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Narratives of the Heart: Haibun
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Physics B Lesson 28: Ideal Gases
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 Riemannian Geometry (PDF)
from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M →...
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from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M → TM is denoted by C∞(TM). Example 4.9. We have seen earlier that the 3-sphere S3 in H ∼= C2 carries a group structure · given by (z, w) · (α, β) = (zα− wβ̄, zβ + wᾱ). This makes (S3, ·) into a Lie group with neutral element e = (1
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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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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=19
→ p · q̄ and a real valued norm given by |p|2 = p · p̄. Then the 3-dimensional unit sphere <span class="highlight">S3</span> in H ∼= R4 with the restricted multiplication forms a compact Lie subgroup (<span class="highlight">S3</span>, ·) of (H∗, ·). They are both non-abelian. We shall now introduce some of the classical real and complex matrix Lie groups. As a reference on this topic we recommend the wonderful book: A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser (2002). Example 2.31. Let Nil3 be the subset of R3×3 given by Nil3 = { 1 x z0 1 y 0 0 1
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=22
R3 and the Riemann sphere Ĉ are diffeomorphic. Exercise 2.8. Find a proof of Proposition 2.24. Exercise 2.9. Let the spheres S1, <span class="highlight">S3</span> and the Lie groups SO(n), O(n), SU(n), U(n) be equipped with their standard differentiable structures introduced above. Use Proposition 2.24 to prove the fol- lowing diffeomorphisms S1 ∼= SO(2), <span class="highlight">S3</span> ∼= SU(2), SO(n)×O(1) ∼= O(n), SU(n)×U(1) ∼= U(n). Exercise 2.10. Find a proof of Corollary 2.28. Exercise 2.11. Let (G, ∗) and (H, ·) be two Lie groups. Prove that the product
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=32
embedding if and only if k = ±1. Example 3.23. Let q ∈ <span class="highlight">S3</span> be a quaternion of unit length and φq : S 1 → <span class="highlight">S3</span> be the map defined by φq : z 7→ qz. For w ∈ S1 let γw : R → S1 be the curve given by γw(t) = weit. Then γw(0) = w, γ̇w(0) = iw and φq(γw(t)) = qwe it. By differentiating we yield dφq(γ̇w(0)) = d dt (φq(γw(t)))|t=0 = d dt (qweit)|t=0 = qiw. Then |dφq(γ̇w(0))| = |qwi| = |q||w| = 1 6= 0 implies that the differen- tial dφq is injective. It is easily checked that the immersion φq is an embedding. In the next
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=35
pi : Rn → Rm given by pi : (x1, . . . , xn) 7→ (x1, . . . , xm). Its differential dpix at a point x is surjective since dpix(v1, . . . , vn) = (v1, . . . , vm). This means that the projection is a submersion. An important sub- mersion between spheres is given by the following. Example 3.30. Let <span class="highlight">S3</span> and S2 be the unit spheres in C2 and C× R ∼= R3, respectively. The Hopf map φ : <span class="highlight">S3</span> → S2 is given by φ : (x, y) 7→ (2xȳ, |x|2 − |y|2). For p ∈ <span class="highlight">S3</span> the Hopf circle Cp through p is given by Cp = {eiθ(x, y)| θ ∈ R
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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.
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=41
from above that (TM,M, pi) together with the maximal bundle atlas B̂ defined by B is a differen- tiable vector bundle. Definition 4.8. Let M be a differentiable manifold, then a section X : M → TM of the tangent bundle is called a vector field. The set of smooth vector fields X : M → TM is denoted by C∞(TM). Example 4.9. We have seen earlier that the 3-sphere <span class="highlight">S3</span> in H ∼= C2 carries a group structure · given by (z, w) · (α, β) = (zα− wβ̄, zβ + wᾱ). This makes (<span class="highlight">S3</span>, ·) into a Lie group with neutral element e = (1
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=83
www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=83
(− � , � )→ O(n) is a geodesic if and only if γt · γ̈ = γ̈t · γ. Exercise 7.3. Find a proof for Proposition 7.23. Exercise 7.4. Find a proof for Corollary 7.24. Exercise 7.5. For the real parameter θ ∈ (0, pi/2) define the 2- dimensional torus T 2θ by T 2θ = {(cos θeiα, sin θeiβ) ∈ <span class="highlight">S3</span>| α, β ∈ R}. Determine for which θ ∈ (0, pi/2) the torus T 2θ is a minimal submanifold of the 3-dimensional sphere <span class="highlight">S3</span> = {(z1, z2) ∈ C2| |z1|2 + |z2|2 = 1}. Exercise 7.6. Find a proof for Corollary 7.27. Exercise 7.7. Determine the totally
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www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=95
zkw̄k and let Tm = {z ∈ Cm| |z1| = ... = |zm| = 1} be the m-dimensional torus in Cm with the induced metric. Find an isometric immersion φ : Rm → Tm, determine all geodesics on Tm and prove that the torus is flat. Exercise 8.6. Find a proof for Proposition 8.17. Exercise 8.7. Let the Lie group <span class="highlight">S3</span> ∼= SU(2) be equipped with the metric g(Z,W ) = 1 2 Re(trace(Z̄tW )). (i) Find an orthonormal basis for TeSU(2). (ii) Prove that (SU(2), g) has constant sectional curvature +1. Exercise 8.8. Let Sm be the unit sphere in
Arctic Animals 4-8
Be au fo rt Se a Am u r R iv er Se a of O kh ot sk Be rin g Se a G ul f o f A la sk a Ba ffi n Ba y Ch uk ch i S ea Yu k o n R i v e r CA N A D A A LA S K A Ar c ti c C ir c le H ud so n Ba y Ri ve r Sa gu en ay N or th P o le RU S S IA G R E E N LA N D Ar c tic O ce an NO RW AY SW ED EN FIN LA ND I...
