorem for Symmetric Matrices . 644 43.4 Applying the Spectral Theorem to an IVP . . . . . . 645 43.5 The General Spectral Theorem

for Symmetric Matrices .......... 646 43.6 The Norm of a Symmetric Matrix. . . . . 648 43.7 Extension to Non-Symmetric Real Matrices 649

XXII Contents Volume 2

44 Solving Linear Algebraic Systems 44.1 Introduction .......... . 44.2 Direct Methods . . . . . . . . . 44.3 Direct Methods for Special Systems . 44.4 Iterative Methods . . . . . . . . . . . 44.5 Estimating the Error of the Solution 44.6 The Conjugate Gradient Method 44.7 GMRES ............. .

651 651 651 658 661 671 674 676

45 Linear Algebra Tool Bag 685 45.1 Linear Algebra in 1R2 685 45.2 Linear Algebra in 1R3 . 686 45.3 Linear Algebra in IRn . 686 45.4 Linear Transformations and Matrices . 687 45.5 The Determinant and Volume 688 45.6 Cramer's Formula. 688 45.7 Inverse............. 689 45.8 Projections ..... . . . . . 689 45.9 The Fundamental Theorem of Linear Algebra 689 45.10 The QR-Decomposition .. 689 45.11 Change of Basis. . . . . . . . 690 45.12 The Least Squares Method . 690 45.13 Eigenvalues and Eigenvectors 690 45.14 The Spectral Theorem . . . . 690 45.15 The Conjugate Gradient Method for Ax = b . 690

46 The Matrix Exponential exp(xA) 691 46.1 Computation of exp(xA) when A Is Diagonalizable 692 46.2 Properties of exp(Ax) 694 46.3 Duhamel's Formula . . . . . . . . . . . . . . 694

47 Lagrange and the Principle of Least Action* 697 47.1 Introduction............ 697 47.2 A Mass-Spring System . . . . . . . 699 47.3 A Pendulum with Fixed Support . 700 47.4 A Pendulum with Moving Support 47.5 The Principle of Least Action .. 47.6 Conservation of the Total Energy 47.7 The Double Pendulum ..... . 47.8 The Two-Body Problem .... . 47.9 Stability of the Motion of a Pendulum

701 701 703 703 704 705

48 N-Body Systems* 48.1 Introduction. 48.2 48.3 48.4

48.5 48.6 48.7 48.8 48.9

Masses and Springs . . The N -Body Problem Masses, Springs and Dashpots: Small Displacements . . . . Adding Dashpots . . . . . . A Cow Falling Down Stairs The Linear Oscillator. . . . The Damped Linear Oscillator Extensions . . .

49 The Crash Model* 49.1 Introduction ......... . 49.2 The Simplified Growth Model 49.3 The Simplified Decay Model . 49.4 The Full Model . . . . . . . .

50 Electrical Circuits* 50.1 Introduction ..

Contents Volume 2

50.2 Inductors, Resistors and Capacitors. 50.3 Building Circuits: Kirchhoff's Laws 50.4 Mutual Induction ......... .

51 String Theory* 51.1 Introduction. 51.2 A Linear System 51.3 A Soft System .. 51.4 A Stiff System .. 51.5 Phase Plane Analysis.

XXIII

709 709 710 712

713 714 715 716 717 719

721 721 722 724 725

729 729 730 731 732

735 735 736 737 737 738

52 Piecewise Linear Approximation 741 52.1 Introduction....................... 741 52.2 Linear Interpolation on [0, 1] . . . . . . . . . . . . . . 742 52.3 The Space of Piecewise Linear Continuous Functions 747 52.4 The L2 Projection into Vh . . . . . . . . . . . . 749

53 FEM for Two-Point Boundary Value Problems 755 53.1 Introduction............... 755 53.2 Initial Boundary-Value Problems . . . 758 53.3 Stationary Boundary Value Problems. 759 53.4 The Finite Element Method . . . . . . 759

XXIV Contents Volume 2

53.5 The Discrete System of Equations. . . . . . . 762 53.6 Handling Different Boundary Conditions . . . 765 53.7 Error Estimates and Adaptive Error Control. 768 53.8 Discretization of Time-Dependent

Reaction-Diffusion-Convection Problems . . . 773 53.9 Non-Linear Reaction-Diffusion-Convection Problems 773

References 777

Index 779

Contents Volume 1

Derivatives and Geometry in ]R3 1

1 What is Mathematics? 3 1.1 Introduction . . . . 3 1.2 The Modern World 3 1.3 The Role of Mathematics 6 1.4 Design and Production of Cars 11 1.5 Navigation: From Stars to GPS 11 1.6 Medical Tomography . . . . . . 11 1.7 Molecular Dynamics and Medical Drug Design 12 1.8 Weather Prediction and Global Warming. 13 1.9 Economy: Stocks and Options . . . . . . 13 1.10 Languages ................. 14 1.11 Mathematics as the Language of Science 15 1.12 The Basic Areas of Mathematics 16 1.13 What Is Science? .......... 17 1.14 What Is Conscience? ........ 17 1.15 How to View this Book as a Friend 18

2 The Mathematics Laboratory 21 2.1 Introduction . . . 21 2.2 Math Experience ...... 22

XXVI Contents Volume 1

3 Introduction to Modeling 3.1 Introduction ..... . 3.2 The Dinner Soup Model 3.3 The Muddy Yard Model 3.4 A System of Equations . 3.5 Formulating and Solving Equations .

