Based on author Siavash Shahshahani's extensive teaching experience, this volume presents a thorough, rigorous course on the theory of differentiable manifolds. Geared toward advanced undergraduates and graduate students in mathematics, the treatment's prerequisites include a strong background in undergraduate mathematics, including multivariable calculus, linear algebra, Based on author Siavash Shahshahani's extensive teaching experience, this volume presents a thorough, rigorous course on the theory of differentiable manifolds. Geared toward advanced undergraduates and graduate students in mathematics, the treatment's prerequisites include a strong background in undergraduate mathematics, including multivariable calculus, linear algebra, elementary abstract algebra, and point set topology. More than 200 exercises offer students ample opportunity to gauge their skills and gain additional insights. The four-part treatment begins with a single chapter devoted to the tensor algebra of linear spaces and their mappings. Part II brings in neighboring points to explore integrating vector fields, Lie bracket, exterior derivative, and Lie derivative. Part III, involving manifolds and vector bundles, develops the main body of the course. The final chapter provides a glimpse into geometric structures by introducing connections on the tangent bundle as a tool to implant the second derivative and the derivative of vector fields on the base manifold. Relevant historical and philosophical asides enhance the mathematical text, and helpful Appendixes offer supplementary material.
An Introductory Course on Differentiable Manifolds
Based on author Siavash Shahshahani's extensive teaching experience, this volume presents a thorough, rigorous course on the theory of differentiable manifolds. Geared toward advanced undergraduates and graduate students in mathematics, the treatment's prerequisites include a strong background in undergraduate mathematics, including multivariable calculus, linear algebra, Based on author Siavash Shahshahani's extensive teaching experience, this volume presents a thorough, rigorous course on the theory of differentiable manifolds. Geared toward advanced undergraduates and graduate students in mathematics, the treatment's prerequisites include a strong background in undergraduate mathematics, including multivariable calculus, linear algebra, elementary abstract algebra, and point set topology. More than 200 exercises offer students ample opportunity to gauge their skills and gain additional insights. The four-part treatment begins with a single chapter devoted to the tensor algebra of linear spaces and their mappings. Part II brings in neighboring points to explore integrating vector fields, Lie bracket, exterior derivative, and Lie derivative. Part III, involving manifolds and vector bundles, develops the main body of the course. The final chapter provides a glimpse into geometric structures by introducing connections on the tangent bundle as a tool to implant the second derivative and the derivative of vector fields on the base manifold. Relevant historical and philosophical asides enhance the mathematical text, and helpful Appendixes offer supplementary material.
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Fantastic book for learning about topological groups. The first reading can go a bit slow due to the terse notation, but coming at it with some background in functional analysis and abstract algebra, then this book can get very exciting. It begins by constructing the antisymmetric product algebra of differential forms using the multilinear dual-basis tensor product. It goes on to study vector fields and tensor fields locally. The rest of the book discusses manifolds along with an accompanying ch Fantastic book for learning about topological groups. The first reading can go a bit slow due to the terse notation, but coming at it with some background in functional analysis and abstract algebra, then this book can get very exciting. It begins by constructing the antisymmetric product algebra of differential forms using the multilinear dual-basis tensor product. It goes on to study vector fields and tensor fields locally. The rest of the book discusses manifolds along with an accompanying chapter on topological groups. This is not an intuitive book and requires a very careful reading.
Saman –
Euclidean spaces are the most natural spaces in Mathematics, one dimensional Euclidean space is a line, 2 dimensional Euclidean space is a plane, 3 dimensional Euclidean space is the space as you see around yourself. Although we can't imagine higher dimensions visually, mathematically we can generalize this notion to higher dimensions. However manifolds are even more complicated objects than n-dimensional Euclidean spaces. Considering the world around us a 3-dimensional Euclidean space can be as Euclidean spaces are the most natural spaces in Mathematics, one dimensional Euclidean space is a line, 2 dimensional Euclidean space is a plane, 3 dimensional Euclidean space is the space as you see around yourself. Although we can't imagine higher dimensions visually, mathematically we can generalize this notion to higher dimensions. However manifolds are even more complicated objects than n-dimensional Euclidean spaces. Considering the world around us a 3-dimensional Euclidean space can be as wrong as assuming that the earth we are living on is a plane. These spaces on a small scale look like a Euclidean space, but on bigger scale they can be very different. We call such spaces Manifolds. This book is an introductory book on mathematical study of these objects. The book is rigid, computational and not intuitive. It can be a good book for both math and physics students, containing many exercises and complete proofs.
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