Differential Equations, Dynamical Systems, and Linear Algebra (Pure and Applied Mathematics (Academic Press), 60.) (英語) ハードカバー – 1974/5/12
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This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.
Carlos Frederico Trotta Matt, Ph.D., Mechanical Engineer
Electric Power Research Center, Rio de Janeiro, Brazil
subsets of real Cartesian space in such a way that extension to manifolds is simple and natural." Therefore, I concentrate upon those two aims.
(1) Firstly, it is my opinion that parts of the book can profitably be utilized in the sophomore year. With the background presupposed by the author
(that is: perusal of Serge Lang's Calculus texts), there is much here within reach. Look at the Problems ! I would be surprised (if not, shocked)
if all of the Chapter One problems (page 12; 1 through 8) posed any issue for a student who has perused (at least) one of Lang's textbooks.
(2) Now, the first two chapters, granted, are fairly easy--even if, the second chapter is merely a synopsis of Newtonian Mechanics.
One typo mars the discussion (First Edition--Page 2, Figure A, the last graph is mislabeled " 'a' greater than zero," change that to "less than").
And, as we read, bottom of page two: " the sign of 'a' is crucial here"....it pays to label the accompanying Figure properly.
And, as is evident before one proceeds too far, the "chain rule" will be utilized over and over again.
(3) Third Chapter begins Linear Algebra: Linearity (page 30) segues to "natural correspondence between operators and n-X-n matrices" (page 31).
"Eigenvalues and eigenvectors are very important." (page 42). Theorem Two is the capstone of the chapter: "By using this theorem we get much
information about the general solutions directly from knowledge of the eigenvalues, without solving the differential equation." (page 50).
Now, before continuing with the text, re-read the first three chapters. Do every problem (or, at the least re-do every example). The concluding Section of Chapter Three--Complex Eigenvalues (pages 55-59)--will be extremely important for the next chapter:
(4) Linear Systems With Constant Coefficients and Complex Eigenvalues. This will be a short interlude of ten pages. It is prelude to the more abstract Chapter Five:
(5) Linear Systems and Exponentials of Operators. Here you will get a survey of introductory Topology. Norms and Operators, follow. Take note of:
The correspondence, page 85, between complex numbers and 2X2 matrices.(By the way, only square matrices are utilized in the text). Also,
the series expansion of exponential and trigonometric functions should be second-nature. Homogeneous,then, nonhomogeneous equations.
Autonomous (no explicit time-dependence) followed by nonautonomous (page 99). Take-away: Variation of Parameters. Another clue: Liebniz.
And, read: "The eigenvector theory of real linear operator is rarely treated in texts, and is important for theory of linear differential equations."
Thus, for the next thirty-three pages that is what you will get--Eigenvalues, nilpotence, Cayley-Hamilton, semi-simple, canonical forms, and more.
Read: "Operators on function spaces have many uses for both theoretical and practical work in differential equations." (page 143).
(6) More Theory in the next two chapters (seven and eight). Equilibrium states, defined. Glance at Problem #4 (page 150). Surely, many physics
students have been exposed previously to this equation. Now, it is placed into mathematical context. Learn the meaning of "dense" (page 153,
and Problem #1, page 157). Chapter Eight--as the authors' write: "more difficult" and "suggest omitting the proofs." That makes good sense !
If you are unfamiliar with things such as Lipschitz condition (or, manipulating epsilons and deltas), then, a review of such is recommended.
Uniform Convergence, too, should be firmly grasped. (Or, glance at Problems #1,4, and 5, page 177). Serge Lang, again, is prime reference.
(7) Stability, next. Again, Physics students should find this material interesting (pendulums in gravity-field, Maxwell on Saturn's rings).
We continue with Stability and Equilibria: Glance at the example of page 201, graph--by hand-- this multivariable function. True, it is easier if you
use a computer for that (Maple, Mathematica, etc.), but, how about doing it once by hand. My initial exposure to differential equations proceeded
by way of "computer labs" (this was late 1990's)--unfortunately, time spent with computer graphics meant less time spent learning mathematics.
By the way, terms such as kinetic energy, potential energy, and conservative forces should already be firmly grasped. (See: Symon).
(8) Chapters ten to twelve--Applications: Circuits (background in Feynman Volume One), periodic solutions, ecology (predator/prey).
Theory, again, in Chapter Thirteen (fear not, glance at Problem #1, page 285, that is not difficult). Next, we will review Classical Mechanics:
(9) A review --or, introduction to--Hamiltonian Mechanics. Read: "Our present goal is to put Newton's equations into the framework of this book."
This will be a nice prelude to the more advanced Textbooks of Arnold:Mathematical Methods Of Classical Mechanics and Ordinary Differential
Equations--both of those textbooks should be consulted in any event !
(10) Concluding Chapters, more theory. That is, perturbations.
The authors write: "this book is only an introduction to the subject of dynamical systems" and "Appendix One (elementary facts) should have been
seen before"....And so, if Appendix One contains anything that is 'new' to you, then, learn it first (before tackling this book). And, if you are searching for "manifolds" turn elsewhere (Arnold). Now, my concluding thoughts regarding the book:
This is not such a terrible book. As with other textbooks that attempt a trilogy of topics under one cover (As, say, Weinberger's Partial Differential Equations), it is difficult to please every group. Here we have three topics--differential equations and dynamical systems and linear algebra.
As usual, you run the risk of pleasing no one when you attempt to please everyone.
Any one of those topics can fill an entire book. Here, in span of 350 pages, we are exposed to all three topics. And, to make matters more involved,
if said student (at sophomore level) is unaccustomed to mathematical proofs , then much in the later chapters will be difficult to follow.
Be that as it may, if the rudiments of Calculus, as laid out in the textbook's of Serge Lang, are taken as prerequisite, this text will be much easier.
One makes comparison to the beautiful textbooks of Hubbard and West, they write of Hirsch and Smale: " this is the first book bringing modern developments of differential equations to a broad audience " and "Smale has profoundly influenced the authors."
(Differential Equations, Part One, page 307, 1991). As with their recommendation of the text, I do the same.
My first course in Differential Equations left me with a potpourri of "tricks" and computer graphics, as but memories.
This text will hopefully instill less of that "trickery" with more mathematical connections and more understanding.
But my opinions won't be as helpful to the Amazon math shopper as a simple listing of what's in the book. So here's the table of contents.
Chapter 1: First Examples
Chapter 2: Newton's Equation and Kepler's Law
Chapter 3: Linear Systems with Constant Coefficiants and Real Eigenvalues
Chapter 4: Linear Systems with Constant Coefficients and Complex Eigenvalues
Chapter 5: Linear Systems and Exponentials of Operators
Chapter 6: Linear Systems and Canonical Forms of Operators
Chapter 7: Contractions and Generic Properties of Operators
Chapter 8: Fundamental Theory
Chapter 9: Stability of Equilibria
Chapter 10: Differential Equations for Electric Circuits
Chapter 11: The Poincare-Bendixson Theorem
Chapter 12: Ecology
Chapter 13: Periodic Attractors
Chapter 14: Classical Mechanics
Chapter 15: Nonautonomous Equations and Differentiability of Flows
Chapter 16: Perturbation Theory and Structural Stability
Appendix I: Elementary Facts
Appendix II: Polynomials
Appendix III: On Canonical Forms
Appendix IV: The Inverse Function Theorem
Answers to Selected Problems