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Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at www.cambridge.org/9781107042193.

Geometry, Differential --- Manifolds (Mathematics) --- Calculus of tensors --- Forms (Mathematics)

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Arising from the fourth Dagstuhl conference entitled Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (2011), this book offers a broad and vivid view of current work in this emerging field. Topics covered range from applications of the analysis of tensor fields to research on their mathematical and analytical properties. Part I, Tensor Data Visualization, surveys techniques for visualization of tensors and tensor fields in engineering, discusses the current state of the art and challenges, and examines tensor invariants and glyph design, including an overview of common glyphs. The second Part, Representation and Processing of Higher-order Descriptors, describes a matrix representation of local phase, outlines mathematical morphological operations techniques, extended for use in vector images, and generalizes erosion to the space of diffusion weighted MRI. Part III, Higher Order Tensors and Riemannian-Finsler Geometry, offers powerful mathematical language to model and analyze large and complex diffusion data such as High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Kurtosis Imaging (DKI). A Part entitled Tensor Signal Processing presents new methods for processing tensor-valued data, including a novel perspective on performing voxel-wise morphometry of diffusion tensor data using kernel-based approach, explores the free-water diffusion model, and reviews proposed approaches for computing fabric tensors, emphasizing trabecular bone research. The last Part, Applications of Tensor Processing, discusses metric and curvature tensors, two of the most studied tensors in geometry processing. Also covered is a technique for diagnostic prediction of first-episode schizophrenia patients based on brain diffusion MRI data. The last chapter presents an interactive system integrating the visual analysis of diffusion MRI tractography with data from electroencephalography.

Calculus of tensors. --- Calculus of tensors --- Information visualization. --- Data processing. --- Data visualization --- Visualization of information --- Information science --- Visual analytics --- Absolute differential calculus --- Analysis, Tensor --- Calculus, Absolute differential --- Calculus, Tensor --- Tensor analysis --- Tensor calculus --- Geometry, Differential --- Geometry, Infinitesimal --- Vector analysis --- Spinor analysis --- Visualization. --- Differential equations, partial. --- Global differential geometry. --- Computer vision. --- Computer graphics. --- Partial Differential Equations. --- Differential Geometry. --- Computer Imaging, Vision, Pattern Recognition and Graphics. --- Computer Graphics. --- Theoretical, Mathematical and Computational Physics. --- Automatic drafting --- Graphic data processing --- Graphics, Computer --- Computer art --- Graphic arts --- Electronic data processing --- Engineering graphics --- Image processing --- Machine vision --- Vision, Computer --- Artificial intelligence --- Pattern recognition systems --- Partial differential equations --- Visualisation --- Imagination --- Visual perception --- Imagery (Psychology) --- Digital techniques --- Mathematics. --- Partial differential equations. --- Differential geometry. --- Optical data processing. --- Mathematical physics. --- Optical computing --- Visual data processing --- Bionics --- Integrated optics --- Photonics --- Computers --- Differential geometry --- Physical mathematics --- Physics --- Math --- Science --- Optical equipment --- Mathematics

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The aim of this work is to provide a proof of the nonlinear gravitational stability of the Minkowski space-time. More precisely, the book offers a constructive proof of global, smooth solutions to the Einstein Vacuum Equations, which look, in the large, like the Minkowski space-time. In particular, these solutions are free of black holes and singularities. The work contains a detailed description of the sense in which these solutions are close to the Minkowski space-time, in all directions. It thus provides the mathematical framework in which we can give a rigorous derivation of the laws of gravitation proposed by Bondi. Moreover, it establishes other important conclusions concerning the nonlinear character of gravitational radiation. The authors obtain their solutions as dynamic developments of all initial data sets, which are close, in a precise manner, to the flat initial data set corresponding to the Minkowski space-time. They thus establish the global dynamic stability of the latter.Originally published in 1994.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Space and time --- Generalized spaces --- Nonlinear theories --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Nonlinear problems --- Nonlinearity (Mathematics) --- Calculus --- Mathematical analysis --- Mathematical physics --- Geometry of paths --- Minkowski space --- Spaces, Generalized --- Weyl space --- Calculus of tensors --- Geometry, Differential --- Geometry, Non-Euclidean --- Hyperspace --- Relativity (Physics) --- Space of more than three dimensions --- Space-time --- Space-time continuum --- Space-times --- Spacetime --- Time and space --- Fourth dimension --- Infinite --- Metaphysics --- Philosophy --- Space sciences --- Time --- Beginning --- Mathematics --- Angular momentum operator. --- Asymptotic analysis. --- Asymptotic expansion. --- Big O notation. --- Boundary value problem. --- Cauchy–Riemann equations. --- Coarea formula. --- Coefficient. --- Compactification (mathematics). --- Comparison theorem. --- Corollary. --- Covariant derivative. --- Curvature tensor. --- Curvature. --- Cut locus (Riemannian manifold). --- Degeneracy (mathematics). --- Degrees of freedom (statistics). --- Derivative. --- Diffeomorphism. --- Differentiable function. --- Eigenvalues and eigenvectors. --- Eikonal equation. --- Einstein field equations. --- Equation. --- Error term. --- Estimation. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Exponential map (Lie theory). --- Exponential map (Riemannian geometry). --- Exterior (topology). --- Foliation. --- Fréchet derivative. --- Geodesic curvature. --- Geodesic. --- Geodesics in general relativity. --- Geometry. --- Hodge dual. --- Homotopy. --- Hyperbolic partial differential equation. --- Hypersurface. --- Hölder's inequality. --- Identity (mathematics). --- Infinitesimal generator (stochastic processes). --- Integral curve. --- Intersection (set theory). --- Isoperimetric inequality. --- Laplace's equation. --- Lie algebra. --- Lie derivative. --- Linear equation. --- Linear map. --- Logarithm. --- Lorentz group. --- Lp space. --- Mass formula. --- Mean curvature. --- Metric tensor. --- Minkowski space. --- Nonlinear system. --- Normal (geometry). --- Null hypersurface. --- Orthonormal basis. --- Partial derivative. --- Poisson's equation. --- Projection (linear algebra). --- Quantity. --- Radial function. --- Ricci curvature. --- Riemann curvature tensor. --- Riemann surface. --- Riemannian geometry. --- Riemannian manifold. --- Sard's theorem. --- Scalar (physics). --- Scalar curvature. --- Scale invariance. --- Schwarzschild metric. --- Second derivative. --- Second fundamental form. --- Sobolev inequality. --- Sobolev space. --- Stokes formula. --- Stokes' theorem. --- Stress–energy tensor. --- Symmetric tensor. --- Symmetrization. --- Tangent space. --- Tensor product. --- Theorem. --- Trace (linear algebra). --- Transversal (geometry). --- Triangle inequality. --- Uniformization theorem. --- Unit sphere. --- Vector field. --- Volume element. --- Wave equation. --- Weyl tensor.