Some Remarks on the History and Objectives of the Calculus of Variations -- 1. Direct Methods of the Calculus of Variations -- 1.1 The Fundamental Theorem of the Calculus of Variations -- 1.2 Applying the Fundamental Theorem in Banach Spaces -- 1.3 Minimising Special Classes of Functions -- 1.4 Some Remarks on Linear Optimisation -- 1.5 Ritz's Approximation Method -- 2. Differential Calculus in Banach Spaces -- 2.1 General Remarks -- 2.2 The Fréchet Derivative -- 2.3 The Gâteaux Derivative -- 2.4 nth Variation -- 2.5 The Assumptions of the Fundamental Theorem of Variational Calculus -- 2.6 Convexity of f and Monotonicity of f ? -- 3. Extrema of Differentiable Functions -- 3.1 Extrema and Critical Values -- 3.2 Necessary Conditions for an Extremum -- 3.3 Sufficient Conditions for an Extremum -- 4. Constrained Minimisation Problems (Method of Lagrange Multipliers) -- 4.1 Geometrical Interpretation of Constrained Minimisation Problems -- 4.2 Ljusternik's Theorems -- 4.3 Necessary and Sufficient Conditions for Extrema Subject to Constraints -- 4.4 A Special Case -- 5. Classical Variational Problems -- 5.1 General Remarks -- 5.2 Hamilton's Principle in Classical Mechanics -- 5.3 Symmetries and Conservation Laws in Classical Mechanics -- 5.4 The Brachystochrone Problem -- 5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory -- 5.6 Noether's Theorem in Classical Field Theory -- 5.7 The Principle of Symmetric Criticality -- 6. The Variational Approach to Linear Boundary and Eigenvalue Problems -- 6.1 The Spectral Theorem for Compact Self-Adjoint Operators. Courant's Classical Minimax Principle. Projection Theorem -- 6.2 Differential Operators and Forms -- 6.3 The Theorem of Lax-Milgram and Some Generalisations -- 6.4 The Spectrum of Elliptic Differential Operators in a Bounded Domain. Some Problems from Classical Potential Theory -- 6.5 Variational Solution of Parabolic Differential Equations. The Heat Conduction Equation. The Stokes Equations -- 7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators -- 7.1 Forms and Operators - Boundary Value Problems -- 7.2 Surjectivity of Coercive Monotonic Operators. Theorems of Browder and Minty -- 7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution -- 8. Nonlinear Elliptic Eigenvalue Problems -- 8.1 Introduction -- 8.2 Determination of the Ground State in Nonlinear Elliptic Eigenvalue Problems -- 8.3 Ljusternik-Schnirelman Theory for Compact Manifolds -- 8.4 The Existence of Infinitely Many Solutions of Nonlinear Elliptic Eigenvalue Problems -- 9. Semilinear Elliptic Differential Equations. Some Recent Results on Global Solutions -- 9.1 Introduction -- 9.2 Technical Preliminaries -- 9.3 Some Properties of Weak Solutions of Semilinear Elliptic Equations -- 9.4 Best Constant in Sobolev Inequality -- 9.5 The Local Case with Critical Sobolev Exponent -- 9.6 The Constrained Minimisation Method Under Scale Covariance -- 9.7 Existence of a Minimiser I: Some General Results -- 9.8 Existence of a Minimiser II: Some Examples -- 9.9 Nonlinear Field Equations in Two Dimensions -- 9.10 Conclusion and Comments -- 9.11 Complementary Remarks -- 10. Thomas-Fermi Theory -- 10.1 General Remarks -- 10.2 Some Results from the Theory of Lp Spaces (1 ? p ? ?) -- 10.3 Minimisation of the Thomas-Fermi Energy Functional -- 10.4 Thomas-Fermi Equations and the Minimisation Problem for the TF Functional -- 10.5 Solution of TF Equations for Potentials of the Form$$V\left( x \right) = \Sigma _{j = 1}^k\frac{{{z_j}}}{{\left/ {x - {x_j}} \right/}}$$ -- 10.6 Remarks on Recent Developments in Thomas-Fermi and Related Theories -- Appendix A. Banach Spaces -- Appendix B. Continuity and Semicontinuity -- Appendix C. Compactness in Banach Spaces -- D.1 Definition and Properties -- D.2 Poincaré's Inequality -- D.3 Continuous Embeddings of Sobolev Spaces -- D.4 Compact Embeddings of Sobolev Spaces -- Appendix E -- E.1 Bessel Potentials -- E.2 Some Properties of Weakly Differentiable Functions -- E.3 Proof of Theorem 9.2.3 -- References -- Index of Names.