Variational Methods in Mathematics, Science and Engineering

1. Front Matter

   Pages 1-9

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2. Preface

   1. Preface

      • Karel Rektorys

      Pages 11-14

3. Notation Frequently Used

   1. Notation Frequently Used

      • Karel Rektorys

      Pages 15-16

4. Introduction

   1. Introduction

      • Karel Rektorys

      Pages 17-20

5. Hilbert Space

   1. Inner Product of Functions. Norm, Metric

      • Karel Rektorys

      Pages 21-31

   2. The Space L₂

      • Karel Rektorys

      Pages 32-37

   3. Convergence in the Space L₂(G) (Convergence in the Mean). Complete Space. Separable Space

      • Karel Rektorys

      Pages 38-46

   4. Orthogonal Systems in L₂(G)

      • Karel Rektorys

      Pages 47-65

   5. Hilbert Space

      • Karel Rektorys

      Pages 66-80

   6. Some Remarks to the Preceding Chapters. Normed Space, Banach Space

      • Karel Rektorys

      Pages 81-85

   7. Operators and Functionals, especially in Hilbert Spaces

      • Karel Rektorys

      Pages 86-112

6. Variational Methods

   1. Theorem on the Minimum of a Quadratic Functional and its Consequences

      • Karel Rektorys

      Pages 113-120

   2. The Space H _A

      • Karel Rektorys

      Pages 121-132

   3. Existence of the Minimum of the Functional F in the Space H _A . Generalized Solutions

      • Karel Rektorys

      Pages 133-145

   4. The Method of Orthonormal Series. Example

      • Karel Rektorys

      Pages 146-152

   5. The Ritz Method

      • Karel Rektorys

      Pages 153-160

   6. The Galerkin Method

      • Karel Rektorys

      Pages 161-165

   7. The Least Squares Method. The Courant Method

      • Karel Rektorys

      Pages 166-171

   8. The Method of Steepest Descent. Example

      • Karel Rektorys

      Pages 172-177

   9. Summary of Chapters 9 to 16

      • Karel Rektorys

      Pages 178-185

The impulse which led to the writing of the present book has emerged from my many years of lecturing in special courses for selected students at the College of Civil Engineering of the Tech­ nical University in Prague, from experience gained as supervisor and consultant to graduate students-engineers in the field of applied mathematics, and - last but not least - from frequent consultations with technicians as well as with physicists who have asked for advice in overcoming difficulties encountered in solving theoretical problems. Even though a varied combination of problems of the most diverse nature was often in question, the problems discussed in this book stood forth as the most essential to this category of specialists. The many discussions I have had gave rise to considerations on writing a book which should fill the rather unfortunate gap in our literature. The book is designed, in the first place, for specialists in the fields of theoretical engineering and science. However, it was my aim that the book should be of interest to mathematicians as well. I have been well aware what an ungrateful task it may be to write a book of the present type, and what problems such an effort can bring: Technicians and physicists on the one side, and mathematicians on the other, are often of diametrically opposing opinions as far as books con­ ceived for both these categories are concerned.

• Mathematica
• applied mathematics
• design
• differential equation
• elasticity
• finite element method
• operator
• partial differential equation
• science

• Prague, Czech Republic

  Karel Rektorys

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