Manifolds, Tensor Analysis, and ApplicationsSpringer Science & Business Media, 13.08.1993 - 656 Seiten The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid me chanics, electromagnetism, plasma dynamics and control thcory arc given in Chapter 8, using both invariant and index notation. The current edition of the book does not deal with Riemannian geometry in much detail, and it does not treat Lie groups, principal bundles, or Morse theory. Some of this is planned for a subsequent edition. Meanwhile, the authors will make available to interested readers supplementary chapters on Lie Groups and Differential Topology and invite comments on the book's contents and development. Throughout the text supplementary topics are given, marked with the symbols ~ and {l:;J. This device enables the reader to skip various topics without disturbing the main flow of the text. Some of these provide additional background material intended for completeness, to minimize the necessity of consulting too many outside references. We treat finite and infinite-dimensional manifolds simultaneously. This is partly for efficiency of exposition. Without advanced applications, using manifolds of mappings, the study of infinite-dimensional manifolds can be hard to motivate. |
Inhalt
CHAPTER 7 | 458 |
Integration on Manifolds | 464 |
Differential Forms | 538 |
CHAPTER 8 | 560 |
References | 631 |
Index | 643 |
Supplementary ChaptersAvailable from the authors as they are produced | 649 |
Andere Ausgaben - Alle anzeigen
Manifolds, Tensor Analysis, and Applications Ralph Abraham,Jerrold E. Marsden,Tudor Ratiu Eingeschränkte Leseprobe - 2012 |
Manifolds, Tensor Analysis, and Applications Ralph Abraham,J.E. Marsden,Tudor Ratiu Keine Leseprobe verfügbar - 2012 |
Häufige Begriffe und Wortgruppen
algebra applied assume Banach space basis boundary bounded called chart choose closed compact complete components condition connected consider constant contains continuous converges coordinates covering defined Definition denote dense derivative diffeomorphism differentiable element equals equations equivalent example Exercise exists fact fiber Figure finite dimensional fixed flow formula function given gives Hence holds identity implies induced integral curve inverse isomorphism Lemma linear locally manifold means metric neighborhood norm open set operator oriented origin projection Proof Proposition Let prove relation representative respectively result satisfying sequence Show smooth solution split structure submanifold submersion subset Supplement Suppose taking tangent tensor theorem topological space topology trivial unique vector bundle vector field vector space write zero
Beliebte Passagen
Seite 27 - A metric space is compact iff it is complete and totally bounded.
Seite 587 - ... fundamental postulates forming the basis of the mechanics of rigid bodies, formulated by Newton in 1687. The first law is concerned with the principle of inertia and states that if a body in motion is not acted upon by an external force, its momentum remains constant (law of conservation of momentum). The second law asserts that the rate of change of momentum of a body is proportional to the force acting upon the body and is in the direction of the applied force.
Seite 250 - M the curve 7m is defined on the entire real axis R, then the vector-field X is said to be complete. The support of a vector-field X defined on a manifold M is defined to be the closure of the set {me M\X(m) = 0}.
Seite 50 - However, || • || does become a norm if we identify functions which differ only on a set of measure zero in [a, b], ie, which are equal almost everywhere.
Seite 32 - ILC) if for each neighborhood U of X, there is a neighborhood V of X such that each loop in V—X which is null homotopic in V is null homotopic in UX.
Seite 587 - ... differences in soil water potential (combined gravitational and pressure potential). Consequently, groundwater flow may be described by the classic equations of steady state potential flow such as the Laplace Equation. This equation combines the two basic laws of groundwater flow, viz Darcy's Law and the Law of Conservation of Mass (also known as the Continuity Equation). For two dimensional flow these equations are: Darcy's Law: vv =-l Eq.3.7 Continuity Equation: dx dy = 0 Eq.
Seite 5 - A, denoted int(A) is the union of all open sets contained in A. The boundary of A, denoted bd(A) is defined by bd(A) = cl(A)ncl(S\A).
Seite 38 - A is the intersection of a countable family of open dense subsets of X.
Seite 300 - X is called stable if for any neighborhood U of M there is a neighborhood V of M such that ir(R*, V)c: U.
Seite 468 - L2(flp(M)); it remains bilinear and positive definite, because as usual, in the definition of L2, functions that differ only on a set of measure zero are identified. Using the Hodge star operator *, we can introduce the codifferential operator...
Verweise auf dieses Buch
Geometric Models for Noncommutative Algebras Ana Cannas da Silva,Alan Weinstein Eingeschränkte Leseprobe - 1999 |