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 Riemannian Geometry (PDF)
ψk : z 7→ zk. For which k ∈ N0 are φk, ψk immersions, submersions or embeddings. Exercise 3.7. Prove that the map φ : Rm → Cm given by φ : (x1, . . . , xm) 7→ (eix1 , . . . , eixm) is a parametrization of the m-dimensional torus Tm in Cm. Exercise 3.8. Find a...
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ψk : z 7→ zk. For which k ∈ N0 are φk, ψk immersions, submersions or embeddings. Exercise 3.7. Prove that the map φ : Rm → Cm given by φ : (x1, . . . , xm) 7→ (eix1 , . . . , eixm) is a parametrization of the m-dimensional torus Tm in Cm. Exercise 3.8. Find a proof for Theorem 3.26. Exercise 3.9. Prove that the Hopf-map φ : S3 → S2 with φ : (x, y) 7→ (2xȳ, |x|2 − |y|2) is a submersion.
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=18
www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=18
16 2. DIFFERENTIABLE MANIFOLDS Example 2.25. The result of Proposition 2.24 can be used to show that the following maps are all smooth. (i) φ1 : S 2 ⊂ R3 → <span class="highlight">S3</span> ⊂ R4, φ1 : (x, y, z) 7→ (x, y, z, 0), (ii) φ2 : S 3 ⊂ C2 → S2 ⊂ C×R, φ2 : (z1, z2) 7→ (2z1z̄2, |z1|2−|z2|2), (iii) φ3 : R1 → S1 ⊂ C, φ3 : t 7→ eit, (iv) φ4 : Rm+1 \ {0} → Sm, φ4 : x 7→ x/|x|, (v) φ5 : Rm+1 \ {0} → RPm, φ5 : x 7→ [x], (vi) φ6 : S m → RPm, φ6 : x 7→ [x]. In differential geometry we are especially interested in differentiable manifolds
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=36
www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=36
ψk : z 7→ zk. For which k ∈ N0 are φk, ψk immersions, submersions or embeddings. Exercise 3.7. Prove that the map φ : Rm → Cm given by φ : (x1, . . . , xm) 7→ (eix1 , . . . , eixm) is a parametrization of the m-dimensional torus Tm in Cm. Exercise 3.8. Find a proof for Theorem 3.26. Exercise 3.9. Prove that the Hopf-map φ : <span class="highlight">S3</span> → S2 with φ : (x, y) 7→ (2xȳ, |x|2 − |y|2) is a submersion.
From Stargazers to Starships: an introduction for teachers
The Sun and associated physics --Sunlight on Earth, the source of weather and climate (S1 - S1B) --Sun's layers (S2) --Sun's magnetism and sunspots (S3) --Colors of sunlight, include spectra, E-M waves, photons (S4 - S5) --Observing Sun in X-rays, etc. (S6) --Solar energy from fus...
Stanford Encyclopedia of Philosophy: revision theory of
four non-quote names, α, β, γ and λ and no predicates other than T. Also recall that M = <D, I > is as follows: D=SentL I(α)=A=Tβ ∨ Tγ I(β)=B=Tα I(γ)=C=¬Tα I(λ)=X=¬Tλ The following table i...
Stanford Encyclopedia of Philosophy: many, problem of
takes a quite different tack. S1. For some j, oj is a typical cloud. S2. Anything that differs minutely from a typical cloud is a cloud. S3. ok differs minutely from oj. C. ok is a cloud. Since we only care about the conditional if oj is a cloud, so is ok, it is clearly acceptable to...
Stanford Encyclopedia of Philosophy: and modern physics
film is young; it has just been placed in the camera and is ready to be exposed. It is then exposed to the object that comes out of the time machine. (That object in fact is a later stage of the film itself). By the time we come to stage S2 of the life of the film, it has been developed and is about...
Asteroid Heating: A Shocking View
to 10,000 times the atmospheric pressure at the surface of the Earth. Shock Stage Effects resulting from general shock pressure Effects resulting from local P-T excursions Shock Pressure (Gpa) Minimum temp. increase (oC) S1 unshockedSharp optical extinction as viewed in microscope. Small number of...
