Embed Size (px)
Citation preview
Applied Mathematics: Body and Soul
Springer-Verlag Berlin Heidelberg GmbH
K. Eriksson • D. Estep· C.Johnson
Applied Mathematics: Body and Soul [VOlUME 3]
Calculus in Several Dimensions
t Springer
Kenneth Eriksson Claes Johnson
Chalmers University of Technology Department ofMathematics 41296 Göteborg, Sweden e-mail: [email protected]
Cataloging-in-Publieation Data applied for
Donald Estep
Colorado State University Department ofMathematics FortCollins, CO 80523-1874 USA e-mail: [email protected]
A eatalog record for this book is available from the Library of Congress.
Bibliographie information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publieation in the Deutsche Nationalbibliografie; detailed bibliographie 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 978-3-642-05660-4 ISBN 978-3-662-05800-8 (eBook)
DOI 10.1007/978-3-662-05800-8
This work is subjeet to copyright. All rights are reserved, whether thewhole or part of the material is eoneerned, speeifieally the rights of translation, reprinting, reuse of illustrations, reeitation, broadeasting, reproduetion on microfilm or in any other way, and storage in data banks. Duplieation of this publieation or parts thereof is permitted only under the provisions ofthe German Copyright Law ofSeptember 9, 1965, in its eurrent version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg New York 2004
Originally published by Springer-Verlag Berlin Heidelberg in 2004.
Softcover reprint of the hardcover 1 st edition 2004
The use of general deseriptive names, registered names, trademarks ete. in this publieation does not imply, even in the absence of a speeifie 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 Mathematies, Chalmers University ofTeclmology Typesetting: Le-TeX Jelonek, Schmidt & Vöckler GbR, Leipzig Printed on acid-free pape SPIN 10999530 46/3111ek-5 4 321
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
ladmit 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 techIliques lie the new fields of Computational Mathematical Modeling (CMM), including Computational Mechanics, Physics, Fluid Dynamies, Electromagnetics and Chemistry, all based on solving systems of differential equations using computers, combined with geometrie modelingjComputer Aided Design (CAD). Computational modeling is also finding revolutionary new applications in biology, medicine, environment al 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 CMMjCAD 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 aga 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-lU covering the basics of calculus and linear algebra. AM IV- offers a continuation with aseries 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 Laboratory) , 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 fuH 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 undergraduatejbeginning 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 literat ure, 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 weIl 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 algebraicj differential equations.
• Emphasis is put on giving the student asolid understanding of basic mathematical concepts such as real numbers, Cauchy sequences, Lipschitz continuity, and constructive tools for solving algebraicjdifferential 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 1-111 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 computationallevel), 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 refiect 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 asound 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 weIl. 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: areal 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 j'(x) if
lim f(x) - ~(x) x--+x X - X
exists and equals j'(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 j'(x) if there is a constant Kf(x) such that for all x elose to x
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 pro ces ses 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 elose 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 Hölder 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 areal 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 formalistsjlogicists: 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 apart 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 li ne 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 go es 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 logicalline 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 logicalline 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.phi.chalmers.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 ho pe 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 Bergström, Niklas Eriksson, Johan Hoffman, Mats Larson, Stig Larsson, Märten 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 an for you body and soul (Green, Body and Soul)
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 ............. .
54.3 Different Parameterizations of a Curve 54.4 Surfaces in IRn , n 2" 3 . . . . . . . . . . 54.5 Lipschitz Continuity . . . . . . . . . . 54.6 Differentiability: Jacobian, Gradient and Tangent 54.7 The Chain Rule ................. . 54.8 The Mean Value Theorem ........... . 54.9 Direction of Steepest Descent and the Gradient 54.10 A Minimum Point Is a Stationary Point 54.11 The Method of Steepest Descent 54.12 Directional Derivatives ..... . 54.13 Higher Order Partial Derivatives 54.14 Taylor's Theorem ........ . 54.15 The Contraction Mapping Theorem. 54.16 Solving f(x) = 0 with f : IRn ----+ IRn
54.17 The Inverse Function Theorem 54.18 The Implicit Function Theorem .. 54.19 Newton's Method ......... . 54.20 Differentiation Under the Integral Sign
790 791 792 792 794 798 799 800 802 802 803 804 805 806 808 809 810 811 812
XVIII Contents Volume 3
55 Level CurvesjSurfaces and the Gradient 55.1 Level Curves .......... . 55.2 Loeal Existenee of Level Curves . 55.3 Level Curves and the Gradient . 55.4 Level Surfaees . . . . . . . . . . . 55.5 Loeal Existenee of Level Surfaees 55.6 Level Surfaees and the Gradient.