4 A Very Short Calculus Course 4.1 Introduction . . . . . . 4.2 Algebraic Equations 4.3 Differential Equations 4.4 Generalization ..... 4.5 Leibniz' Teen-Age Dream 4.6 Summary 4.7 Leibniz...........

5 Natural Numbers and Integers 5.1 Introduction ......... . 5.2 The Natural Numbers ... . 5.3 5.4 5.5 5.6 5.7 5.8 5.9

Is There a Largest Natural Number? The Set N of All Natural Numbers Integers ............... . Absolute Value and the Distance Between Numbers. Division with Remainder . . . . . . . Factorization into Prime Factors .. Computer Representation of Integers

6 Mathematical Induction 6.1 Induction ............. . 6.2 Changes in a Population of Insects

7 Rational Numbers 7.1 Introduction ................ . 7.2 How to Construct the Rational Numbers. 7.3 On the Need for Rational Numbers .... 7.4 Decimal Expansions of Rational Numbers 7.5 Periodic Decimal Expansions of Rational Numbers 7.6 Set Notation. . . . . . . . . . . . . . . . . 7.7 The Set Q of All Rational Numbers .... 7.8 The Rational Number Line and Intervals. 7.9 Growth of Bacteria .. 7.10 Chemical Equilibrium .......... .

25 25 25 28 29 30

33 33 34 34 39 41 43 44

47 47 48 51 52 53 56 57 58 59

63 63 68

71 71 72 75 75 76 80 81 82 83 85

Contents Volume 1

8 Pythagoras and Euclid 8.1 Introduction ................. . 8.2 Pythagoras Theorem ............ . 8.3 The Sum of the Angles of a Triangle is 1800

8.4 Similar Triangles .............. . 8.5 When Are Two Straight Lines Orthogonal? 8.6 The GPS Navigator ............. . 8.7 Geometric Definition of sin( v) and cos( v) 8.8 Geometric Proof of Addition Formulas for cos( v) 8.9 Remembering Some Area Formulas 8.10 Greek Mathematics ........... . 8.11 The Euclidean Plane Q2 ........ . 8.12 From Pythagoras to Euclid to Descartes 8.13 Non-Euclidean Geometry ....... .

9 What is a Function? 9.1 Introduction .......... . 9.2 Functions in Daily Life .... . 9.3 Graphing Functions of Integers 9.4 Graphing Functions of Rational Numbers 9.5 A Function of Two Variables . 9.6 Functions of Several Variables.

10 Polynomial functions 10.1 Introduction ... 10.2 Linear Polynomials 10.3 Parallel Lines . . . 10.4 Orthogonal Lines . 10.5 Quadratic Polynomials 10.6 Arithmetic with Polynomials 10.7 Graphs of General Polynomials 10.8 Piecewise Polynomial Functions

11 Combinations of functions 11.1 Introduction ............. . 11.2 Sum of Two Functions and Product

of a Function with a Number .... 11.3 Linear Combinations of Functions . . 11.4 Multiplication and Division of Functions 11.5 Rational Functions ...... . 11.6 The Composition of Functions.

12 Lipschitz Continuity 12.1 Introduction ... 12.2 The Lipschitz Continuity of a Linear Function .

XXVII

87 87 87 89 91 91 94 96 97 98 98 99

100 101

103 103 106 109 112 114 116

119 119 120 124 124 125 129 135 137

141 141

142 142 143 143 145

149 149 150

XXVIII Contents Volume 1

12.3 The Definition of Lipschitz Continuity 12.4 Monomials ............. . 12.5 Linear Combinations of Functions. 12.6 Bounded Functions ..... 12.7 The Product of Functions . . . 12.8 The Quotient of Functions ... 12.9 The Composition of Functions. 12.10 Functions of Two Rational Variables 12.11 Functions of Several Rational Variables.

13 Sequences and limits 13.1 A First Encounter with Sequences and Limits 13.2 Socket Wrench Sets .......... . 13.3 J.P. Johansson's Adjustable Wrenches 13.4 The Power of Language:

From Infinitely Many to One ..... 13.5 The E - N Definition of a Limit . . . . 13.6 A Converging Sequence Has a Unique Limit 13.7 Lipschitz Continuous Functions and Sequences 13.8 Generalization to Functions of Two Variables . 13.9 Computing Limits ............... . 13.10 Computer Representation of Rational Numbers 13.11 Sonya Kovalevskaya .............. .

14 The Square Root of Two 14.1 Introduction ................. . 14.2 V2 Is Not a Rational Number! ...... . 14.3 Computing V2 by the Bisection Algorithm. 14.4 The Bisection Algorithm Converges! .... 14.5 First Encounters with Cauchy Sequences .. 14.6 Computing V2 by the Deca-section Algorithm.

