The Mathematics of Sonya Kovalevskaya

ISBN-13: 9781461297666 / Angielski / Miękka / 2011 / 236 str.

R. Cooke

Opis

This book is the result of a decision taken in 1980 to begin studying the history of mathematics in the nineteenth century. I hoped by doing it to learn some thing of value about Kovalevskaya herself and about the mathematical world she inhabited. Having been trained as a mathematician, I also hoped to learn something about the proper approach to the history of the subject. The decision to begin the study with Kovalevskaya, apart from the intrinsic interest of Kovalevskaya herself, was primarily based upon the fact that the writing on her in English had been done by people who were interested in sociological and psychological aspects of her life. None of these writings discussed her mathematical work in much detail. This omission seemed to me a serious one in biographical studies of a woman whose primary significance was her mathematical work. In regard to both the content of nineteenth century mathematics and the nature of the history of mathematics I learned a great deal from writing this book. The attempt to put Kovalevskaya's work in historical context involved reading dozens of significant papers by great mathematicians. In many cases, I fear, the purport of these papers is better known to many of my readers than to me. If I persevered despite misgivings, my excuse is that this book is, after all, primarily about Kovalevskaya. If specialists in Euler, Cauchy, etc."

Spis treści

I: Childhood and Education.- 1. Biography: 1850–1874.- 1.1. Introduction.- 1.2. Kovalevskaya’s Ancestors.- 1.3. Kovalevskaya’s Family.- 1.4. Kovalevskaya’s Uncles.- 1.5. Aniuta.- 1.6. Early Mathematical Training.- 1.7. In Search of Higher Education.- 1.8. Vladimir Kovalevsky.- 1.9. Study Abroad.- 1.10. Weierstrass.- 1.11. Kovalevskaya and Weierstrass.- 2. Partial Differential Equations.- 2.1. Introduction.- 2.2. Differential Equations in the Complex Domain.- 2.3. The Year 1842: Cauchy.- 2.4. The Year 1842: Weierstrass.- 2.5. The Problem Posed to Kovalevskaya.- 2.6. Kovalevskaya’s Paper.- 2.7. Unpublished Work.- 2.8. Publishing the Result.- 2.9. Evaluation of the Work.- 3. Degenerate Abelian Integrals.- 3.1. Introduction.- 3.2. Euler.- 3.3. Legendre.- 3.4. Abel.- 3.5. Cauchy.- 3.6. Jacobi.- 3.7. Göpel and Rosenhain.- 3.8. Weierstrass’ Early Work.- 3.9. Hermite.- 3.10. Riemann.- 3.11. Weierstrass’ Later Work.- 3.12. Kovalevskaya’s Paper.- 3.13. Evaluation.- 4. The Shape of Saturn’s Rings.- 4.1. Introduction.- 4.2. Laplace’s Work: Physical Assumptions.- 4.3. Laplace’s Work: Mathematical Assumptions.- 4.4. The Potential of the Ring.- 4.5. Final Computations.- 4.6. Extensions.- 4.7. Reformulation of the Problem.- 4.8. Kovalevskaya’s Method.- 4.9. Execution of the Program.- 4.10. Significance of the Paper.- 4.11. Personal Notes.- II: Mature Life.- 5. Biography: 1875–1891.- 5.1. Return to Russia.- 5.2. Mittag-Leffler.- 5.3. The Year 1880.- 5.4. A New Project.- 5.5. Crisis.- 5.6. The Year 1883.- 5.7. Stockholm.- 5.8. The Academic Year 1884–85.- 5.9. The Academic Year 1885–86.- 5.10. Life as an Untenured Professor.- 5.11. New Distractions.- 5.12. The Bordin Competition.- 5.13. Reactions.- 5.14. Final Days.- 6. The Lamé Equations.- 6.1. Introduction.- 6.2. Huyghens.- 6.3. Fresnel.- 6.4. Lamé.- 6.5. Weierstrass.- 6.6. Kovalevskaya.- 7. The Euler Equations.- 7.1. Introduction.- 7.2. Euler.- 7.3. Lagrange.- 7.4. Some Nineteenth-Century Work.- 7.5. Weierstrass.- 7.6. The Mathematical Mermaid.- 7.7. Kovalevskaya.- 7.8. Conclusions.- 7.9. Klein.- 7.10. Evaluation.- 8. Bruns’ Theorem.- 8.1. Introduction.- 8.2. Kovalevskaya’s Paper.- 8.3. Commentary.- 9. Evaluations.- 9.1. Introduction.- 9.2. Poincaré.- 9.3. Markov.- 9.4. Kovalevskaya.- 9.5. The Moscow Mathematicians.- 9.6. Volterra.- 9.7. Women in Mathematics.- 9.8. Loria.- 9.9. Mittag-Leffler.- 9.10. Klein.- 9.11. Golubev.- 9.12. Kovalevskaya as a Mathematician.- III: Appendices and Bibliography.- Appendices.- Appendix 1. The Method of Majorants.- Appendix 2. Some Complex Analysis.- Appendix 3. Weierstrass’ Formula.- Appendix 4. Derivation of Euler’s Equations.- Appendix 5. Calculus of Variations.- Appendix 6. Jacobi’s Last-Multiplier Method.