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prefacebackground notationchapter 1  topology  1.1 topological spaces  1.2 metric spaces  1.3 continuity  1.4 subspaces, products, and quotients  1.5 compactness  1.6 connectedness  1.7 baire spaceschapter 2  banach spaces and differential calculus  2.1 banach spaces  2.2 linear and multilinear mappings  2.3 the derivative  2.4 properties of the derivative  2.5 the inverse and implicit function theoremschapter 3  manifolds and vector bundles  3.1 manifolds  3.2 submanifolds, products, and mappings  3.3 the tangent bundle  3.4 vector bundles  3.5 submersions, immersions and transversalitychapter 4  vector fields and dynamical systems  4.1 vector fields and rows  4.2 vector fields as differential operators  4.3 an introduction to dynamical systems  4.4 frobenius' theorem and foliationschapter 5  tensors  5.1 tensors in linear spaces  5.2 tensor bundles and tensor fields  5.3 the lie derivative: algebraic approach  5.4 the lie derivative: dynamic approach  5.5 partitions of unitychapter 6  differential forms  6.1 exterior algebra  6.2 determinants, volumes. and the hodge star operator  6.3 differential forms  6.4 the exterior derivative, interior product. and lie derivative  6.5 orientation, volume elements, and the codifferentialchapter 7  integration on manifolds  7.1 the definition of the integral  7.2 stokes theorem  7.3 the classical theorems of green, gauss, and stokes  7.4 induced flows on function spaces and ergodicity  7.5 introduction to hodge-derham theory and topological applications of differential formschapter 8  applications  8.1 hamiltonian mechanics  8.2 fluid mechanics  8.3 electromagnetism  8.3 the lie-poisson bracket in continuum mechanics and plasma physics  8.4 constraints and controlreferencesindexsupplementary chapters--available from the authors as they are produced  s-1 lie groups  s-2 introduction to differential topology  s-3 topics in riemannian geometry