15 Real numbers 15.1 Introduction ................ . 15.2 Adding and Subtracting Real Numbers .. 15.3 Generalization to f(x, x) with f Lipschitz 15.4 Multiplying and Dividing Real Numbers 15.5 The Absolute Value ............ . 15.6 Comparing Two Real Numbers ..... . 15.7 Summary of Arithmetic with Real Numbers 15.8 Why V2V2 Equals 2 . . . . . . . . . 15.9 A Reflection on the Nature of V2 ... . 15.10 Cauchy Sequences of Real Numbers .. . 15.11 Extension from f : Q -+ Q to f : ~ -+ ~ 15.12 Lipschitz Continuity of Extended Functions

151 154 157 158 159 160 161 162 163

165 165 167 169

169 170 174 175 176 177 180 181

185 185 187 188 189 192 192

195 195 197 199 200 200 200 201 201 202 203 204 205

Contents Volume 1

15.13 Graphing Functions f : lR. -+ lR. ...... . 15.14 Extending a Lipschitz Continuous Function 15.15 Intervals of Real Numbers ....... . 15.16 What Is f(x) if x Is Irrational? .... . 15.17 Continuity Versus Lipschitz Continuity .

XXIX

206 206 207 208 211

16 The Bisection Algorithm for f(x) = 0 215 16.1 Bisection....... 215 16.2 An Example . . . . . 217 16.3 Computational Cost 219

17 Do Mathematicians Quarrel?* 221 17.1 Introduction......... 221 17.2 The Formalists . . . . . . . 224 17.3 The Logicists and Set Theory 224 17.4 The Constructivists. . . . . . 227 17.5 The Peano Axiom System for Natural Numbers 229 17.6 Real Numbers. . . . . . . . . . . . . . . . . . . 229 17.7 Cantor Versus Kronecker. . . . . . . . . . . . . 230 17.8 Deciding Whether a Number is Rational or Irrational. 232 17.9 The Set of All Possible Books . . . . . . . . 233 17.10 Recipes and Good Food . . . . . . . . . . . 234 17.11 The "New Math" in Elementary Education 234 17.12 The Search for Rigor in Mathematics. 235 17.13 A Non-Constructive Proof. 236 17.14 Summary .... 237

18 The Function y = xT 241 18.1 The Function Vx . . . . . . . . . . . 241 18.2 Computing with the Function Vx . . 242 18.3 Is Vx Lipschitz Continuous on lR.+? . 242 18.4 The Function xT for Rational r = ~ . 243 18.5 Computing with the Function x T •• 243 18.6 Generalizing the Concept of Lipschitz Continuity 243 18.7 Turbulent Flow is Holder (Lipschitz) Continuous with Ex-

ponent l . . . . . . . . . . . . . . . . . . . . . . . . . . 244

19 Fixed Points and Contraction Mappings 245 19.1 Introduction........... 245 19.2 Contraction Mappings . . . . . 246 19.3 Rewriting f(x) = 0 as x = g(x) 247 19.4 Card Sales Model. . . . . . . . 248 19.5 Private Economy Model . . . . 249 19.6 Fixed Point Iteration in the Card Sales Model. 250 19.7 A Contraction Mapping Has a Unique Fixed Point 254

XXX Contents Volume 1

19.8 Generalization to g : [a, b] ----7 [a, b] ..... . 19.9 Linear Convergence in Fixed Point Iteration 19.10 Quicker Convergence .. 19.11 Quadratic Convergence.

20 Analytic Geometry in ]R2 20.1 Introduction ..... . 20.2 Descartes, Inventor of Analytic Geometry 20.3 Descartes: Dualism of Body and Soul . 20.4 The Euclidean Plane]R2 . . 20.5 Surveyors and Navigators ...... . 20.6 A First Glimpse of Vectors ...... . 20.7 Ordered Pairs as Points or Vectors/Arrows. 20.8 Vector Addition . . . . . . . . . . . . . . . . 20.9 Vector Addition and the Parallelogram Law 20.10 Multiplication of a Vector by a Real Number 20.11 The Norm of a Vector ..... . 20.12 Polar Representation of a Vector 20.13 Standard Basis Vectors ..... . 20.14 Scalar Product ......... .

256 257 258 259

265 265 266 266 267 269 270 271 272 273 274 275 275 277 278

20.15 Properties of the Scalar Product 278 20.16 Geometric Interpretation of the Scalar Product 279 20.17 Orthogonality and Scalar Product. . 280 20.18 Projection of a Vector onto a Vector 281 20.19 Rotation by 900 • • . • • • • • • • • 283 20.20 Rotation by an Arbitrary Angle e . 285 20.21 Rotation by e Again! . . . . . . 286 20.22 Rotating a Coordinate System. . . 286 20.23 Vector Product . . . . . . . . . . . 287 20.24 The Area of a Triangle with a Corner at the Origin. 290 20.25 The Area of a General Triangle . . . . . . . . . . . . 290 20.26 The Area of a Parallelogram Spanned

by Two Vectors. . . . . . . . . . 291 20.27 Straight Lines . . . . . . . . . . . 292 20.28 Projection of a Point onto a Line 294 20.29 When Are Two Lines Parallel? . 294 20.30 A System of Two Linear Equations

in Two Unknowns ............... 295 20.31 Linear Independence and Basis . . . . . . . . 297 20.32 The Connection to Calculus in One Variable. 298 20.33 Linear Mappings f : ]R2 ----7 ]R. . . . . . . . . . 299 20.34 Linear Mappings f : ]R2 ----7]R2 . . . . . . . . . 299 20.35 Linear Mappings and Linear Systems of Equations 300 20.36 A First Encounter with Matrices . . . 300 20.37 First Applications of Matrix Notation ....... 302