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Be au fo rt Se a Am u r R iv er Se a of O kh ot sk Be rin g Se a G ul f o f A la sk a Ba ffi n Ba y Ch uk ch i S ea Yu k o n R i v e r CA N A D A A LA S K A Ar c ti c C ir c le H ud so n Ba y Ri ve r Sa gu en ay N or th P o le RU S S IA G R E E N LA N D Ar c tic O ce an NO RW AY SW ED EN FIN LA ND ICELA ND M ur m an sk Di ks on Ti ks i Pe ve k Ba rro w Pr ud ho e Ba y Tu kto ya ktu k Ch ur ch ill Go dh av n Re so lut e Re so lut eT hu le Al er t Re yk jav ik sc al e ~ 1 cm = 2 33 k m 23 3
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http://www.seaworld.org/just-for-teachers/guides/pdf/arctic-animals-4-8.pdf#page=8
www.seaworld.org/just-for-teachers/guides/pdf/arctic-animals-4-8.pdf#page...
ro e n la n d ic a siz e : to 1 .7 m (5 .6 ft .) an d 1 30 k g (2 87 lb .), m al es s om ew ha t l ar ge r th an fe m al es d ist rib u tio n : po pu la ti on c en te rs in th e no rt hw es t A tl an ti c O ce an a ro un d N ew fo un d la nd d ie t: pe la gi c cr us ta ce an s an d fis he s su ch a s ca pe lin a nd he rr in g. D ur in g th e su m m er th ey a ls o fe ed o n ar c- tic c od a nd p ol ar c od fo un d at h ig h la <span class="highlight">tit</span> ud es . p re d a to rs : po la r
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www.seaworld.org/just-for-teachers/guides/pdf/arctic-animals-4-8.pdf#page...
Be au fo rt Se a Am u r R iv er Se a of O kh ot sk Be rin g Se a G ul f o f A la sk a Ba ffi n Ba y Ch uk ch i S ea Yu k o n R i v e r CA N A D A A LA S K A Ar c ti c C ir c le H ud so n Ba y Ri ve r Sa gu en ay N or th P o le RU S S IA G R E E N LA N D Ar c tic O ce an NO RW AY SW ED EN FIN LA ND ICELA ND M ur m an sk Di ks on Ti ks i Pe ve k Ba rro w Pr ud ho e Ba y Tu kto ya ktu k Ch ur ch ill Go dh av n Re so lut e Re so lut eT hu le Al er t Re yk <span class="highlight">jav</span> ik sc al e ~ 1 cm = 2 33 k m 23 3
Penguin Teacher Guide: Measure for Measure
and the Visual Text: Using Video in the Classroom. http/www.holycross.edu/departments/ theatre/projects/isp/measure/taechguide/video.html Vineberg presents five broad questions related to how a video presents Shakespeare’s plays to a viewer’s mind, not just through the ears but al...
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and the Visual Text: Using Video in the Classroom. http/www.holycross.edu/departments/ theatre/projects/isp/measure/taechguide/video.html Vineberg presents five broad questions related to how a video presents Shakespeare’s plays to a viewer’s mind, not just through the ears but also through the eyes. A B O U T T H E A U T H O R O F T H I S G U I D E ROBERT SMALL is professor of English Education and former dean of the College of Education and Human Development at Radford University in Southwest Virginia
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http://us.penguingroup.com/static/pdf/teachersguides/measure.pdf#page=16
us.penguingroup.com/static/pdf/teachersguides/measure.pdf#page=16
and the Visual Text: Using Video in the Classroom. http/www.holycross.edu/departments/ theatre/projects/isp/measure/taechguide/<span class="highlight">video.html</span> Vineberg presents five broad questions related to how a video presents Shakespeare’s plays to a viewer’s mind, not just through the ears but also through the eyes. A B O U T T H E A U T H O R O F T H I S G U I D E ROBERT SMALL is professor of English Education and former dean of the College of Education and Human Development at Radford University in Southwest Virginia
iEmily: Tzatziki with Pita (a creamy, tangy dip)
you eat. Chef's tip: In addition to being a bonanza of a scrabble word, tzatziki (say tit-zee-kee) is a classic appetizer from Greece. Perfectly cooling for a hot summer day. Do raw onions make you cry? Every chef has his or her secret to preventing watery eyes when chopping onions. Here a...
Deer Behind Britain's Great Bird Decline?
? James Owen in the United Kingdom for National Geographic News March 3, 2003 This spring leading ornithologists will begin surveying bird life in 350 woods across Britain. The study aims to reveal the reasons for a startling decline in many woodland species. Since the 1970s the woods have bee...
Clemson University
Clemson University clemson.edu About Photo Background Off Background On Restore Page It's Homecoming Week! Check out the list of events here: http://www.clemson.edu/alumni/homecoming-video.html Go Tigers! Photo credit: Pat Wright Download Skip navigation| Acrobat Reader| Flash Player|...
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