815 815 817 817 818 819 819
56 Linearization and Stability of Initial Value Problems 823 56.1 Introduetion............... 823 56.2 Stationary Solutions . . . . . . . . . . . . . 824 56.3 Linearization at a Stationary Solution . . . 824 56.4 Stability Analysis when j'(u) Is Symmetrie 825 56.5 Stability Faetors ............ 826 56.6 Stability of Time-Dependent Solutions 829 56.7 Sum Up . . . . . . . . . . . . . . . . . 829
57 Adaptive Solvers for IVPs 831 57.1 Introduction............... 831 57.2 The eG(I) Method . . . . . . . . . . . 832 57.3 Adaptive Time Step Control for eG(I) 834 57.4 Analysis of eG(I) for a Linear Sealar IVP 834 57.5 Analysis of eG(I) for a General IVP ... 837 57.6 Analysis of Baekward Euler for a General IVP . 838 57.7 Stiff Initial Value Problems .......... 840 57.8 On Explieit Time-Stepping for Stiff Problems 842
58 Lorenz and the Essence of Chaos* 849 58.1 Introduetion............ 849 58.2 The Lorenz System. . . . . . . . 850 58.3 The Aeeuraey of the Computations 852 58.4 Computability of the Lorenz System 854 58.5 The Lorenz Challenge . . . . . . . . 856
59 The Solar System* 859 59.1 Introduetion.. 859 59.2 Newton's Equation . 862 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
863 864 867 868 869
Contents Volume 3 XIX
60 Optimization 871 60.1 Introduction....... 871 60.2 Sorting if D Is Finite . . 872 60.3 What if D Is Not Finite? 873 60.4 60.5 60.6 60.7 60.8 60.9
Existence of a Minimum Point . The Derivative Is Zero at an Interior Minimum Point . The Role of the Hessian . . . . . . . . . . . Minimization Algorithms: Steepest Descent Existence of a Minimum Value and Point 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 jR2 ........... .
874 874 878 878 879 881 882 882
885 885 886
61.3 The Laplacian in Polar Coordinates. 887 61.4 Some Basic Examples ........ 888 61.5 The Laplacian Under Rigid Coordinate Transformations 888 61.6 The Case of jR3 . . . . . . . . . . . . . . 889 61. 7 Basic Examples, Again . . . . . . . . . . 890 61.8 The Laplacian in Spherical Coordinates 891
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 jR2
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 jRn
893 893 894 895
899 899 899 901 902 903 904 905 906 907
64 Double Integrals 911 64.1 Introduction...................... 911 64.2 Double Integrals over the Unit Square . . . . . . . 912 64.3 Double Integrals via One-Dimensional Integration. 915 64.4 Generalization to an Arbitrary Rectangle 918 64.5 Interpreting the Double Integral as a Volume . . . 918
XX Contents Volume 3
64.6 64.7 64.8
Extension to General Domains ..... Iterated Integrals over General Domains The Area of a Two-Dimensional Domain
919 921 922
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 65.3
65.4 65.5 65.6 65.7
Surface Area .................. . The Surface Area of a the Graph of a Function of Two Variables ......... . Surfaces of Revolution . . . . . . . Independence of Parameterization . Surface Integrals ......... . 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 ]R3
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 ]R2
67.1 67.2 67.3
Introduction . . . . . . . . . . The Special Case of a Square The General Case . . . . . . .
68 Gauss' Theorem and Green's Formula in ]R3
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 11 75.1 Introduction ... 75.2 Lipschitz Continuity 75.3 Differentiability .. . 75.4 The Chain Rule .. . 75.5 Mean Value Theorem for f : ]Rn --->]R
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 .. . . . . .
XXI
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
XXII Contents Volume 3
76 Piecewise Linear Polynomials in ]R2 and ]R3 76.1 Introduction............ 76.2 Triangulation of a Domain in ]R2 76.3 Mesh Generation in ]R3 . . . 76.4 76.5 76.6 76.7
Piecewise Linear Functions Max-Norm Error Estimates Sobolev and his Spaces . Quadrature in]R2 ..... .
1037 1037 1038 1041 1042 1044 1047 1048
77 FEM for Houndary Value Problems in ]R2 and ]R3 1051 77.1 Introduction............. 1051 77.2 Richard Courant: Inventor of FEM 1052 77.3 77.4 77.5 77.6 77.7 77.8 77.9 77.10 77.11 77.12 77.13 77.14 77.15 77.16 77.17 77.18
Variational Formulation The cG(l) FEM ...... . Basic Data Structures . . . Solving the Discrete System An Equivalent Minimization Problem. An Energy Norm aPriori Error Estimate An Energy Norm aPosteriori Error Estimate Adaptive Error Control ........... . An Example .................. . Non-Homogeneous Dirichlet Boundary Conditions . An L-shaped Membrane . . . . . . . . . . . Robin and Neumann Boundary Conditions ... Stationary Convection-Diffusion-Reaction . . . Time-Dependent Convection-Diffusion-Reaction The Wave Equation Examples ..
78 Inverse Problems 78.1 Introduction. 78.2 78.3 78.4 78.5 78.6
An Inverse Problem for One-Dimensional Convection An Inverse Problem for One-Dimensional Diffusion An Inverse Problem for Poisson's Equation An Inverse Problem for Laplace's Equation The Backward Heat Equation . . . . . . . .
