Download Analysis and Algebra on Differentiable Manifolds: A Workbook by Pedro M. Gadea, Jaime Muñoz Masqué, Ihor V. Mykytyuk PDF

By Pedro M. Gadea, Jaime Muñoz Masqué, Ihor V. Mykytyuk

This is the second one variation of this most sensible promoting challenge ebook for college kids, now containing over four hundred thoroughly solved workouts on differentiable manifolds, Lie idea, fibre bundles and Riemannian manifolds.

The routines move from easy computations to particularly subtle instruments. a few of the definitions and theorems used all through are defined within the first portion of each one bankruptcy the place they appear.

A 56-page number of formulae is incorporated that are priceless as an aide-mémoire, even for academics and researchers on these topics.

In this second edition:
• seventy six new difficulties
• a bit dedicated to a generalization of Gauss’ Lemma
• a quick novel part facing a few houses of the power of Hopf vector fields
• an multiplied selection of formulae and tables
• a longer bibliography


This booklet may be necessary to complicated undergraduate and graduate scholars of arithmetic, theoretical physics and a few branches of engineering with a rudimentary wisdom of linear and multilinear algebra.

Show description

Read or Download Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers PDF

Similar differential geometry books

From Holomorphic Functions to Complex Manifolds

This creation to the speculation of complicated manifolds covers an important branches and strategies in advanced research of a number of variables whereas thoroughly averting summary thoughts concerning sheaves, coherence, and higher-dimensional cohomology. in basic terms basic tools resembling strength sequence, holomorphic vector bundles, and one-dimensional cocycles are used.

Geometric Function Theory: Explorations in Complex Analysis (Cornerstones)

Advanced variables is an exact, dependent, and desirable topic. awarded from the perspective of recent paintings within the box, this new e-book addresses complex subject matters in complicated research that verge on present components of analysis, together with invariant geometry, the Bergman metric, the automorphism teams of domains, harmonic degree, boundary regularity of conformal maps, the Poisson kernel, the Hilbert rework, the boundary habit of harmonic and holomorphic features, the inhomogeneous Cauchy–Riemann equations, and the corona challenge.

Symmetry in Mechanics: A Gentle, Modern Introduction

"And what's the use," suggestion Alice, "of a ebook with out photos or conversations in it? " -Lewis Carroll This e-book is written for modem undergraduate scholars - now not the correct stu­ dents that arithmetic professors want for (and who sometimes grace our campuses), however the scholars like many the writer has taught: proficient yet ap­ preciating evaluate and reinforcement of prior direction paintings; prepared to work flat out, yet tough context and motivation for the math they're studying.

Differential Geometry and Continuum Mechanics

This booklet examines the fascinating interface among differential geometry and continuum mechanics, now regarded as being of accelerating technological importance. issues mentioned contain isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, using manifolds within the description of microstructure in continuum mechanics, experimental dimension of microstructure, defects, dislocations, floor energies, and nematic liquid crystals.

Additional resources for Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers

Example text

35). Consider the vector v = (d/ds)0 tangent at the origin p = (0, 0) to E and let j : E → R2 be the canonical injection of E in R2 . (i) Compute j∗ v. (ii) Compute j∗ v if E is given by the chart (sin 2s, sin s) → s, s ∈ (−π, π). Solution (i) The origin p corresponds to s = π , so j∗p ≡ As v = d ds |p , ∂ sin 2s ∂s 0 0 ∂ sin s ∂s = s=π 2 . −1 we have j∗p v ≡ ∂ 2 2 (1) = ≡2 −1 −1 ∂x − p ∂ ∂y . p (ii) We now have j∗p ≡ ∂ |p + so j∗p v = 2 ∂x ∂ sin 2s ∂s ∂ sin s ∂s = 2 , 1 s=0 ∂ ∂y |p . 4) x = sin θ cos ϕ, y = sin θ sin ϕ, z = cos θ, 0 < θ < π, 0 < ϕ < 2π, of S 2 .

Denote this point simply by m (see Fig. 20). Since m belongs to the straight line r(x, J (x − p)), we can put m = x + tJ (x − p), with t such that q − m, J (x − p) = 0. Hence, since J is an isometry, we get t= q − x, J (x − p) , x − p, x − p ϕq ◦ ϕp−1 (x) = x + q − x, J (x − p) J (x − p), x − p, x − p which is C ∞ , for the scalar product is a polynomial in the components of its factors, so the components of (ϕq ◦ϕp−1 )(x) are rational functions of the components of x. Consequently, we have proved that {(Up , ϕp )}p∈R2 is an atlas on M, which is thus a 2-dimensional C ∞ manifold when endowed with the differentiable structure corresponding to the given atlas.

4). We have 1 y 1 = sin2 u sin 2v, y 2 = cos u. 2 So we can write (ϕ|S 2 )∗ ≡ 1 2 sin 2u sin 2v − sin u sin2 u cos 2v ; 0 thus rank(ϕ|S 2 )∗ < 2 if and only if either sin u = 0 or cos 2v = 0. 34 1 Differentiable Manifolds Fig. 17 The set of critical points of ϕ|S 2 We have sin u = 0 in both charts. In the first chart, we have cos 2v = 0 for v = π/4, 3π/4, 5π/4, 7π/4. In the second chart, one has cos 2v = 0 for v = −3π/4, −π/4, π/4, 3π/4. The sets of respective critical points coincide: They are the four half-circles in Fig.

Download PDF sample

Rated 4.57 of 5 – based on 43 votes