By Serge Lang (auth.), Serge Lang (eds.)
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 the current e-book nonetheless meets the outdated wishes, yet fulfills new ones.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.).
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Additional resources for Differential and Riemannian Manifolds
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.