Contents Volume 1

20.38 Addition of Matrices . . . . . . . . . . . . . . 20.39 Multiplication of a Matrix by a Real Number 20.40 Multiplication of Two Matrices . . . 20.41 The Transpose of a Matrix. . . . . . 20.42 The Transpose of a 2-Column Vector 20.43 The Identity Matrix .. . . . . . 20.44 The Inverse of a Matrix . . . . . 20.45 Rotation in Matrix Form Again! 20.46 A Mirror in Matrix Form 20.47 Change of Basis Again! . 20.48 Queen Christina ...

21 Analytic Geometry in ]R3

21.1 Introduction ................... . 21.2 Vector Addition and Multiplication by a Scalar 21.3 Scalar Product and Norm ..... . 21.4 Projection of a Vector onto a Vector 21.5 The Angle Between Two Vectors .. 21.6 Vector Product ........... . 21. 7 Geometric Interpretation of the Vector Product 21.8 Connection Between Vector Products in ]R2 and ]R3

21.9 Volume of a Parallelepiped Spanned by Three Vectors .......... .

21.10 The Triple Product a· b xc . ... . 21.11 A Formula for the Volume Spanned

by Three Vectors ........ . 21.12 Lines ............... . 21.13 Projection of a Point onto a Line 21.14 Planes .............. . 21.15 The Intersection of a Line and a Plane 21.16 Two Intersecting Planes Determine a Line 21.17 Projection of a Point onto a Plane 21.18 Distance from a Point to a Plane .... . 21.19 Rotation Around a Given Vector .... . 21.20 Lines and Planes Through the Origin Are Subspaces 21.21 Systems of 3 Linear Equations in 3 Unknowns .. 21.22 Solving a 3 x 3-System by Gaussian Elimination 21.23 3 x 3 Matrices: Sum, Product and Transpose 21.24 Ways of Viewing a System of Linear Equations 21.25 Non-Singular Matrices 21.26 The Inverse of a Matrix . . . . . . . 21.27 Different Bases ........... . 21.28 Linearly Independent Set of Vectors 21.29 Orthogonal Matrices ........ . 21.30 Linear Transformations Versus Matrices

XXXI

303 303 303 305 305 305 306 306 307 308 309

313 313 315 315 316 316 317 319 320

320 321

322 323 324 324 326 327 328 328 329 330 330 332 333 335 336 336 337 337 338 338

XXXII Contents Volume 1

21.31 The Scalar Product Is Invariant Under Orthogonal Transformations

21.32 Looking Ahead to Functions f : ][{3 ---. ][(3

22 Complex Numbers 22.1 Introduction .. 22.2 Addition and Multiplication 22.3 The Triangle Inequality .. 22.4 Open Domains ...... . 22.5 Polar Representation of Complex Numbers. 22.6 Geometrical Interpretation of Multiplication . 22.7 Complex Conjugation ........ . 22.8 Division ................ . 22.9 The Fundamental Theorem of Algebra 22.10 Roots ................. . 22.11 Solving a Quadratic Equation w 2 + 2bw + c = 0 . 22.12 Gosta Mittag-Leffler ............... .

23 The Derivative 23.1 Rates of Change 23.2 Paying Taxes . . 23.3 Hiking...... 23.4 Definition of the Derivative

339 340

345 345 346 347 348 348 348 349 350 350 351 351 352

355 355 356 359 359

23.5 The Derivative of a Linear Function Is Constant 362 23.6 The Derivative of x 2 Is 2x . . . . . . 362 23.7 The Derivative of xn Is nxn- 1 . • . . 364 23.8 The Derivative of ~ Is - ;2 for x =I- 0 365 23.9 The Derivative as a Function .... 365 23.10 Denoting the Derivative of f(x) by Df(x) 365 23.11 Denoting the Derivative of f(x) by ~ . . 367 23.12 The Derivative as a Limit of Difference Quotients. 367 23.13 How to Compute a Derivative? . . . . . . . . . . . 369 23.14 Uniform Differentiability on an Interval ...... 371 23.15 A Bounded Derivative Implies Lipschitz Continuity 372 23.16 A Slightly Different Viewpoint. 374 23.17 Swedenborg . . . . . . . . . . . 374

24 Differentiation Rules 24.1 Introduction ... 24.2 The Linear Combination Rule 24.3 The Product Rule 24.4 24.5 24.6 24.7

The Chain Rule . . . . . . . . The Quotient Rule . . . . . . . . . . . . . . Derivatives of Derivatives: f(n) = D n f = '£f One-Sided Derivatives . . . . . . . . . . . . .

