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 Riemannian Geometry (PDF)
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...
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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 S3 and S2 be the unit spheres in C2 and C× R ∼= R3, respectively. The Hopf map φ : S3 → S2 is given by φ : (x, y) 7→ (2xȳ, |x|2 − |y|2). For p ∈ S3 the Hopf circle Cp through p is given by Cp = {eiθ(x, y)| θ ∈ R
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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=19
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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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=32
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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http://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf#page=35
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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United States Puzzle
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Colorado Schools Unit: Science of the People
29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environmen...
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29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (S3) Students apply thinking skills to their reading, writing, speaking, listening, and viewing. (RW4) BENCHMARK(S) Studetns know and understand how the human body functions in health and disease and factors
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http://www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=31
www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=31
29Goals 2000 Partnership for Educating Colorado Students Lesson 7: Dr. Bernardo Houssay What wil students be learning? STANDARD(S) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (<span class="highlight">S3</span>) Students apply thinking skills to their reading, writing, speaking, listening, and viewing. (RW4) BENCHMARK(S) Studetns know and understand how the human body functions in health and disease and factors
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http://www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=39
www.dpsk12.org/programs/almaproject/pdf/Scienceof%20People.pdf#page=39
37Goals 2000 Partnership for Educating Colorado Students Lesson 9: Dr. Eloy Rodriquez What will students be learning? OUTCOME(S) Students understand the process of scientific investigation and design, conduct, communicate about, and evaluate such investigations. (S1) Students know and understand the characteristics and structure of living things, the process of life, and how living things interact with each other and their environment. (<span class="highlight">S3</span>) Students understand that science involves a particular way of
science_pub2003
Vermont Department of Education (Science Grade Expectations) S27 Science — Physical Science: Vermont Standards and Evidences—Properties of Matter 7.12 a, aa, aaa, b, bb, bbb, c, cc, ccc, 3.10, 3.11, 3.13 [See also connection with S: 14 Physical Science—Physical Change] Grades 3-...
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Vermont Department of Education (Science Grade Expectations) S27 Science — Physical Science: Vermont Standards and Evidences—Properties of Matter 7.12 a, aa, aaa, b, bb, bbb, c, cc, ccc, 3.10, 3.11, 3.13 [See also connection with S: 14 Physical Science—Physical Change] Grades 3-4 Grades 5-6 Properties of Matter S3-4:12 Students demonstrate their understanding of the States of Matter by… • Identifying , describing and comparing the properties of selected solids, liquids and gases. Science
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=5
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
+ Assessment (GEs) + Learning Opportunity + Teaching Opportunity = Curriculum How do we read a GE? S5-6:14 Students demonstrate their understanding of Physical Change by…• Predicting the effect of heating and cooling on the physical state and themass of a substance. Science Concepts:a. Energy is required to transform the physical state of a substance from solidto liquid to gas, while conserving mass. Physical changes are reversible. <span class="highlight">S3</span>-4:14 Students demonstrate their understanding of Physical Change by
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http://education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page=28
education.vermont.gov/new/pdfdoc/pubs/grade_expectations/science.pdf#page...
Vermont Department of Education (Science Grade Expectations) S27 Science — Physical Science: Vermont Standards and Evidences—Properties of Matter 7.12 a, aa, aaa, b, bb, bbb, c, cc, ccc, 3.10, 3.11, 3.13 [See also connection with S: 14 Physical Science—Physical Change] Grades 3-4 Grades 5-6 Properties of Matter <span class="highlight">S3</span>-4:12 Students demonstrate their understanding of the States of Matter by… • Identifying , describing and comparing the properties of selected solids, liquids and gases. Science
Tennessee State Board of Education
d Given the equation that describes the charging of a capacitor, identify the 1/e time constant and predict when the capacitor will be discharged. e Draw a curve that represents the cooling rate of an isolated container of hot liquid located in a room at some cooler temperature (T). #0;93256....
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d Given the equation that describes the charging of a capacitor, identify the 1/e time constant and predict when the capacitor will be discharged. e Draw a curve that represents the cooling rate of an isolated container of hot liquid located in a room at some cooler temperature (T). #0;93256.7.4 Identify workplace applications in electrical and thermal systems where technicians measure or control time constants. 29.
