This book was written for advanced undergraduate math or science majors. Its initial purpose was to illustrate the elementary mathematical theory of ordinary differential equations and their diverse and powerful applications. Historically these have been decisive in many physical problems, some of which have philosophically challenged and indeed altered our civilization's concepts. Because of the importance of the subject, the book is also suitable for a one-semester course for graduate students. The book consists of 12 chapters and six appendices.

Contents:

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  An Introduction to First-Order Ordinary Differential Equations

  
  


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  Planetary Motion

  
  


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  Second-Order Ordinary Differential Equations

  
  


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  Some More Advanced Topics in ODEs

  
  


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  Approximation of Solutions

  
  


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  Minding's Theorem, Sturm's Comparison Theorem and Other Results Concerning the Application of ODE to the Differential Geometry of Surfaces

  
  


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  2 × 2 Linear Systems

  
  


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  Autonomous Dynamical Systems and the Poincare–Bendixson Theorem

  
  


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  Matrix Differential Equations and the Matrix Exponential Function

  
  


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  Classical Partial Differential Equations of the Second Order

  
  


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  An Introduction to the Calculus of Variations

  
  


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  The Gauss Bonnet Theorem for Surfaces in ℝ3

  
  


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  Appendices:

  
  
  
  
  • The Gaussian Distribution
  
  
  • Contraction Mappings and Picard's Existence Theorem
  
  
  • Stokes' Theorem
  
  
  • Real Analytic Functions
  
  
  • Fourier Series
  
  
  • Special Relativity
  
  
  
  

Readership: Advanced undergraduate math or science majors, researchers.