- Membership
- Publications
- Meetings
- Competitions
- Community
- Programs
- Students
- High School Teachers
- Faculty and Departments
- Underrepresented Groups
- MAA Awards
- MAA Grants

- News
- About MAA

Publisher:

Dover Publications

Publication Date:

1974

Number of Pages:

326

Format:

Paperback

Price:

14.95

ISBN:

0486630692

Category:

Monograph

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

There is no review yet. Please check back later.

Preface | ||||||||

Chapter 1. | Introduction | |||||||

Chapter 2. | Background Preliminaries | |||||||

1. Piecewise continuity, piecewise differentiability 2. Partial and total differentiation 3. Differentiation of an integral 4. Integration by parts 5. Euler's theorem on homogeneous functions | ||||||||

6. Method of undetermined lagrange multipliers 7. The line integral 8. Determinants 9. Formula for surface area 10. Taylor's theorem for functions of several variables | ||||||||

11. The surface integral 12. Gradient, laplacian 13. Green's theorem (two dimensions) 14. Green's theorem (three dimensions) | ||||||||

Chapter 3. | Introductory Problems | |||||||

1. A basic lemma 2. Statement and formulation of several problems 3. The Euler-Lagrange equation 4. First integrals of the Euler-Lagrange equation. A degenerate case 5. Geodesics | ||||||||

6. The brachistochrone 7. Minimum surface of revolution 8. Several dependent variables 9. Parametric representation | ||||||||

10. Undetermined end points 11. Brachistochrone from a given curve to a fixed point | ||||||||

Chapter 4. | Isoperimetric Problems | |||||||

1. The simple isoperimetric problem 2. Direct extensions 3. Problem of the maximum enclosed area 4. Shape of a hanging rope. 5. Restrictions imposed through finite or differential equations | ||||||||

Chapter 5. | Geometrical Optics: Fermat's Principle | |||||||

1. Law of refraction (Snell's law) 2. Fermat's principle and the calculus of variations | ||||||||

Chapter 6. | Dynamics of Particles | |||||||

1. Potential and kinetic energies. 2. Generalized coordinates 3. Hamilton's principle. Lagrange equations of motion 3. Generalized momenta. Hamilton equations of motion. | ||||||||

4. Canonical transformations 5. The Hamilton-Jacobi differential equation 6. Principle of least action 7. The extended Hamilton's principle | ||||||||

Chapter 7. | Two Independent Variables: The Vibrating String | |||||||

1. Extremization of a double integral 2. The vibrating string 3. Eigenvalue-eigenfunction problem for the vibrating string | ||||||||

4. Eigenfunction expansion of arbitrary functions. Minimum characterization of the eigenvalue-eigenfunction problem 5. General solution of the vibrating-string equation | ||||||||

6. Approximation of the vibrating-string eigenvalues and eigenfunctions (Ritz method) 7. Remarks on the distinction between imposed and free end-point conditions | ||||||||

Chapter 8. | The Sturm-Liouville Eigenvalue-Eigenfunction Problem | |||||||

1. Isoperimetric problem leading to a Sturm-Liouville system 2. Transformation of a Sturm-Liouville system 3. Two singular cases: Laguerre polynomials, Bessel functions | ||||||||

Chapter 9. | Several Independent Variables: The Vibrating Membrane | |||||||

1. Extremization of a multiple integral 2. Change of independent variables. Transformation of the laplacian 3. The vibrating membrane 4. Eigenvalue-eigenfunction problem for the membrane | ||||||||

5. Membrane with boundary held elastically. The free membrane 6. Orthogonality of the eigenfunctions. Expansion of arbitrary functions 7. General solution of the membrane equation | ||||||||

8. The rectangular membrane of uniform density 9. The minimum characterization of the membrane eigenvalues 10. Consequences of the minimum characterization of the membrane eigenvalues | ||||||||

11. The maximum-minimum characterization of the membrane eigenvalues 12. The asymptotic distribution of the membrane eigenvalues 13. Approximation of the membrane eigenvalues | ||||||||

Chapter 10. | Theory of Elasticity | |||||||

1. Stress and strain 2. General equations of motion and equilibrium 3. General aspects of the approach to certain dynamical problems 4. Bending of a cylindrical bar by couples | ||||||||

5. Transverse vibrations of a bar 6. The eigenvalue-eigenfunction problem for the vibrating bar 7. Bending of a rectangular plate by couples 8. Transverse vibrations of a thin plate | ||||||||

9. The eigenvalue-eigenfunction problem for the vibrating plate 10. The rectangular plate. Ritz method of approximation | ||||||||

Chapter 11. | Quantum Mechanics | |||||||

1. First derivation of the Schrödinger equation for a single particle 2. The wave character of a particle. Second derivation of the Schrödinger equation | ||||||||

3. The hydrogen atom. Physical interpretation of the Schrödinger wave functions 4. Extension to systems of particles. Minimum character of the energy eigenvalues | ||||||||

5. Ritz method: Ground state of the helium atom. Hartree model of the many-electron atom | ||||||||

Chapter 12. | Electrostatics | |||||||

1. Laplace's equation. Capacity of a condenser 2. Approximation of the capacity from below (relaxed boundary conditions) 3. Remarks on problems in two dimensions | ||||||||

4. The existence of minima of the Dirichlet integral | ||||||||

Bibliography; Index |

- Log in to post comments