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Chapter 0. Introduction -- 1. Logic: quantifiers -- 2. The logical connectives -- 3. Negations of quantifiers -- 4. Sets -- 5. Restricted variables -- 6. Ordered pairs and relations -- 7. Functions and mappings -- 8. Product sets; index notation -- 9. Composition -- 10. Duality -- 11. The Boolean operations -- 12. Partitions and equivalence relations -- Chapter 1. Vector Spaces -- 1. Fundamental notions -- 2. Vector spaces and geometry -- 3. Product spaces and Horn(V, W) -- 4. Affine subspaces and quotient spaces -- 5. Direct sums -- 6. Bilinearity -- Chapter 2. Finite-Dimensional Vector Spaces -- 1. Bases -- 2. Dimension -- 3. The dual space -- 4. Matrices -- 5. Trace and determinant -- 6. Matrix computations -- 7. The diagonalization of a quadratic form -- Chapter 3. The Differential Calculus -- 1. Review in R -- 2. Norms -- 3. Continuity -- 4. Equivalent norms -- 5. Infinitesimals -- 6. The differential -- 7. Directional derivatives; the mean-value theorem -- 8. The differential and product spaces -- 9. The differential and Rn -- 10. Elementary application? -- 11. The implicit-function theorem -- 12. Submanifolds and Lagrange multipliers -- 13. Functional dependence -- 14. Uniform continuity and function-valued mappings -- 15. The calculus of variations -- 16. The second differential and the classification of critical points -- 17. The Taylor formula -- Chapter 4. Compactness and Completeness -- 1. Metric spaces; open and closed sets -- 2. Topology -- 3. Sequential convergence -- 4. Sequential compactness -- 5. Compactness and uniformity -- 6. Equicontinuity -- 7. Completeness -- 8. A first look at Banach algebras -- 9. The contraction mapping fixed-point theorem -- 10. The integral of a parametrized arc -- 11. The complex number system -- 12. Weak methods -- Chapter 5. Scalar Product Spaces -- 1. Scalar products -- 2. Orthogonal projection -- 3. Sell-adjoint transformations -- 4. Orthogonal transformations -- 5. Compact transformations -- Chapter 6. Differential Equations -- 1. The fundamental theorem -- 2. Differentiable dependence on parameters -- 3. The linear equation -- 4. The nth-order linear equation -- 5. Solving the inhomogeneous equation -- 6. The boundary-value problem -- 7. Fourier series -- Chapter 7. Multilinear Functionals -- 1. Bilinear functionals -- 2. Multilinear functionals -- 3. Permutations -- 4. The sign of a permutation -- 5. The subspace an of alternating tensors -- 6. The determinant -- 7. The exterior algebra -- 8. Exterior powers of scalar product spaces -- 9. The star operator -- Chapter 8. Integration -- 1. Introduction -- 2. Axioms -- 3. Rectangles and paved sets -- 4. The minimal theory -- 5. The minimal theory (continued) -- 6. Contented sets -- 7. When is a set contented? -- 8. Behavior under linear distortions -- 9. Axioms for integration -- 10. Integration of contented functions -- 11. The change of variables formula -- 12. Successive integration -- 13. Absolutely integrable functions -- 14. Problem set: The Fourier transform -- Chapter 9. Differentiate Manifolds -- 1. Atlases -- 2. Functions, convergence -- 3. Differentiable manifolds -- 4. The tangent space -- 5. Flows and vector fields -- 6. Lie derivatives -- 7. Linear differential forms -- 8. Computations with coordinates -- 9. Riemann metrics -- Chapter 10. The Integral Calculus on Manifolds -- 1. Compactness -- 2. Partitions of unity -- 3. Densities -- 4. Volume density of a Riemann metric -- 5. Fullback and Lie derivatives of densities -- 6. The divergence theorem -- 7. Afore complicated domains -- Chapter 11. Exterior Calculus -- 1. Exterior differential forms -- 2. Oriented manifolds and the integration of exterior differential forms -- 3. The operator d -- 4. Stokes' theorem -- 5. Some illustrations of Stokes' theorem -- 6. The Lie derivative of a differential form -- Appendix I. Vector analysis -- Appendix II. Elementary differential geometry of surfaces in E3 -- Chapter 12. Potential Theory in En -- 1. Solid angle -- 2. Green's formulas -- 3. The maximum principle -- 6. Green's functions -- 5. The Poisson integral formula -- 6. Consequences of the Poisson integral formula -- 7. Harnack's theorem -- 8. Subharmonic functions -- 9. Dirichlet's problem -- 10. Behavior near the boundary -- 11. Dirichlet's principle -- 12. Physical applications -- 13. Problem set: The calculus of residues -- Chapter 13. Classical Mechanics -- 1. The tangent and cotangent bundles -- 2. Equations of variation -- 3. The fundamental linear differential form on T*(M) -- 4. The fundamental exterior two-form on T*(M) -- 5. Hamiltonian mechanics -- 6. The central-force problem -- 7. The two-body problem -- 8. Lagrange's equations -- 9. Variational principles -- 10. Geodesic coordinates -- 11. Euler's equations -- 12. Rigid-body motion -- 13. Small oscillations -- 14. Small oscillations (continued) -- 15. Canonical transformations -- Selected References -- Notation Index -- Index.

An authorised reissue of the long out of print classic textbook, Advanced Calculus by the late Dr Lynn Loomis and Dr Shlomo Sternberg both of Harvard University has been a revered but hard to find textbook for the advanced calculus course for decades.
This book is based on an honors course in advanced calculus that the authors gave in the 1960's. The foundational material, presented in the unstarred sections of Chapters 1 through 11, was normally covered, but different applications of this basic material were stressed from year to year, and the book therefore contains more material than was covered in any one year. It can accordingly be used (with omissions) as a text for a year's course in advanced calculus, or as a text for a three-semester introduction to analysis.

The prerequisites are a good grounding in the calculus of one variable from a mathematically rigorous point of view, together with some acquaintance with linear algebra. The reader should be familiar with limit and continuity type arguments and have a certain amount of mathematical sophistication. As possible introductory texts, we mention Differential and Integral Calculus by R Courant, Calculus by T Apostol, Calculus by M Spivak, and Pure Mathematics by G Hardy. The reader should also have some experience with partial derivatives.

In overall plan the book divides roughly into a first half which develops the calculus (principally the differential calculus) in the setting of normed vector spaces, and a second half which deals with the calculus of differentiable manifolds.

Readership: Undergraduates in mathematics.