Visual Differential Geometry and Forms (eBook)

A Mathematical Drama in Five Acts

Tristan Needham (Autor)

eBook Download: PDF

2021
536 Seiten
Princeton University Press (Verlag)
978-0-691-21989-9 (ISBN)

Infos

An inviting, intuitive, and visual exploration of differential geometry and formsVisual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner.Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book.Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.

Produkt-Details

• Erscheint lt. Verlag: 13.7.2021
• Zusatzinfo: 235 b/w illus.
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Schlagworte: acceleration • A Geometric Approach to Differential Forms • Angular momentum • asymptotic analysis • Asymptotic curve • A Visual Introduction to Differential Forms and Calculus on Manifolds • Bending • bisection • Cartan • Cartesian coordinate system • Complex Analysis • complex number • complex plane • Compounding • conformal map • conjugate points • Contour line • coordinate system • Covariance and contravariance of vectors • Curvature • Curve • curved surfaces • David Bachman • Derivative • Diagram (category theory) • dichotomy • Differentiable function • differential form • Differential Geometry • Differential geometry of curves • differential geometry of surfaces • Dimensional Analysis • Divergence Theorem • Einstein • electromagnetism • Elliptic Integral • Equation • Equations of motion • equilateral triangle • Euclidean Geometry • exterior derivative • Exterior (topology) • Facet (geometry) • Faraday • Gaussian curvature • Generalized Stokes’s Theorem • Generalized Stokes Theorem • Generalized Stokes’ Theorem • General relativity • geodesic • Geodesic Curvature • Geometric • geometric mean • Geometry • Geometry and Topology • Gravitation • Great Circle • holonomy • Hyperbolic Geometry • Inequality (mathematics) • Integral curve • Intersection curve • Inverse-square law • isometry • Jon Pierre Fortney • Limit (mathematics) • Linear map • Line element • Line (geometry) • manifold • Manifolds • Mathematics • Maxwell • Maxwell's equations • Measurement • Metric Space • Metric tensor • moving frames • Newton • Non-Euclidean geometry • Normal (geometry) • Notation • orthogonality • parallelogram • Parallel Transport • Penrose • Prediction • principia • Pseudo-Riemannian Geometry • pseudosphere • Quadratic equation • Quantity • Refractive index • relativity theory • Riemannian Geometry • Riemann sphere • Riemann Tensor • schwarzschild radius • sectional curvature • Space-time • Spacetime • Special relativity • Sphere theorem (3-manifolds) • spherical geometry • Stokes' theorem • Structural Equation Modeling • Surface of revolution • Tangent Space • Tangent vector • tensor calculus • tensor product • Terminology • Terrain • Theorem • Theorema egregium • Theory of Forms • Theory of relativity • Topographic Map • Topology • tractrix • two-dimensional space • Variational Principle • Vector calculus • Vector calculus identities • Vector field • Vector Space • Velocity • Visual
• ISBN-10: 0-691-21989-3 / 0691219893
• ISBN-13: 978-0-691-21989-9 / 9780691219899