79 Optimal Control 79.1 Introduction ............. . 79.2 The Connection Between ~; and ~~
80 Differential Equations Tool Hag 80.1 Introduction .............. . 80.2 The Equation u'(x) = A(x)u(x) ... . 80.3 The Equation u'(x) = A(x)u(x) + f(x)
1053 1053 1059 1060 1061 1062 1063 1065 1067 1068 1068 1070 1072 1073 1074 1074
1079 1079 1081 1083 1085 1088 1089
1093 1093 1095
1097 1097 1098 1098
Contents Volume 3
80.4 The Differential Equation L~=o akDku(x) = 0 80.5 The Damped Linear Oseillator .... . 80.6 The Matrix Exponential ........ . 80.7 Fundamental Solutions of the Laplaeian 80.8 The Wave Equation in 1d ....... . 80.9 Numerieal Methods for IVPs ..... . 80.10 eg(l) for Conveetion-Diffusion-Reaetion 80.11 Svensson's Formula for Laplaee's Equation . 80.12 Optimal Contral .............. .
81 Applications Tool Hag 8l.1 Introduetion ........ . 8l.2 Malthus' Population Model 8l.3 The Logistics Equation ... 8l.4 Mass-Spring-Dashpot System 8l.5 LCR-Cireuit... ...... . 8l.6 8l.7 8l.8 8l.9 8l.10 8l.11 8l.12
Laplaee's Equation for Gravitation The Heat Equation . . . . . . . The Wave Equation ..... . Conveetion-Diffusion-Reaction . Maxwell's Equations . . . . . . The Ineompressible Navier-Stokes Equations . Sehrödinger's Equation ............ .
82 Analytic Functions 82.1 The Definition of an Analytie Function ..... . 82.2 The Derivative as a Limit of Differenee Quotients 82.3 Linear Functions Are Analytie. . . . . . . ... . 82.4 The Funetion j(z) = z2 Is Analytie ....... . 82.5 The Function j(z) = zn Is Analytie for n = 1,2, .. . 82.6 Rules of Differentiation ..... . 82.7 The Funetion j(z) = z-n 82.8 The Cauehy-Riemann Equations 82.9 The Cauehy-Riemann Equations and the Derivative. 82.10 The Cauehy-Riemann Equations in Polar Coordinates 82.11 The Real and Imaginary Parts of an Analytie Funetion 82.12 Conjugate Harmonie Funetions .......... . 82.13 The Derivative of an Analytie Function Is Analytie 82.14 Curves in the Complex Plane ....... . 82.15 Conformal Mappings ............ . 82.16 Translation-rotation-expansion/ eontraction 82.17 Inversion ................... . 82.18 Möbius Transformations .......... . 82.19 w = zl/2, W = eZ , W = log(z) and W = sin(z) . 82.20 Complex Integrals: First Shot ........ .
XXIII
1098 1099 1099 1100 1100 1100 1101 1101 1101
1103 1103 1103 1103 1103 1104 1104 1104 1104 1104 1105 1105 1105
1107 1107 1109 1109 1109 1110 1110 1110 1110 1112 1113 1113 1113 1114 1114 1116 1117 1117 1118 1119 1121
XXIV Contents Volume 3
82.21 Complex Integrals: General Case ..... 82.22 Basic Properties of the Complex Integral. 82.23 Taylor's Formula: First Shot .. . 82.24 Cauchy's Theorem ....... . 82.25 Cauchy's Representation Formula 82.26 Taylor's Formula: Second Shot 82.27 Power Series Representation of Analytic Functions 82.28 Laurent Series ............. . 82.29 Residue Calculus: Simple Poles ... . 82.30 Residue Calculus: Poles of Any Order 82.31 The Residue Theorem ........ . 82.32 Computation of J~7r R(sin(t),cos(t))dt 82 33 C . f Joo p(x) d . omputatlOn 0 -00 q(x) x . .... . 82.34 Applications to Potential Theory in ]R2
83 Fourier Series 83.1 Introduction .............. . 83.2 Warm Up I: Orthonormal Basis in en
83.3 Warm Up II: Series .......... . 83.4 Complex Fourier Series ........ . 83.5 Fourier Series as an Orthonormal Basis Expansion 83.6 Truncated Fourier Series and Best L2-Approximation . 83.7 Real Fourier Series . . . . . . . . . . . 83.8 Basic Properties of Fourier Coefficients 83.9 The Inversion Formula ....... . 83.10 Parseval's and Plancherel's Formulas 83.11 Space Versus Frequency Analysis 83.12 Different Per iods .......... . 83.13 Weierstrass Functions ....... . 83.14 Solving the Heat Equation Using Fourier Series 83.15 Computing Fourier Coefficients with Quadrature 83.16 The Discrete Fourier Transform ......... .
