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 top promoting challenge e-book for college students, now containing over four hundred thoroughly solved routines on differentiable manifolds, Lie idea, fibre bundles and Riemannian manifolds.

The workouts cross from uncomplicated computations to particularly refined instruments. a few of the definitions and theorems used all through are defined within the first portion of every one bankruptcy the place they appear.

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

In this 2d edition:
• 76 new difficulties
• a part dedicated to a generalization of Gauss’ Lemma
• a brief novel part facing a few houses of the strength of Hopf vector fields
• an elevated selection of formulae and tables
• an prolonged bibliography

Audience

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

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

Example text

Solution (i) Notice that x 2 +y 2 = e2u > 0; so that f (u, v) ∈ R2 \{(0, 0)} for all (u, v) ∈ R2 . We have ∂x ∂u ∂y ∂u ∂x ∂v ∂y ∂v = eu cos v eu sin v −eu sin v , eu cos v hence ∂(x, y)/∂(u, v) = e2u > 0 for all (u, v) ∈ R2 . (ii) By (i), f is a local diffeomorphism at every point of R2 . So f can be taken as a local coordinate map on a neighbourhood of every point. (iii) The map f is not a diffeomorphism as it is not injective. We have f (u, v) = f (u , v ) if and only if u = u and v − v = 2kπ , k ∈ Z.

In order to see that ϕ is a diffeomorphism from R3 to ϕ(R3 ), it suffices to prove that the determinant of its Jacobian matrix Jϕ never vanishes. We have ⎛ 0 det Jϕ = det ⎝2e2x 1 2e2y 0 −1 ⎞ 2e2z −2e2z ⎠ = −4 e2y+2z + e2x+2z = 0. 64 Consider the C ∞ function f : R3 → R3 defined by f (x, y, z) = (x cos z − y sin z, x sin z + y cos z, z). Prove that f |S 2 is a diffeomorphism from the unit sphere S 2 onto itself. Solution For each (x, y, z) ∈ S 2 , one has f (x, y, z) ∈ S 2 , so that (f |S 2 )(S 2 ) ⊂ S 2 .

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 .

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