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http://www.state.tn.us/education/cte/ca/doc/pt1and2_standards.pdf#page=59
www.state.tn.us/education/cte/ca/doc/pt1and2_standards.pdf#page=59
d Given the equation that describes the charging of a capacitor, identify the 1/e time constant and predict when the capacitor will be discharged. e Draw a curve that represents the cooling rate of an isolated container of <span class="highlight">hot</span> liquid located in a room at some cooler temperature (T). #0;93256.7.4 Identify workplace applications in electrical and thermal systems where technicians measure or control time constants. 29.
7th Grade„Grade Level Expectations
Critically…by identifying the underlying purposes of media messages • Evaluate the difference between a stated purpose and an underlying reason in media messages (e.g., TV commercials, radio, Internet, video games, advertisements) • Identify underlying purposes (e.g., profit...
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Critically…by identifying the underlying purposes of media messages • Evaluate the difference between a stated purpose and an underlying reason in media messages (e.g., TV commercials, radio, Internet, video games, advertisements) • Identify underlying purposes (e.g., profit vs. nonprofit, humanitarianism, support of artistry) of media messages • Analyze the difference between a stated purpose and an underlying reason in media messages (e.g., TV commercials, radio, Internet, video games
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http://www.doe.k12.de.us/infosuites/staff/ci/content_areas/files/ela/11th%20Grade%20GLEs.pdf#page=25
www.doe.k12.de.us/infosuites/staff/ci/content_areas/files/ela/11th%20Grad...
Critically…by identifying the underlying purposes of media messages • Evaluate the difference between a stated purpose and an underlying reason in media messages (e.g., TV commercials, radio, Internet, <span class="highlight">video</span> games, advertisements) • Identify underlying purposes (e.g., profit vs. nonprofit, humanitarianism, support of artistry) of media messages • Analyze the difference between a stated purpose and an underlying reason in media messages (e.g., TV commercials, radio, Internet, <span class="highlight">video</span> games
English Voluntary State Curriculum
logical structure appropriate to the subject, audience, purpose, and medium ADP G4 #0;z Develop ideas, opinions, and/or arguments with effective evidence and support #0;z Apply technology (e.g., software, audio, digital video, graphics) to address audience understanding by communicating ideas...
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logical structure appropriate to the subject, audience, purpose, and medium ADP G4 #0;z Develop ideas, opinions, and/or arguments with effective evidence and support #0;z Apply technology (e.g., software, audio, digital video, graphics) to address audience understanding by communicating ideas and information best understood visually ADP C8, G4 #0;z Alter a text to present the same content to a different audience via the same or different media ECLG 4.3.3 Grades 11 and 12 The student will #0;z Establish
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http://mdk12.org/share/hsvsc/source/VSC_english_hs.pdf#page=15
mdk12.org/share/hsvsc/source/VSC_english_hs.pdf#page=15
logical structure appropriate to the subject, audience, purpose, and medium ADP G4 #0;z Develop ideas, opinions, and/or arguments with effective evidence and support #0;z Apply technology (e.g., software, audio, digital <span class="highlight">video</span>, graphics) to address audience understanding by communicating ideas and information best understood visually ADP C8, G4 #0;z Alter a text to present the same content to a different audience via the same or different media ECLG 4.3.3 Grades 11 and 12 The student will #0;z Establish
7th Grade„Grade Level Expectations
characters, actions, ideas or information in text(s) 2.7b critically…by identifying the underlying purposes of media messages • Identify underlying purposes (e.g., profit vs. nonprofit, humanitarianism, support of artistry) of media messages • Analyze the difference between a stat...
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characters, actions, ideas or information in text(s) 2.7b critically…by identifying the underlying purposes of media messages • Identify underlying purposes (e.g., profit vs. nonprofit, humanitarianism, support of artistry) of media messages • Analyze the difference between a stated purpose and an underlying reason in media messages (e.g., TV commercials, radio, Internet, video games, advertisements)
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http://www.doe.k12.de.us/infosuites/staff/ci/content_areas/files/ela/9th%20Grade%20GLEs.pdf#page=22
www.doe.k12.de.us/infosuites/staff/ci/content_areas/files/ela/9th%20Grade...
characters, actions, ideas or information in text(s) 2.7b critically…by identifying the underlying purposes of media messages • Identify underlying purposes (e.g., profit vs. nonprofit, humanitarianism, support of artistry) of media messages • Analyze the difference between a stated purpose and an underlying reason in media messages (e.g., TV commercials, radio, Internet, <span class="highlight">video</span> games, advertisements)
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