84 Fourier Transforms 84.1 Basic Properties of the Fourier Transform ..... 84.2 The Fourier Transform [(0 Tends to 0 as I~I -> 00
84.3 Convolution...... 84.4 84.5 84.6 84.7 84.8 84.9 84.10
The Inversion Formula . . . . . . . . . . . . . . . . Parseval's Formula ................. . Solving the Heat Equation Using the Fourier Transform Fourier Se ries and Fourier Transforms The Sampling Theorem ... The Laplace Transform . . . . Wavelets and the Haar Basis
1122 1123 1123 1124 1125 1127 1128 1130 1131 1133 1133 1134
1135
1136
1143 1143 1146 1146 1147 1148 1149 1149 1152 1157 1159 1160 1161 1161 1162 1164 1164
1167 1169 1171 1171 1171 1173 1173 1174 1175 1176 1177
Contents Volume 3
85 Analytic Functions Tool Hag 85.1 Differentiability and Analyticity ............ . 85.2 The Cauchy-Riemann Equations ........... . 85.3 The Real and Imaginary Parts of an Analytic Function 85.4 Conjugate Harmonie Functions ........... . 85.5 Curves in the Complex Plane ............ . 85.6 An Analytic Function Defines a Conformal Mapping 85.7 Complex Integrals ........ . 85.8 Cauchy's Theorem ........ . 85.9 Cauchy's Representation Formula . 85.10 Taylor's Formula ... 85.11 The Residue Theorem .
86 Fourier Analysis Tool Hag 86.1 Properties of Fourier Coefficients 86.2 Convolution ......... . 86.3 86.4 86.5 86.6 86.7 86.8
Fourier Series Representation Parseval's Formula ..... Discrete Fourier Transforms . Fourier Transforms . . . . . . Properties of Fourier Transforms The Sampling Theorem .....
87 Incompressible Navier-Stokes: Quick and Easy 87.1 Introduction .................. . 87.2 87.3 87.4 87.5 87.6 87.7 87.8 87.9 87.10 87.11 87.12
The Incompressible Navier-Stokes Equations . The Basic Energy Estimate for Navier-Stokes Lions and his School . . . . . . . . . . . . Turbulence: Lipschitz with Exponent 1/37 Existence and Uniqueness of Solutions Numerical Methods ........ . The Stabilized cG(l)dG(O) Method The cG(l)cG(l) Method ... . The cG(l)dG(l) Method ... . Neumann Boundary Conditions Computational Examples
References
Index
xxv
1181 1181 1181 1182 1182 1182 1183 1183 1183 1183 1184 1184
1185 1185 1185 1186 1186 1186 1186 1187 1187
1189 1189 1190 1191 1192 1193 1194 1194 1195 1196 1197 1197 1199
1205
1207
Contents Volume 1
Derivatives and Geometry in ]R3
1 What is Mathematics? 1.1 1ntroduction .... 1.2 The Modern World 1.3 The Role of Mathematics 1.4 Design and Production of Cars 1.5 Navigation: From Stars to GPS 1.6 Medical Tomography . . . . . . 1.7 Molecular Dynamics and Medical Drug Design 1.8 Weather Prediction and Global Warming . 1.9 Economy: Stocks and Options ..... . 1.10 Languages ................ . 1.11 Mathematics as the Language of Science 1.12 The Basic Areas of Mathematics 1.13 What 1s Science? ......... . 1.14 What 1s Conscience? ....... . 1.15 How to View this Book as a Friend
2 The Mathematics Laboratory 2.1 1ntroduction ... 2.2 Math Experience ..... .
1
3 3 3 6
11 11 11 12 13 13 14 15 16 17 17 18
21 21 22
XXVIII Contents Volume 1
3 Introduction to Modeling 3.1 Introduction ..... . 3.2 The Dinner Soup Model 3.3 The Muddy Yard Model 3.4 3.5
A System of Equations . 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 4.6 4.7
Leibniz' Teen-Age Dream Summary Leibniz .......... .
25 25 25 28 29 30
33 33 34 34 39 41 43 44
5 Natural Numbers and Integers 47 5.1 Introduction . . . . . . . . . . 47 5.2 The Natural Numbers . . . . 48 5.3 Is There a Largest Natural Number? 51 5.4 The Set N of All Natural Numbers . 52 5.5 Integers................ 53 5.6 Absolute Value and the Distance Between Numbers . 56 5.7 5.8 5.9
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 N umbers 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 Ql of All Rational Numbers ... . 7.8 The Rational Number Line and Intervals . 7.9 Growth of Bacteria .. 7.10 Chemical Equilibrium .......... .
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 Introduetion ................. . 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 Geometrie Definition of sin( v) and eos( v) 8.8 Geometrie Proof of Addition Formulas for eos( v) 8.9 Remembering So me Area Formulas 8.10 Greek Mathematies ........... . 8.11 The Euelidean Plane Q2 . . . . . . . . . 8.12 From Pythagoras to Euclid to Deseartes 8.13 Non-Euelidean Geometry ....... .