377 377 378 379 380 381 382 383

Contents Volume 1

24.8 Quadratic Approximation ...... . 24.9 The Derivative of an Inverse Function 24.10 Implicit Differentiation . 24.11 Partial Derivatives 24.12 A Sum Up So Far.

25 Newton's Method 25.1 Introduction. 25.2 Convergence of Fixed Point Iteration 25.3 Newton's Method .......... . 25.4 Newton's Method Converges Quadratically. 25.5 A Geometric Interpretation of Newton's Method 25.6 What Is the Error of an Approximate Root? . 25.7 Stopping Criterion .......... . 25.8 Globally Convergent Newton Methods ....

26 Galileo, Newton, Hooke, Malthus and Fourier 26.1 Introduction....... 26.2 Newton's Law of Motion 26.3 Galileo's Law of Motion 26.4 Hooke's Law ...... . 26.5 Newton's Law plus Hooke's Law 26.6 Fourier's Law for Heat Flow ... 26.7 Newton and Rocket Propulsion . 26.8 Malthus and Population Growth 26.9 Einstein's Law of Motion. 26.10 Summary

References

Index

XXXIII

384 387 388 389 390

393 393 393 394 395 396 397 400 400

403 403 404 404 407 408 409 410 412 413 414

417

419

Contents Volume 3

Calculus in Several Dimensions 787

54 Vector-Valued Functions of Several Real Variables 789 54.1 Introduction............... 789 54.2 Curves in IRn .............. 790 54.3 Different Parameterizations of a Curve 791 54.4 Surfaces in IRn, n ~ 3 . . . . . . . . . . 792 54.5 Lipschitz Continuity . . . . . . . . . . 792 54.6 Differentiability: Jacobian, Gradient and Tangent 794 54.7 The Chain Rule . . . . . . . . . . . . . . . . . . 798 54.8 The Mean Value Theorem . . . . . . . . . . . . 799 54.9 Direction of Steepest Descent and the Gradient 800 54.10 A Minimum Point Is a Stationary Point 802 54.11 The Method of Steepest Descent 802 54.12 Directional Derivatives . . . . . . 803 54.13 Higher Order Partial Derivatives 804 54.14 Taylor's Theorem. . . . . . . . . 805 54.15 The Contraction Mapping Theorem. 806 54.16 Solving f(x) = 0 with f : IRn --+ IRn 808 54.17 The Inverse Function Theorem 809 54.18 The Implicit Function Theorem. . 810 54.19 Newton's Method. . . . . . . . . . 811 54.20 Differentiation Under the Integral Sign. 812

XXXVI Contents Volume 3

55 Level Curves/Surfaces and the Gradient 55.1 Level Curves .......... . 55.2 Local Existence of Level Curves . 55.3 Level Curves and the Gradient . 55.4 Level Surfaces .......... . 55.5 Local Existence of Level Surfaces 55.6 Level Surfaces and the Gradient.

815 815 817 817 818 819 819

56 Linearization and Stability of Initial Value Problems 823 56.1 Introduction............... 823 56.2 Stationary Solutions . . . . . . . . . . . . . 56.3 Linearization at a Stationary Solution . . . 56.4 Stability Analysis when j'(u) Is Symmetric 56.5 Stability Factors ........... . 56.6 Stability of Time-Dependent Solutions 56.7 Sum Up ................ .

57 Adaptive Solvers for IVPs 57.1 Introduction .............. . 57.2 The cG(l) Method .......... . 57.3 Adaptive Time Step Control for cG(l) 57.4 Analysis of cG(l) for a Linear Scalar IVP 57.5 Analysis of cG(l) for a General IVP ... 57.6 Analysis of Backward Euler for a General IVP . 57.7 Stiff Initial Value Problems ......... . 57.8 On Explicit Time-Stepping for Stiff Problems

58 Lorenz and the Essence of Chaos* 58.1 Introduction ........... . 58.2 The Lorenz System . . . . . . . . 58.3 The Accuracy of the Computations . 58.4 Computability of the Lorenz System 58.5 The Lorenz Challenge . . . . . . . .

59 The Solar System* 59.1 Introduction .. 59.2 Newton's Equation . 59.3 Einstein's Equation . 59.4 The Solar System as a System of ODEs 59.5 Predictability and Computability . . . . 59.6 Adaptive Time-Stepping . . . . . . . . . 59.7 Limits of Computability and Predictability

824 824 825 826 829 829

831 831 832 834 834 837 838 840 842

849 849 850 852 854 856

859 859 862 863 864 867 868 869

60 Optimization 60.1 Introduction ....... . 60.2 Sorting if n Is Finite . . . 60.3 What if n Is Not Finite? . 60.4 Existence of a Minimum Point.

Contents Volume 3

60.5 The Derivative Is Zero at an Interior Minimum Point . 60.6 The Role of the Hessian .......... . 60.7 Minimization Algorithms: Steepest Descent 60.8 Existence of a Minimum Value and Point . 60.9 Existence of Greatest Lower Bound ..... 60.10 Constructibility of a Minimum Value and Point 60.11 A Decreasing Bounded Sequence Converges!

