By Serge Lang (auth.), Serge Lang (eds.)

ISBN-10: 0387943382

ISBN-13: 9780387943381

ISBN-10: 3540943382

ISBN-13: 9783540943389

This is the 3rd model of a publication on differential manifolds. the 1st model seemed in 1962, and was once written on the very starting of a interval of significant enlargement of the topic. on the time, i discovered no passable publication for the principles of the topic, for a number of purposes. I increased the publication in 1971, and that i extend it nonetheless extra this day. particularly, i've got extra 3 chapters on Riemannian and pseudo Riemannian geometry, that's, covariant derivatives, curvature, and a few purposes as much as the Hopf-Rinow and Hadamard-Cartan theorems, in addition to a few calculus of diversifications and purposes to quantity kinds. i've got rewritten the sections on sprays, and i've given extra examples of using Stokes' theorem. i've got additionally given many extra references to the literature, all of this to increase the viewpoint of the ebook, which i'm hoping can be utilized between issues for a common path best into many instructions. the current e-book nonetheless meets the outdated wishes, yet fulfills new ones. on the most simple point, the booklet provides an creation to the elemental thoughts that are utilized in differential topology, differential geometry, and differential equations. In differential topology, one stories for example homotopy sessions of maps and the opportunity of discovering appropriate differentiable maps in them (immersions, embeddings, isomorphisms, etc.).

**Read Online or Download Differential and Riemannian Manifolds PDF**

**Similar mathematics_1 books**

Dieses erfolgreiche einf? hrende Lehrbuch liegt nun in der 10. Auflage vor. Es zeichnet sich durch eine exakte und anschauliche Darstellung aus. Der Lehrstoff ist klar gegliedert und intestine strukturiert. Er wird durch eine F? lle von Beispielen und Abbildungen veranschaulicht und vertieft. Zahlreiche Aufgaben mit L?

**Probabilistic Expert Systems (CBMS-NSF Regional Conference Series in Applied Mathematics)**

Probabilistic professional structures emphasizes the elemental computational ideas that make probabilistic reasoning possible in professional structures. the main to computation in those platforms is the modularity of the probabilistic version. Shafer describes and compares the central architectures for exploiting this modularity within the computation of past and posterior chances.

**Surveys in Differential-Algebraic Equations III**

The current quantity includes survey articles on a number of fields of Differential-Algebraic Equations (DAEs), that have common functions in managed dynamical platforms, specially in mechanical and electric engineering and a powerful relation to (ordinary) differential equations. the person chapters supply studies, shows of the present country of analysis and new strategies in - Flexibility of DAE formulations - Reachability research and deterministic international optimization - Numerical linear algebra equipment - Boundary price difficulties the consequences are offered in an obtainable type, making this booklet appropriate not just for energetic researchers but additionally for graduate scholars (with a great wisdom of the fundamental ideas of DAEs) for self-study.

- Rational approximations and orthogonality
- Large Scale Scientific Computation. Proceedings of a Conference Conducted by the Mathematics Research Center, the University of Wisconsin–Madison, May 17–19, 1983
- Theory of Linear Operators in Hilbert Space: Volume II
- Billard - théorie et jeu
- Some Modern Mathematics for Physicists and Other Outsiders. An Introduction to Algebra, Topology, and Functional Analysis

**Additional resources for Differential and Riemannian Manifolds**

**Example text**

Math. Anal. Appl. 382, 814–821 (2011) 15. : On two different concept of superquadracity. Inequalities and Applications, pp. 209–215. Birkhauser, Basel (2010) 16. : Notes on a theorem of Hilbert. Math. Z. 6, 314–317 (1920) 17. : On the Jensen-Steffensen inequality and superquadracity. Anale Universitatii, Oradea, Fasc. Mathematica, Tom XVIII, pp. 269–275 (2011) 18. : Refinement of Hardy’s inequalities via superquadratic and subquadratic functions. J. Math. Anal. Appl. 339, 1305–1312 (2008) 19. : Hardy type inequalities via convexity-the journey so far.

Theorem 1 (Variation of Constants). 0; 1 and s; t 2 Œ0; 1/. t; /f . t; / is the Cauchy function for (3). 1 Taylor’s Formula and Integral Inequalities for Conformable Fractional Derivatives 27 Proof. t; /f . t; /f . t; /f . t; /f . t; /f . t/; s t u and the proof is complete. Theorem 2 (Taylor Formula). 0; 1 and n 2 N. n C 1/ times ˛-fractional differentiable on Œ0; 1/, and s; t 2 Œ0; 1/. s/ C kŠ ˛ nŠ s ˛ kD0 ˛ Ãn DnC1 ˛ f . /d˛ : Proof. t/. t/ C 1 nŠ Z tÂ s ˛ Ãn t˛ ˛ g. R. s/ ˛ m/Š m for 0 Ä m Ä n.

We consequently have that w also solves (5), and thus u Á w by uniqueness. u t Corollary 1. 0; 1 and s; r 2 Œ0; 1/ be fixed. n k/Š t˛ s˛ Ãk Â ˛ s˛ r˛ Ãn ˛ Proof. t/ D Taylor’s formula. It can also be shown directly. k : 1 nŠ t˛ r˛ n ˛ in t u 2 Steffensen Inequality In this section we prove a new ˛-fractional version of Steffensen’s inequality and of Hayashi’s inequality. The results in this and subsequent sections differ from those in [10, 12, 13, 15]. Lemma 1. 0; 1 and a; b 2 R with 0 Ä a < b.