9 What is a Function? 9.1 Introduetion ... 9.2 9.3 9.4 9.5 9.6
Funetions in Daily Life . . . . . Graphing Functions of Integers Graphing Functions of Rational Numbers A Funetion of Two Variables Functions of Several Variables
10 Polynomial functions 10.1 Introduetion ... 10.2 Linear Polynomials 10.3 Parallel Lines ... 10.4 10.5 10.6 10.7 10.8
Orthogonal Lines . Quadratie Polynomials Arithmetie with Polynomials Graphs of General Polynomials Pieeewise Polynomial Functions
11 Combinations of functions 11.1 Introduetion ............. . 11.2 Sum of Two Functions and Produet
of a Function with a Number .... 11.3 11.4 11.5 11.6
Linear Combinations of Functions. . Multiplieation and Division of Functions Rational Functions . . . . . . . The Composition of Functions .
12 Lipschitz Continuity 12.1 Introduetion ... 12.2 The Lipsehitz Continuity of a Linear Function .
XXIX
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
XXX 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 Ras 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 J2 Is Not a Rational Number! ...... . 14.3 Computing J2 by the Bisection Algorithm . 14.4 The Bisection Algorithm Converges! .... 14.5 First Encounters with Cauchy Sequences .. 14.6 Computing J2 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 J2J2 Equals 2 . . . . . . . . . 15.9 A Reflection on the Nature of J2 ... . 15.10 Cauchy Sequences of Real Numbers .. . 15.11 Extension from f : Q ---> Q to f : IR. ---> IR. 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 : ~ ---7 ~ •••••••
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 .
16 The Bisection Algorithm for f(x) = 0 16.1 Bisection ...... . 16.2 An Example . . . .. 16.3 Computational Cost
17 Do Mathematicians Quarrel?* 17.1 Introduction ........ . 17.2 The Formalists ...... . 17.3 The Logicists and Set Theory 17.4 The Constructivists. . . . . . 17.5 17.6 17.7 17.8 17.9 17.10 17.11 17.12 17.13
The Peano Axiom System for Natural Numbers Real Numbers .................. . Cantor Versus Kronecker ............ . Deciding Whether a Number is Rational or Irrational. The Set of All Possible Books . . . . . . . . Recipes and Good Food . . . . . . . . . . . The "New Math" in Elementary Education The Search for Rigor in Mathematics . A Non-Constructive Proof
17.14 Summary ...
XXXI
206 206 207 208 211
215 215 217 219
221 221 224 224 227 229 229 230 232 233 234 234 235 236 237
18 The Function y = x T 241 18.1 The Function Vx . . . . . . . . . . . 241 18.2 Computing with the Function Vx . . 242 18.3 Is Vx Lipschitz Continuous on ~+? . 242 18.4 The Function x T for Rational r = p. . 243
q 18.5 Computing with the Function x T • • 243 18.6 Generalizing the Concept of Lipschitz Continuity 243 18.7 Turbulent Flow is Hölder (Lipschitz) Continuous with Ex-
ponent ~ . . . . . . . . . . . . . . . . . . . . . . . . . . 244
19 Fixed Points and Contraction Mappings 245 19.1 Introduction........... 245 19.2 19.3 19.4 19.5 19.6 19.7
Contraction Mappings . . . . . Rewriting f(x) = 0 as x = g(x) Card Sales Model . . . . . . . . Private Economy Model . . . . Fixed Point Iteration in the Card Sales Model . A Contraction Mapping Has a U nique Fixed Point
246 247 248 249 250 254
XXXII
19.8 19.9 19.10 19.11
Contents Volume 1
Generalization to 9 : [a, b] ----* [a, b] ..... . Linear Convergence in Fixed Point Iteration Quicker Convergence . . Quadratic Convergence .
20 Analytic Geometry in ~2 20.1 Introduction ..... . 20.2 Descartes, Inventor of Analytic Geometry 20.3 Descartes: Dualism of Body and Soul. 20.4 The Euclidean Plane ~2 ..
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 aReal Number 20.11 The Norm of a Vector ..... . 20.12 Polar Representation of a Vector 20.13 Standard Basis Vectors ..... .