61 The Divergence, Rotation and Laplacian 61.1 Introduction ............. . 61.2 The Case of]R2 . . . . . . . . . . . . 61.3 The Laplacian in Polar Coordinates. 61.4 Some Basic Examples ....... . 61.5 The Laplacian Under Rigid Coordinate Transformations 61.6 The Case of]R3 . . . . . . . . . . . . . . 61.7 Basic Examples, Again ......... . 61.8 The Laplacian in Spherical Coordinates

62 Meteorology and Coriolis Forces* 62.1 Introduction ....................... . 62.2 A Basic Meteorological Model . . . . . . . . . . . . . . 62.3 Rotating Coordinate Systems and Coriolis Acceleration

63 Curve Integrals 63.1 Introduction ......... . 63.2 The Length of a Curve in ]R2 63.3 Curve Integral. . . . . . 63.4 Reparameterization.... 63.5 Work and Line Integrals . 63.6 Work and Gradient Fields 63.7 Using the Arclength as a Parameter 63.8 The Curvature of a Plane Curve 63.9 Extension to Curves in ]Rn

64 Double Integrals 64.1 Introduction ..................... . 64.2 Double Integrals over the Unit Square ...... . 64.3 Double Integrals via One-Dimensional Integration. 64.4 Generalization to an Arbitrary Rectangle .. 64.5 Interpreting the Double Integral as a Volume . . .

XXXVII

871 871 872 873 874 874 878 878 879 881 882 882

885 885 886 887 888 888 889 890 891

893 893 894 895

899 899 899 901 902 903 904 905 906 907

911 911 912 915 918 918

XXXVIII Contents Volume 3

64.6 Extension to General Domains ...... 919 64.7 Iterated Integrals over General Domains . 921 64.8 The Area of a Two-Dimensional Domain. 922 64.9 The Integral as the Limit of a General Riemann Sum . 922 64.10 Change of Variables in a Double Integral. . . . . . . . 923

65 Surface Integrals 65.1 Introduction ................... . 65.2 Surface Area .................. . 65.3 The Surface Area of a the Graph of a Function

of Two Variables . . . . . . . . . . 65.4 Surfaces of Revolution . . . . . . . 65.5 Independence of Parameterization. 65.6 Surface Integrals ......... . 65.7 Moment of Inertia of a Thin Spherical Shell

66 Multiple Integrals 66.1 Introduction ................. . 66.2 Triple Integrals over the Unit Cube .... . 66.3 Triple Integrals over General Domains in ~3 66.4 The Volume of a Three-Dimensional Domain 66.5 Triple Integrals as Limits of Riemann Sums 66.6 Change of Variables in a Triple Integral 66.7 Solids of Revolution .... 66.8 Moment of Inertia of a Ball ...... .

67 Gauss' Theorem and Green's Formula in ~2 67.1 Introduction ......... . 67.2 The Special Case of a Square 67.3 The General Case. . . . . . .

68 Gauss' Theorem and Green's Formula in ~3 68.1 George Green (1793-1841) ......... .

929 929 929

932 932 933 934 935

939 939 939 940 941 942 943 945 946

949 949 950 950

959 962

69 Stokes' Theorem 965 69.1 Introduction................... 965 69.2 The Special Case of a Surface in a Plane . . . 967 69.3 Generalization to an Arbitrary Plane Surface 968 69.4 Generalization to a Surface Bounded by a Plane Curve 969

70 Potential Fields 973 70.1 Introduction................. 973 70.2 An Irrotational Field Is a Potential Field. 974 70.3 A Counter-Example for a Non-Convex n . 976

Contents Volume 3

71 Center of Mass and Archimedes' Principle* 71.1 Introduction ..... . 71.2 Center of Mass . . . . . . . 71.3 Archimedes' Principle ... 71.4 Stability of Floating Bodies

72 Newton's Nightmare*

73 Laplacian Models 73.1 Introduction. 73.2 Heat Conduction . . 73.3 The Heat Equation . 73.4 Stationary Heat Conduction: Poisson's Equation 73.5 Convection-Diffusion-Reaction . 73.6 Elastic Membrane ................. . 73.7 Solving the Poisson Equation . . . . . . . . . . . 73.8 The Wave Equation: Vibrating Elastic Membrane . 73.9 Fluid Mechanics ... 73.10 Maxwell's Equations ............ . 73.11 Gravitation ................. . 73.12 The Eigenvalue Problem for the Laplacian. 73.13 Quantum Mechanics ............ .

74 Chemical Reactions* 74.1 Constant Temperature 74.2 Variable Temperature 74.3 Space Dependence

75 Calculus Tool Bag II 75.1 Introduction ... 75.2 Lipschitz Continuity 75.3 Differentiability... 75.4 The Chain Rule . . . 75.5 Mean Value Theorem for f : ~n -+ ~ .

75.6 A Minimum Point Is a Stationary Point 75.7 Taylor's Theorem ....... . 75.8 Contraction Mapping Theorem 75.9 Inverse Function Theorem . 75.10 Implicit Function Theorem. 75.11 Newton's Method ... 75.12 Differential Operators 75.13 Curve Integrals .. 75.14 Multiple Integrals .. . 75.15 Surface Integrals .. . 75.16 Green's and Gauss' Formulas 75.17 Stokes' Theorem ...... .

