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Change and Variations: A History of Differential Equations to 1900Suurem pilt
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This book presents a history of differential equations, both ordinary and partial, as well as the calculus of variations, from the origins of the subjects to around 1900. Topics treated include the wave equation in the hands of dAlembert and Euler; Fouriers solutions to the heat equation and the contribution of Kovalevskaya; the work of Euler, Gauss, Kummer, Riemann, and Poincaré on the hypergeometric equation; Greens functions, the Dirichlet principle, and Schwarzs solution of the Dirichlet problem; minimal surfaces; the telegraphists equation and Thomsons successful design of the trans-Atlantic cable; Riemanns paper on shock waves; the geometrical interpretation of mechanics; and aspects of the study of the calculus of variations from the problems of the catenary and the brachistochrone to attempts at a rigorous theory by Weierstrass, Kneser, and Hilbert. Three final chapters look at how the theory of partial differential equations stood around 1900, as they were treated by Picard and Hadamard. There are also extensive, new translations of original papers by Cauchy, Riemann, Schwarz, Darboux, and Picard. The first book to cover the history of differential equations and the calculus of variations in such breadth and detail, it will appeal to anyone with an interest in the field. Beyond secondary school mathematics and physics, a course in mathematical analysis is the only prerequisite to fully appreciate its contents. Based on a course for third-year university students, the book contains numerous historical and mathematical exercises, offers extensive advice to the student on how to write essays, and can easily be used in whole or in part as a course in the history of mathematics. Several appendices help make the book self-contained and suitable for self-study.

Arvustused

This book is a very good example of a text for a course in the history of mathematics. the author provides for students and readers a historical overview of how mathematics, physics, celestial mechanics and difficult problems to tackle from differential equations as well as applications were intertwined, and the resulting dialogues between mathematicians, physicists and astronomers. This book is a successful attempt to fill in some of the gaps on the history of differential equations. (Clara Silvia Roero, Mathematical Reviews, September, 2022)

1 The First Ordinary Differential Equations
1(16)
1.1 Introduction
1(1)
1.2 Origins: Inverse Tangent Problems
1(3)
1.2.1 Debeaune's Problem
2(1)
1.2.2 Other Inverse Tangent Problems
3(1)
1.3 From Inverse Tangent Problems to Differential Equations
4(4)
1.4 Differential Equations
8(2)
1.5 Linear Ordinary Differential Equations
10(6)
1.5.1 A Note on the Adjoint Equation
14(2)
1.6 Exercises
16(1)
2 Variational Problems and the Calculus
17(10)
2.1 Introduction
17(1)
2.2 Bernoulli's Problems
18(4)
2.3 The Bernoullis' Brachistochrones
22(2)
2.4 Geodesies on Surfaces
24(1)
2.5 Exercises
25(2)
3 The Vibrating String and the Partial Differential Calculus
27(12)
3.1 Introduction
27(1)
3.2 Early Investigations into the Partial Differential Calculus
27(2)
3.3 D'Alembert: The Vibrating String and the Wave Equation
29(6)
3.3.1 D'Alembert's Breakthrough
30(4)
3.3.2 Mersenne's Law and Modes
34(1)
3.4 Euler Rewrites the Wave Equation
35(1)
3.5 Formal Complex Methods
36(2)
3.6 Exercises
38(1)
4 Rational Mechanics
39(16)
4.1 Introduction
39(1)
4.2 Fluid Mechanics
39(6)
4.2.1 Recent Discoveries About the Euler Equations
45(1)
4.3 Euler and the Propagation of Sound
45(3)
4.4 Euler's Vision of Mechanics
48(3)
4.5 Darboux's Account
51(2)
4.6 Exercises
53(2)
5 The Early Theory of Partial Differential Equations
55(16)
5.1 Introduction
55(1)
5.2 Euler's General Theory of Partial Differential Equations