256 257 258 259
265 265 266 266 267 269 270 271 272 273 274 275 275 277
20.14 Scalar Product . . . . . . . . . . 278 20.15 Properties of the Scalar Product 278 20.16 Geometrie 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 90° . . . . . . . . . . . 283 20.20 Rotation by an Arbitrary Angle 8 . 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 U nknowns ............... 295 20.31 Linear Independence and Basis . . . . . . . . 297 20.32 The Connection to Calculus in One Variable. 298 20.33 Linear Mappings f : ~2 ----* ~. • . • • • • • • • 299 20.34 Linear Mappings f : ~2 ----* ~2 . . . . . . . . . 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 Multiplieation of a Matrix by aReal Number 20.40 Multiplieation of Two Matriees ... 20.41 The Transpose of a Matrix. . . . . . 20.42 The Transpose of a 2-Column Veetor 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
XXXIII
303 303 303 305 305 305 306 306 307 308 309
21 Analytic Geometry in ]R3 313 21.1 Introduetion.................... 313 21.2 Veetor Addition and Multiplieation by a Sealar 315 21.3 Sealar Produet and Norm . . . . . . 315 21.4 Projection of a Vector onto a Vector 316 21.5 The Angle Between Two Veetors . . 316 21.6 Veetor Produet . . . . . . . . . . . . 317 21.7 Geometrie Interpretation of the Veetor Produet 319 21.8 Conneetion Between Veetor Products in ]R2 and ]R3 320 21.9 Volume of a Parallelepiped Spanned
by Three Veetors . . . . . . . . . . . 320 21.10 The Tripie Produet a . b xc. . . . . 321 21.11 A Formula for the Volume Spanned
by Three Veetors . . . . . . . . . 322 21.12 Lines . . . . . . . . . . . . . . . . 323 21.13 Projeetion of a Point onto a Line 324 21.14 Planes . . . . . . . . . . . . . . . 324 21.15 The Interseetion of a Line and a Plane 326 21.16 Two Interseeting Planes Determine a Line 327 21.17 Projeetion of a Point onto a Plane 328 21.18 Distanee from a Point to a Plane . . . . . 328 21.19 Rotation Around a Given Veetor . . . . . 329 21.20 Lines and Planes Through the Origin Are Subspaees 330 21.21 Systems of 3 Linear Equations in 3 Unknowns . . 330 21.22 Solving a 3 x 3-System by Gaussian Elimination 332 21.23 3 x 3 Matriees: Sum, Produet and Transpose 333 21.24 Ways of Viewing a System of Linear Equations 335 21.25 Non-Singular Matriees 336 21.26 The Inverse of a Matrix . . . . . . . 336 21.27 Different Bases ........... . 21.28 Linearly Independent Set of Veetors 21.29 Orthogonal Matriees ........ . 21.30 Linear Transformations Versus Matrices
337 337 338 338
XXXIV Contents Volume 1
21.31 The Scalar Product Is Invariant Under Orthogonal Transformations 339
21.32 Looking Ahead to Functions f : ]R3 -+ ]R3 340
22 Complex N umbers 345 22.1 Introduction......... 345 22.2 Addition and Multiplication 346 22.3 The Triangle Inequality .. 347 22.4 Open Domains ....... 348 22.5 Polar Representation of Complex Numbers . 348 22.6 Geometrical Interpretation of Multiplication . 348 22.7 Complex Conjugation ......... 349 22.8 Division................. 350 22.9 The Fundamental Theorem of Algebra 350 22.10 Roots .................. 351 22.11 Solving a Quadratic Equation w 2 + 2bw + c = 0 . 351 22.12 Gösta Mittag-LefRer . . . . . . . . . . . . . . . . 352
23 The Derivative 355 23.1 Rates of Change 355 23.2 Paying Taxes . . 356 23.3 Hiking...... 359 23.4 Definition of the Derivative 359 23.5 23.6 23.7 23.8 23.9 23.10 23.11 23.12 23.13 23.14 23.15 23.16 23.17
The Derivative of a Linear Function Is Constant The Derivative of x 2 Is 2x . . . . . . The Derivative of x n Is nxn - 1 ....
The Derivative of 1 Is - -l;, for x --I- 0 x x r The Derivative as a Function .... Denoting the Derivative of f (x) by D f (x) Denoting the Derivative of f(x) by 1,; .. The Derivative as a Limit of Difference Quotients . How to Compute a Derivative? .......... . Uniform Differentiability on an Interval ..... . A Bounded Derivative Implies Lipschitz Continuity A Slightly Different Viewpoint . Swedenborg . . . . . . . . . . .
24 Differentiation Rules 24.1 Introduction ......... . 24.2 The Linear Combination Rule 24.3 The Product Rule 24.4 The Chain Rule . . . . . . . . 24.5 The Quotient Rule . . . . . . . . . . . . . . 24.6 Derivatives of Derivatives: f(n) = Dn f = ~ 24.7 One-Sided Derivatives . . . . . . . . . . . . .
362 362 364 365 365 365 367 367 369 371 372 374 374
377 377 378 379 380 381 382 383
Contents Volume 1
24.8 Quadratie Approximation . . . . . . . 24.9 The Derivative of an Inverse Function 24.10 Implieit Differentiation. 24.11 Partial Derivatives 24.12 A Sum Up So Far .