Table of Moons
J. Kleyna22,707,00061,297.7 Suttungr2000Gladman et al.19,470,00071016.8 S/2004 S07 2004D. Jewitt, S. Sheppard, J. Kleyna19,800,00061,103 S/2004 S12 2004D. Jewitt, S. Sheppard, J. Kleyna19,650,00051,048 S/2004 S13 2004D. Jewitt, S. Sheppard, J. Kleyna18,450,0006906 S/2004 S17 2004D. Jewitt, S. Sheppa...
Microsoft Word - 3969.doc
represent characters in familiar stories (PK-LL-S1) (ELA-1-E4) 8. Listen to a story and state orally what the story is about (PK-LL-R1) (PK-LL-R2) (PK-LL-L1) (ELA-1-E5) 9. Answer simple questions about a story read aloud (PK-LL-S3) (PK-LL-R4) (ELA-1-E5) 10. Share related life experiences afte...
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represent characters in familiar stories (PK-LL-S1) (ELA-1-E4) 8. Listen to a story and state orally what the story is about (PK-LL-R1) (PK-LL-R2) (PK-LL-L1) (ELA-1-E5) 9. Answer simple questions about a story read aloud (PK-LL-S3) (PK-LL-R4) (ELA-1-E5) 10. Share related life experiences after stories are read aloud (PK-LL-L1) (PK-LL-S1) (ELA-1-E6) 11. Orally express thoughts about characters or events in a story (PK-LL-S1) (PK-LL-S2) (PKS-LL-R2) (ELA-1-E6) Standard 7: 12. Demonstrate understanding
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http://www.doe.state.la.us/lde/uploads/3906.pdf#page=1
www.doe.state.la.us/lde/uploads/3906.pdf#page=1
represent characters in familiar stories (PK-LL-S1) (ELA-1-E4) 8. Listen to a story and state orally what the story is about (PK-LL-R1) (PK-LL-R2) (PK-LL-L1) (ELA-1-E5) 9. Answer simple questions about a story read aloud (PK-LL-<span class="highlight">S3</span>) (PK-LL-R4) (ELA-1-E5) 10. Share related life experiences after stories are read aloud (PK-LL-L1) (PK-LL-S1) (ELA-1-E6) 11. Orally express thoughts about characters or events in a story (PK-LL-S1) (PK-LL-S2) (PKS-LL-R2) (ELA-1-E6) Standard 7: 12. Demonstrate understanding
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HIGH SCHOOL MATHEMATICS CONTENT EXPECTATIONS page 3 of 19 11/07 MICHIGAN DEPARTMENT OF EDUCATION STANDARDS (and number of core expectations in each standard) A1: Expressions, Equations, and Inequalities (16) A2: Functions (16) A3: Families of Functions (27) G1: Figures and Their Properties (29) G...
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HIGH SCHOOL MATHEMATICS CONTENT EXPECTATIONS page 3 of 19 11/07 MICHIGAN DEPARTMENT OF EDUCATION STANDARDS (and number of core expectations in each standard) A1: Expressions, Equations, and Inequalities (16) A2: Functions (16) A3: Families of Functions (27) G1: Figures and Their Properties (29) G2: Relationships Between Figures (10) G3: Transformations of Figures in the Plane (5) S1: Univariate Data- Examining Distributions (9) S2: Bivariate Data— Examining Relationships (6) S3: Samples
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http://www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=5
www.michigan.gov/documents/Math11-14-open1_142202_7.pdf#page=5
HIGH SCHOOL MATHEMATICS CONTENT EXPECTATIONS page 3 of 19 11/07 MICHIGAN DEPARTMENT OF EDUCATION STANDARDS (and number of core expectations in each standard) A1: Expressions, Equations, and Inequalities (16) A2: Functions (16) A3: Families of Functions (27) G1: Figures and Their Properties (29) G2: Relationships Between Figures (10) G3: Transformations of Figures in the Plane (5) S1: Univariate Data- Examining Distributions (9) S2: Bivariate Data— Examining Relationships (6) <span class="highlight">S3</span>: Samples
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