55(9)
5.2.1 Second-Order Partial Differential Equations
60(4)
5.3 The Introduction of Characteristics by d'Alembert
64(1)
5.4 Laplace
65(3)
5.4.1 Lagrange's Method
68(1)
5.5 Exercises
68(3)
6 Lagrange's General Theory of Partial Differential Equations
71(10)
6.1 Introduction
71(1)
6.2 Clairaut's Paradox
71(3)
6.3 Lagrange
74(5)
6.3.1 Lagrange [ 173]
74(2)
6.3.2 Lagrange [ 173]
76(2)
6.3.3 Lagrange [ 175]
78(1)
6.4 Exercises
79(2)
7 The Calculus of Variations
81(14)
7.1 Introduction
81(1)
7.2 The Euler--Lagrange Equations Discovered
81(4)
7.3 Maupertuis and the Principle of Least Action
85(2)
7.4 Euler's Later Approach
87(2)
7.5 Brachistochrone and the Calculus of Variations
89(2)
7.6 Generalised Coordinates
91(1)
7.7 Exercises
92(3)
8 Monge and Solutions to Partial Differential Equations
95(16)
8.1 Introduction
95(1)
8.2 Monge and First-Order Partial Differential Equation
95(6)
8.2.1 A Comparison with the Modern Account
97(1)
8.2.2 The General First-Order Case
98(3)
8.3 Monge on General First-Order Equation
101(3)
8.4 Monge on Second-Order Partial Differential Equation
104(2)
8.5 Lagrange at the Ecole Poly technique, 1806
106(4)
8.5.1 Lacroix's Traiti (1798)
109(1)
8.6 Exercises
110(1)
9 Revision
111(2)
9.1 Revision and Assessment 1
111(2)
9.1.1 Comments
111(2)
10 The Heat Equation
113(16)
10.1 Introduction
113(1)
10.2 Fourier and His Series
113(9)
10.2.1 Dirichlet on the Convergence of Fourier Series
120(1)
10.2.2 Fourier Integrals
120(2)
10.3 The Analysis of Fourier Integrals
122(2)
10.4 Stokes and Laplace Transform
124(3)
10.5 Exercises
127(2)
11 Gauss and the Hypergeometric Equation
129(14)
11.1 Introduction
129(1)
11.2 Elliptic Integrals
129(2)
11.3 Gauss
131(5)
11.3.1 The Hypergeometric Equation
133(3)
11.4 Kummer and His 24 Solutions
136(3)
11.5 The Method of Undetermined Coefficients
139(2)
11.6 Exercises
141(2)
12 Existence Theorems
143(10)
12.1 Introduction
143(1)
12.2 Cauchy and Ordinary Differential Equations
143(9)
12.2.1 Later Developments
152(1)
12.3 Exercises
152(1)
13 Riemann and Complex Function Theory
153(6)
13.1 Introduction
153(1)
13.2 Complex Function Theory
153(2)
13.3 The Riemann Mapping Theorem
155(2)
13.4 A Look Ahead
157(1)
13.5 Exercises
158(1)
14 Riemann and the Hypergeometric Equation
159(12)
14.1 Introduction
159(5)
14.1.1 Ordinary Differential Equations and Many-Valued Functions
159(4)
14.1.2 The Riemann Sphere
163(1)
14.2 Riemann's P-Functions
164(2)
14.3 Riemann's Arguments
166(3)
14.4 Exercises
169(2)
15 Schwarz and the Complex Hypergeometric Equation
171(8)
15.1 Introduction
171(1)
15.2 Quotients of Solutions
171(5)
15.3 Exercises
176(3)
16 Complex Ordinary Differential Equations: Poincare
179(12)
16.1 Introduction
179(1)
16.2 Poincare and Linear Ordinary Differential Equations
179(5)
16.3 Poincare's Breakthrough and Non-Euclidean Geometry
184(3)
16.4 Non-Euclidean Geometry
187(2)
16.4.1 Summary
189(1)
16.5 Exercises
189(2)
17 More General Partial Differential Equations
191(8)
17.1 Introduction
191(1)
17.2 Cauchy's Method in 1819
192(1)
17.3 Cauchy and the General Partial Differential Equation
193(2)
17.4 Kovalevskaya's Theorem and Her Counter-Example
195(3)
17.5 Exercises
198(1)
18 Green's Functions and Dirichlet's Principle
199(10)
18.1 Introduction
199(1)
18.2 Green's Theorems and Green's Functions
199(3)
18.3 Dirichlet Principle and Problem
202(1)
18.4 Riemann on Green's Theorem
203(2)
18.5 Riemann on the Dirichlet Principle
205(2)
18.6 Exercises
207(2)
19 Attempts on Laplace's Equation
209(8)
19.1 Introduction
209(1)
19.2 Weierstrass, Prym, and Schwarz
209(5)
19.2.1 Schwarz's Alternating Method
213(1)
19.3 Harnack
214(2)
19.4 Exercises
216(1)
20 Applied Wave Equations
217(8)
20.1 Introduction
217(1)
20.2 The Trans-Atlantic Cable
217(4)
20.3 Poincare's Solution
221(3)
20.3.1 Conclusion
223(1)
20.4 Exercises
224(1)
21 Revision
225(2)
21.1 Revision and Assessment 2
225(2)