25 Newton's Method 25.1 Introduetion. 25.2 Convergenee of Fixed Point Iteration 25.3 Newton's Method .......... . 25.4 Newton's Method Converges Quadratieally . 25.5 A Geometrie 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 26.3 26.4 26.5 26.6
Newton's Law of Motion Galileo's Law of Motion Hooke's Law ...... . Newton's Law plus Hooke's Law 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
xxxv
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 2
Integrals and Geometry in :!Rn 427
27 The Integral 429 27.1 Primitive Functions and Integrals. . . . . . . . . . . 429 27.2 Primitive Function of f(x) = x m 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) = x r for r =I- -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
XXXVIII Contents Volume 2
28.4 Integrating Piecewise Lipschitz Continuous Functions 455
28.5 Linearity............. 456 28.6 Monotonicity . . . . . . . . . . 457 28.7 The Triangle Inequality for Integrals 457 28.8 Differentiation and Integration
are Inverse Operations . . . . . . . . 458 28.9 Change of Variables or Substitution. 459 28.10 Integration by Parts . . . . . . . . . 461 28.11 The Mean Value Theorem. . . . . . 462 28.12 Monotone Functions and the Sign of the Derivative 464 28.13 A Function with Zero Derivative is Constant. . . . 464 28.14 A Bounded Derivative Implies Lipschitz Continuity . 465 28.15 Taylor's Theorem 465 28.16 October 29, 1675 468 28.17 The Hodometer . 469
29 The Logarithm log(x) 473 29.1 The Definition of log(x) ..... . 473 29.2 The Importance of the Logarithm . 474 29.3 Important Properties oflog(x) 475
30 N umerieal 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 IR . . . 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 IR . . . . . . 501
32 Trigonometrie Functions 505 32.1 The Defining Differential Equation . . . . . . . . . . . 505 32.2 Trigonometrie Identities . . . . . . . . . . . . . . . . . 509 32.3 The Functions tan(x) and eot(x) and Their Derivatives 510 32.4 Inverses of Trigonometrie Functions . 511 32.5 The Functions sinh(x) and eosh(x) . . . . . . . . 513 32.6 The Hanging Chain. . . . . . . . . . . . . . . . . 514 32.7 Comparing u" + k2u(x) = 0 and u" - k2u(x) = 0 515
Contents Volume 2 XXXIX
33 The Functions exp(z), log(z), sin(z) and eos(z) for z E C 517 33.1 Introduetion.......... 517 33.2 Definition of exp(z) . . . . . . 517 33.3 Definition of sin(z) and eos(z) 518 33.4 de Moivres Formula . 518 33.5 Definition of log(z) . . 519
34 Techniques of Integration 34.1 Introduetion......
521 521
34.2 Rational Funetions: The Simple Cases 522 34.3 Rational Funetions: Partial Fraetions . 523 34.4 Products of Polynomial and Trigonometrie
or Exponential Functions ................ 528 34.5 Combinations of Trigonometrie and Root Functions. . 528 34.6 Produets of Exponential and Trigonometrie 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) = >.(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
361mproper Integrals 541 36.1 Introduetion.............. 541 36.2 Integrals Over Unbounded Intervals . 541 36.3 Integrals of Unbounded Functions . 543
37 Series 37.1 37.2 37.3 37.4 37.5 37.6 37.7 37.8
Introduetion . . . . . . . . . . . . . . . Definition of Convergent Infinite Series Positive Series ........ . Absolutely Convergent Series . . . . . Alternating Series. . . . . . . . . . . . The Series 2::1 t Theoretieally Diverges! Abel . Galois ................... .
38 Scalar Autonomous Initial Value Problems 38.1 Introduction ........... . 38.2 An Analytieal Solution Formula . 38.3 Construction of the Solution ...
547 547 548 549 552 552 553 555 556
559 559 560 563
XL Contents Volume 2
39 Separable Scalar Initial Value Problems 567 39.1 Introduction............... 567 39.2 An Analytical Solution Formula . . . . 568 39.3 Volterra-Lotka's Predator-Prey Model 570 39.4 A Generalization . . . . . . . . . 571
40 The General Initial Value Problem 575 40.1 Introduction............ 575 40.2 Determinism and Materialism . . 577 40.3 Predictability and Computability 577 40.4 Construction of the Solution. . . 579 40.5 Computational Work . . . . . . . 580 40.6 Extension to Second Order Initial Value Problems 581 40.7 Numerical Methods . . . . . . . . . . . . . . . . . . 582
41 Calculus Tool Bag I 41.1 Introduction ... 41.2 41.3 41.4 41.5 41.6 41.7 41.8 41.9 41.10 41.11 41.12 41.13
Rational Numbers ......... . Real Numbers. Sequences and Limits Polynomials and Rational Functions Lipschitz Continuity Derivatives ........... . Differentiation Rules . . . . . . . Solving f(x) = 0 with f : ]R --; ]R Integrals ..... The Logarithm . . . . . . . . The Exponential . . . . . . . The Trigonometrie Functions List of Primitive Functions. .