XXXIX

977 977 978 981 983

987

993 993 993 996 997 999 999

1001 1003 1003 1009 1013 1017 1019

1025 1025 1028 1028

1031 1031 1031 1031 1032 1032 1032 1032 1033 1033 1033 1033 1033 1034 1035 1035 1036 1036

XL Contents Volume 3

76 Piecewise Linear Polynomials in ]R2 and ]R3 1037 76.1 Introduction............ 1037 76.2 Triangulation of a Domain in ]R2 1038 76.3 Mesh Generation in]R3 . . . 1041 76.4 Piecewise Linear Functions 1042 76.5 Max-Norm Error Estimates 1044 76.6 Sobolev and his Spaces. 1047 76.7 Quadrature in]R2 . . . . . . 1048

77 FEM for Boundary Value Problems in ]R2 and ]R3 1051 77.1 Introduction............. 1051 77.2 Richard Courant: Inventor of FEM 1052 77.3 Variational Formulation 1053 77.4 The cG(I) FEM. . . . . . . 1053 77.5 Basic Data Structures . . . 1059 77.6 Solving the Discrete System 1060 77.7 An Equivalent Minimization Problem. 1061 77.8 An Energy Norm a Priori Error Estimate 1062 77.9 An Energy Norm a Posteriori Error Estimate 1063 77.10 Adaptive Error Control ............ 1065 77.11 An Example. . . . . . . . . . . . . . . . . . . 1067 77.12 Non-Homogeneous Dirichlet Boundary Conditions. 1068 77.13 An L-shaped Membrane . . . . . . . . . . . 1068 77.14 Robin and Neumann Boundary Conditions. . . . 1070 77.15 Stationary Convection-DifFusion-Reaction . . . . 1072 77.16 Time-Dependent Convection-Diffusion-Reaction . 1073 77.17 The Wave Equation 1074 77.18 Examples . . . . . . . . . . . . . . . . . . . . . . 1074

78 Inverse Problems 1079 78.1 Introduction........................ 1079 78.2 An Inverse Problem for One-Dimensional Convection. 1081 78.3 An Inverse Problem for One-Dimensional Diffusion 1083 78.4 An Inverse Problem for Poisson's Equation 1085 78.5 An Inverse Problem for Laplace's Equation 1088 78.6 The Backward Heat Equation . . . . . . . . 1089

79 Optimal Control 1093 79.1 Introduction.............. 1093 79.2 The Connection Between ~: and ~~ 1095

80 Differential Equations Tool Bag 1097 80.1 Introduction............... 1097 80.2 The Equation u'(x) = ..\(x)u(x) . . . . 1098 80.3 The Equation u'(x) = ..\(x)u(x) + f(x) 1098

Contents Volume 3 XLI

80.4 80.5 80.6 80.7 80.8 80.9 80.10 80.11 80.12

The Differential Equation E~=o akDku(x) = 0 The Damped Linear Oscillator ..... The Matrix Exponential . . . . . . . . . Fundamental Solutions of the Laplacian The Wave Equation in 1d . . . . . . . . Numerical Methods for IVPs ..... . cg( 1) for Convection-Diffusion-Reaction Svensson's Formula for Laplace's Equation. Optimal Control .............. .

81 Applications Tool Bag 81.1 Introduction ........ . 81.2 Malthus' Population Model 81.3 The Logistics Equation. . . 81.4 Mass-Spring-Dashpot System 81.5 LCR-Circuit.......... 81.6 Laplace's Equation for Gravitation 81. 7 The Heat Equation . . . . . . . 81.8 The Wave Equation ..... . 81.9 Convection-Diffusion-Reaction. 81.10 Maxwell's Equations ..... . 81.11 The Incompressible Navier-Stokes Equations. 81.12 Schrodinger's Equation ............ .

1098 1099 1099 1100 1100 1100 1101 1101 1101

1103 1103 1103 1103 1103 1104 1104 1104 1104 1104 1105 1105 1105

82 Analytic Functions 1107 82.1 The Definition of an Analytic Function. . . . . . . 1107 82.2 The Derivative as a Limit of Difference Quotients . 1109 82.3 Linear Functions Are Analytic. . . . . . . . . . . . 1109 82.4 The Function J(z) = z2 Is Analytic. . . . . . . . . 1109 82.5 The Function J(z) = zn Is Analytic for n = 1,2,... 1110 82.6 Rules of Differentiation. . . . . . 1110 82.7 The Function J(z) = z-n .............. 1110 82.8 The Cauchy-Riemann Equations .......... 1110 82.9 The Cauchy-Riemann Equations and the Derivative. 1112 82.10 The Cauchy-Riemann Equations in Polar Coordinates 1113 82.11 The Real and Imaginary Parts of an Analytic Function 1113 82.12 Conjugate Harmonic Functions . . . . . . . . . . . 1113 82.13 The Derivative of an Analytic Function Is Analytic 1114 82.14 Curves in the Complex Plane . . . . . . . . 1114 82.15 Conformal Mappings . . . . . . . . . . . . . 1116 82.16 Translation-rotation-expansion/contraction 1117 82.17 Inversion. . . . . . . . . . . . . . . . . . . . 1117 82.18 Mobius Transformations . . . . . . . . . . . 1118 82.19 w = zl/2, W = eZ , W = log(z) and W = sin(z). 1119 82.20 Complex Integrals: First Shot . . . . . . . . . 1121

