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| Preface | |
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| Introduction | |
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| Background Preliminaries | |
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| Piecewise continuity, piecewise differentiability | |
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| Partial and total differentiation | |
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| Differentiation of an integral | |
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| Integration by parts | |
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| Euler's theorem on homogeneous functions | |
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| Method of undetermined lagrange multipliers | |
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| The line integral | |
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| Determinants | |
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| Formula for surface area | |
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| Taylor's theorem for functions of several variables | |
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| The surface integral | |
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| Gradient, laplacian | |
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| Green's theorem (two dimensions) | |
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| Green's theorem (three dimensions) | |
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| Introductory Problems | |
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| A basic lemma | |
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| Statement and formulation of several problems | |
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| The Euler-Lagrange equation | |
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| First integrals of the Euler-Lagrange equation | |
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| A degenerate case | |
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| Geodesics | |
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| The brachistochrone | |
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| Minimum surface of revolution | |
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| Several dependent variables | |
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| Parametric representation | |
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| Undetermined end points | |
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| Brachistochrone from a given curve to a fixed point | |
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| Isoperimetric Problems | |
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| The simple isoperimetric problem | |
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| Direct extensions | |
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| Problem of the maximum enclosed area | |
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| Shape of a hanging rope | |
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| Restrictions imposed through finite or differential equations | |
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| Geometrical Optics: Fermat's Principle | |
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| Law of refraction (Snell's law) | |
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| Fermat's principle and the calculus of variations | |
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| Dynamics of Particles | |
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| Potential and kinetic energies | |
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| Generalized coordinates | |
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| Hamilton's principle | |
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| Lagrange equations of motion | |
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| Generalized momenta | |
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| Hamilton equations of motion | |
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| Canonical transformations | |
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| The Hamilton-Jacobi differential equation | |
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| Principle of least action | |
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| The extended Hamilton's principle | |
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| Two Independent Variables: The Vibrating String | |
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| Extremization of a double integral | |
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| The vibrating string | |
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| Eigenvalue-eigenfunction problem for the vibrating string | |
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| Eigenfunction expansion of arbitrary functions | |
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| Minimum characterization of the eigenvalue-eigenfunction problem | |
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| General solution of the vibrating-string equation | |
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| Approximation of the vibrating-string eigenvalues and eigenfunctions (Ritz method) | |
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| Remarks on the distinction between imposed and free end-point conditions | |
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| The Sturm-Liouville Eigenvalue-Eigenfunction Problem | |
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| Isoperimetric problem leading to a Sturm-Liouville system | |
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| Transformation of a Sturm-Liouville system | |
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| Two singular cases: Laguerre polynomials, Bessel functions | |
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| Several Independent Variables: The Vibrating Membrane | |
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| Extremization of a multiple integral | |
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| Change of independent variables | |
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| Transformation of the laplacian | |
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| The vibrating membrane | |
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| Eigenvalue-eigenfunction problem for the membrane | |
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| Membrane with boundary held elastically | |
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| The free membrane | |
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| Orthogonality of the eigenfunctions | |
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| Expansion of arbitrary functions | |
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| General solution of the membrane equation | |
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| The rectangular membrane of uniform density | |
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| The minimum characterization of the membrane eigenvalues | |
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| Consequences of the minimum characterization of the membrane eigenvalues | |
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| The maximum-minimum characterization of the membrane eigenvalues | |
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| The asymptotic distribution of the membrane eigenvalues | |
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| Approximation of the membrane eigenvalues | |
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| Theory of Elasticity | |
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| Stress and strain | |
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| General equations of motion and equilibrium | |
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| General aspects of the approach to certain dynamical problems | |
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| Bending of a cylindrical bar by couples | |
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| Transverse vibrations of a bar | |
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| The eigenvalue-eigenfunction problem for the vibrating bar | |
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| Bending of a rectangular plate by couples | |
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| Transverse vibrations of a thin plate | |
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| The eigenvalue-eigenfunction problem for the vibrating plate | |
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| The rectangular plate | |
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| Ritz method of approximation | |
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| Quantum Mechanics | |
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| First derivation of the Schr�dinger equation for a single particle | |
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| The wave character of a particle. Sec | |