41.14 Series ............ . 41.15 The Differential Equation ü + >.(x)u(x) = f(x) 41.16 Separable Scalar Initial Value Problems
42 Analytic Geometry in ]Rn 42.1 Introduction and Survey of Basic Objectives . 42.2 Body /Soul and Artificial Intelligence . 42.3 The Vector Space Structure of]Rn ... 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 42.9 42.10 42.11
Linear Independence . . . . . . . . . . . . . . Reducing a Set of Vectors to Get a Basis . . . Using Column Echelon Form to Obtain a Basis Using Column Echelon Form to Obtain R(A) .
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 XLI
42.12 Using Row Echelon Form to Obtain N(A) 610 42.13 Gaussian Elimination. . . . . . . . 612 42.14 A Basis for IRn Contains n Vectors 612 42.15 Coordinates in Different Bases. . . 614 42.16 Linear Functions f : IRn -+ IR ... 615 42.17 Linear Transformations f : IRn -+ IRm 615 42.18 Matriees . . . . . . . . . . . . . . . . 616 42.19 Matrix Calculus . . . . . . . . . . . . 617 42.20 The Transpose of a Linear Transformation. 619 42.21 Matrix Norms . . . . . . . . . . . . . . . . . 620 42.22 The Lipsehitz Constant of a Linear Transformation 621 42.23 Volume in IRn: Determinants and Permutations 621 42.24 Definition of the Volume V(al, ... ,an) 623 42.25 The Volume V(al, a2) in IR2 . . . . 624 42.26 The Volume V(al, a2, a3) in IR3 . . 624 42.27 The Volume V(al, a2, a3, a4) in IR4 625 42.28 The Volume V(al, ... , an) in IRn . 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 Magie Formula det AB = det A det B . 626 42.32 Test of Linear Independenee . . . . . . . . . 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 Charaeterization of the Projection 631 42.37 Orthogonal Deeomposition: Pythagoras Theorem 632 42.38 Properties of Projections . . . . . . . . . . . . . . 633 42.39 Orthogonalization: The Gram-Sehmidt Proeedure . 633 42.40 Orthogonal Matriees . . . . . . . . . . . . . . . . . 634 42.41 Invarianee of the Sealar Product
Under Orthogonal Transformations. . . . . . 634 42.42 The QR-Deeomposition ............ 635 42.43 The Fundamental Theorem of Linear Algebra 635 42.44 Change of Basis: Coordinates and Matriees 637 42.45 Least Squares Methods . . . . . . . . . . . . 638
43 The Spectral Theorem 641 43.1 Eigenvalues and Eigenveetors . . . . . . . . . . . . . 641 43.2 Basis of Eigenveetors . . . . . . . . . . . . . . . . . . 643 43.3 An Easy Speetral Theorem for Symmetrie Matriees . 644 43.4 Applying the Speetral Theorem to an IVP . . . . . . 645 43.5 The General Spectral Theorem
for Symmetrie Matriees .......... 646 43.6 The Norm of a Symmetrie Matrix. . . . . 648 43.7 Extension to Non-Symmetrie Real Matriees 649
XLII Contents Volume 2
44 Solving Linear Aigebraic 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 ............. .
45 Linear Algebra Tool Bag 45.1 Linear Algebra in jR2
45.2 Linear Algebra in jR3 .
45.3 45.4 45.5 45.6 45.7 45.8 45.9 45.10 45.11 45.12 45.13 45.14 45.15
Linear Algebra in jRn .
Linear Transformations and Matrices . The Determinant and Volume Cramer's Formula . Inverse ............ . Projections ......... . The Fundamental Theorem of Linear Algebra The QR-Decomposition .. Change of Basis . . . . . . . . The Least Squares Method . Eigenvalues and Eigenvectors The Spectral Theorem . . . . The Conjugate Gradient Method for Ax = b .
651 651 651 658 661 671 674 676
685 685 686 686 687 688 688 689 689 689 689 690 690 690 690 690
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 701 47.5 The Principle of Least Action . . 701 47.6 Conservation of the Total Energy 703 47.7 The Double Pendulum . . . . . . 703 47.8 The Two-Body Problem ..... 47.9 Stability of the Motion of a Pendulum
704 705
Contents Volume 2
48 N-Body Systems* 48.1 Introduction...... 48.2 Masses and Springs .. 48.3 The N-Body Problem 48.4 Masses, Springs and Dashpots:
Small Displacements . . . . 48.5 Adding Dashpots ..... . 48.6 A Cow Falling Down Stairs 48.7 The Linear Oscillator. . . . 48.8 The Damped Linear Oscillator 48.9 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 .. 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.
XLIII
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 L 2 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
XLIV Contents Volume 2
53.5 53.6 53.7 53.8
53.9
The Discrete System of Equations. . . . . . . Handling Different Boundary Conditions ... Error Estimates and Adaptive Error Control . Discretization of Time-Dependent Reaction-Diffusion-Convection Problems ... Non-Linear Reaction-Diffusion-Convection Problems
References
Index
762 765 768
773 773
777
779