XLII Contents Volume 3

82.21 Complex Integrals: General Case . . . . . 1122 82.22 Basic Properties of the Complex Integral. 1123 82.23 Taylor's Formula: First Shot. . . . 1123 82.24 Cauchy's Theorem . . . . . . . . . 1124 82.25 Cauchy's Representation Formula . 1125 82.26 Taylor's Formula: Second Shot .. 1127 82.27 Power Series Representation of Analytic Functions 1128 82.28 Laurent Series. . . . . . . . . . . . . . 1130 82.29 Residue Calculus: Simple Poles . . . . 1131 82.30 Residue Calculus: Poles of Any Order 1133 82.31 The Residue Theorem . . . . . . . . . 1133 82.32 Computation of Jo27r R(sin(t), cos(t)) dt 1134 82.33 Computation of J~oo ~ dx . . . . . . 1135

82.34 Applications to Potential Theory in IR2 . 1136

83 Fourier Series 1143 83.1 Introduction............... 1143 83.2 Warm Up I: Orthonormal Basis in en 1146 83.3 Warm Up II: Series . . . . . . . . . . . 1146 83.4 Complex Fourier Series. . . . . . . . . 1147 83.5 Fourier Series as an Orthonormal Basis Expansion 1148 83.6 Truncated Fourier Series and Best L2-Approximation . 1149 83.7 Real Fourier Series . . . . . . . . . . . . 1149 83.8 Basic Properties of Fourier Coefficients . 1152 83.9 The Inversion Formula. . . . . . . . 1157 83.10 Parseval's and Plancherel's Formulas 1159 83.11 Space Versus Frequency Analysis 1160 83.12 Different Periods . . . . . . . . . . . 1161 83.13 Weierstrass Functions ........ 1161 83.14 Solving the Heat Equation Using Fourier Series 1162 83.15 Computing Fourier Coefficients with Quadrature 1164 83.16 The Discrete Fourier Transform. . . . . . . . . . 1164

84 Fourier Transforms 1167 84.1 Basic Properties of the Fourier Transform . . . . . 1169 84.2 The Fourier Transform J(~) Tends to 0 as I~I ---7 00 1171 84.3 Convolution...... 1171 84.4 The Inversion Formula . . . . . . . . . . . . . . . . 1171 84.5 Parseval's Formula . . . . . . . . . . . . . . . . . . 1173 84.6 Solving the Heat Equation Using the Fourier Transform 1173 84.7 Fourier Series and Fourier Transforms 1174 84.8 The Sampling Theorem ... 1175 84.9 The Laplace Transform. . . . 1176 84.10 Wavelets and the Haar Basis 1177

Contents Volume 3 XLIII

85 Analytic Functions Tool Bag 1181 85.1 Differentiability and analyticity . . . . . . . . . . . .. 1181 85.2 The Cauchy-Riemann Equations ............ 1181 85.3 The Real and Imaginary Parts of an Analytic Function 1182 85.4 Conjugate Harmonic Functions . . . . . . . . . . . . 1182 85.5 Curves in the Complex Plane . . . . . . . . . . . . . 1182 85.6 An Analytic Function Defines a Conformal Mapping 1183 85.7 Complex Integrals ......... 1183 85.8 Cauchy's Theorem . . . . . . . . . 1183 85.9 Cauchy's Representation Formula . 1183 85.10 Taylor's Formula . . . 1184 85.11 The Residue Theorem . . . . . . . 1184

86 Fourier Analysis Tool Bag 1185 86.1 Properties of Fourier Coefficients 1185 86.2 Convolution.......... 1185 86.3 Fourier Series Representation 1186 86.4 Parseval's Formula . . . . . 1186 86.5 Discrete Fourier Transforms . 1186 86.6 Fourier Transforms . . . . . . 1186 86.7 Properties of Fourier Transforms 1187 86.8 The Sampling Theorem ..... 1187

87 Incompressible Navier-Stokes: Quick and Easy 1189 87.1 Introduction................... 1189 87.2 The Incompressible Navier-Stokes Equations. 1190 87.3 The Basic Energy Estimate for Navier-Stokes 1191 87.4 Lions and his School . . . . . . . . . . . . 1192 87.5 Turbulence: Lipschitz with Exponent 1/3? 1193 87.6 Existence and Uniqueness of Solutions 1194 87.7 Numerical Methods. . . . . . . . . . 1194 87.8 The Stabilized cG(I)dG(O) Method. 1195 87.9 The cG(I)cG(I) Method. . . . 1196 87.10 The cG(I)dG(I) Method. . . . 1197 87.11 Neumann Boundary Conditions 1197 87.12 Computational Examples 1199

References 1205

Index 1207