# Analysis and Algebra on Differentiable Manifolds: A Workbook by Prof. P. M. Gadea, Prof. J. Muñoz Masqué (auth.) By Prof. P. M. Gadea, Prof. J. Muñoz Masqué (auth.)

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

Best analysis books

Econometric Analysis of Financial Markets

This selection of papers represents the cutting-edge within the applicationof fresh econometric easy methods to the research of monetary markets. From a methodological perspective the most emphasis is on cointegration research and ARCH modelling. In cointegration research the hyperlinks among long-runcomponents of time sequence are studied.

Magnetic Domains: The Analysis of Magnetic Microstructures

The wealthy international of magnetic microstructure or magnetic domain names - extending from the "nano-world" to noticeable dimensions - is systematically coated during this publication. the topic, that may be known as "mesomagnetism", varieties the hyperlink among atomic foundations and technical functions of magnetic fabrics, starting from laptop garage platforms to the magnetic cores of electric equipment.

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

Example text

T ) as above and moreover ϕi j (t 1 , . . ,t n ) = t1 t i−1 t i+1 t j−1 1 t j tn , . . , , , . . , , , , . . , ti ti ti ti ti ti ti = (x1 , . . , xn ). The equations x1 = t1 t i−1 i t i+1 i−1 , . . , ti ti t t j−1 1 tj tn x j−2 = i , x j−1 = i , x j = i , . . , xn = i , t t t t correspond to differentiable functions on Ui j . ) (3) Rn+1 − {0} is an open submanifold of Rn+1 . 7 Submersions. Quotient Manifolds ϕi ◦ π ◦ id−1 : 49 π −1 (Ui ) → Rn (x1 , . . , xn+1 ) → x1 xi−1 xi+1 xn+1 , .

3 3 R EMARK . f∗ cannot be injective at any point since dim R2 > dim R. 2. Let f : R2 → R2 , (x, y) → (x2 − 2y, 4x3 y2 ), g : R2 → R3 , (x, y) → (x2 y + y2 , x − 2y3 , yex ). (1) Compute f∗(1,2) and g∗(x,y) . (2) Find g∗ 4 ∂ ∂ − ∂x ∂y . 3 Differentiable Functions and Mappings λ ∂ ∂ ∂ +μ +ν ∂x ∂y ∂z to be the image of some vector by g∗ . ⎛ Solution. (1) f∗(1,2) ≡ ⎝ 2 −2 48 23 ⎞ ⎠, 16 g(0,0) ⎛ ⎞ 2x y x2 + 2y ⎜ ⎟ g∗(x,y) ≡ ⎜ −6y2 ⎟ ⎝ 1 ⎠. x x ye e (2) g∗ 4 ∂ ∂ − ∂x ∂y (0,1) ⎛ ⎞ 0 2 ⎛ ⎞ ⎜ ⎟ 4 ⎟⎝ ⎠ ≡⎜ ⎝1 −6⎠ −1 1 1 ⎛ ⎞ −2 ⎜ ⎟ ∂ ∂ ∂ ⎟ =⎜ ≡ −2 + 10 + 3 ⎝ 10⎠ ∂x ∂y ∂z 3 .

By virtue of Sard’s theorem, f (Rk ) = L has zero measure. 6. Let M1 and M2 be two C∞ manifolds. Give an example of differentiable mapping f : M1 → M2 such that all the points of M1 are critical points and the set of critical values has zero measure. Solution. Let f : M1 → M2 defined by f (p) = q, for every p ∈ M1 and q a fixed point of M2 . Then the rank of f is zero, hence all the points of M1 are critical. On the other hand, the set of critical values is reduced to the point q, and the set {q} has obviously zero measure.