22 Riemann's Shockwave Paper
227(10)
22.1 Introduction
227(1)
22.2 Riemann's Paper
227(5)
22.3 Darboux on Riemann's Approach to the Shockwave Equation
232(1)
22.4 Telegraphy
233(2)
22.5 Exercises
235(2)
23 The Example of Minimal Surfaces
237(14)
23.1 Introduction
237(1)
23.2 Euler and Lagrange
237(3)
23.3 Meusnier, Monge, and Legendre
240(3)
23.4 Riemann and Weierstrass
243(5)
23.5 Simple Solutions of the Plateau Problem
248(2)
23.6 Exercises
250(1)
24 Partial Differential Equations and Mechanics
251(12)
24.1 Introduction
251(1)
24.2 Hamiltonian Dynamics
252(4)
24.3 Hamilton's and Jacobi's Theories of Dynamics
256(5)
24.3.1 Jacobi
258(3)
24.4 First-Order Partial Differential Equation Theory
261(1)
24.5 Exercises
262(1)
25 Geometrical Interpretations of Mechanics
263(12)
25.1 Introduction
263(1)
25.2 Gaussian Curvature
263(4)
25.2.1 Liouville's Contributions
265(2)
25.3 Geometrising Mechanics
267(3)
25.4 The Connection to Hamilton-Jacobi Theory
270(3)
25.5 Exercises
273(2)
26 The Calculus of Variations in the nineteenth Century
275(14)
26.1 Introduction
275(1)
26.2 After Lagrange
275(2)
26.3 Weierstrass's Theory
277(5)
26.3.1 Two Examples
280(2)
26.4 Hilbert's Problem 23 and the Theory of the Calculus of Variations
282(5)
26.5 Exercises
287(2)
27 Poincare and Mathematical Physics
289(8)
27.1 Introduction
289(1)
27.2 The Classical Classification of Linear Partial Differential Equations
289(4)
27.3 Poincare and the Dirichlet Problem
293(3)
27.4 Exercises
296(1)
28 Elliptic Equations and Regular Variational Problems
297(6)
28.1 Introduction
297(1)
28.2 Picard on Second-Order Linear Elliptic Equations
297(2)
28.3 Hilbert's Problems 19 and 20
299(3)
28.4 Exercises
302(1)
29 Initial Value Conditions for Hyperbolic Partial Differential Equations
303(10)
29.1 Introduction
303(1)
29.2 Picard on Second-Order Linear Hyperbolic Equations
303(1)
29.3 Hadamard and Mathematical Physics
304(3)
29.4 The Cauchy Problem
307(4)
29.4.1 Commentary and Concluding Remarks
310(1)
29.5 Exercises
311(2)
30 Revision
313(2)
30.1 Revision and Assessment 3
313(2)
31 Translations
315(34)
31.1 Cauchy: Note on the Integration of First-Order Partial Differential Equations in Any Number of Variables
315(6)
31.2 Riemann's Lectures on Partial Differential Equations and Physics
321(2)
31.2.1 Riemann, Introduction to Partial Differential Equations
321(2)
31.3 Extracts from Schwarz, "Ueber eine Abbildungsaufgaben", 1869
323(4)
31.3.1 The Schwarz--Christoffel Transformation
326(1)
31.4 An Extract from Schwarz, On the Alternating Method
327(5)
31.5 Schwarz on the Hypergeometric Equation (1873)---A Summary
332(2)
31.6 Darboux on the Solution of Riemann's Equation (1887)
334(5)
31.7 Picard and Elliptic Partial Differential Equations (1890)
339(3)
31.8 Picard and Hyperbolic Partial Differential Equations (1890)
342(7)
Appendix A Newton's Principia Mathematica 349(8)
Appendix B Characteristics 357(6)
Appendix C The First-Order Non-linear Partial Differential Equation 363(6)
Appendix D Green's Theorem and Heat Conduction 369(10)
Appendix E Complex Analysis 379(6)
Appendix F Mobius Transformations 385(8)
Appendix G Lipschitz and Picard 393(4)
Appendix H The Assessment 397(8)
References 405(12)
Index 417
Jeremy Gray is an Emeritus Professor at the Open University and an Honorary Professor of Mathematics at the University of Warwick. He is an Inaugural Fellow of the American Mathematical Society, and for his work in the history of mathematics he was a recipient in 2009 of the AMSs Albert Leon Whiteman Memorial Prize, and in 2016 of the Otto Neugebauer Prize of the European Mathematical Society. He is the author or co-author of 13 books on the subject, including Platos Ghost: The Modernist Transformation of Mathematics (2008), Henri Poincaré: a scientific biography (2012), and three books on the history of algebra, analysis, and geometry on the 19th century.
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