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¹ÌºÐ±âÇÏÇÐ 2. Á¦Àϱ⺻Çü½Ä; Á¦À̱⺻Çü½Ä
    ±è°ü¼®  2019-06-16 16:51:19, Á¶È¸¼ö : 952
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4. The First Fundamental Form <- Figure I-24 ÂüÁ¶

    ∘  If 𝛂(t) = 𝐗 (u(t), v(t)), a¡Â t ¡Âb, <- a curve on a surface; s = s(t): arc length, total length L = s(b) = ¡òba¡«𝛂'(t)¡«dt,
        (ds/dt)2 = ¡«𝛂'¡«2 =  𝛂' ∙ 𝛂' = (u'𝐗1 + v'𝐗2) ∙ (u'𝐗1 + v'𝐗2) = u'2(𝐗1 ∙ 𝐗1) + 2u'v'(𝐗1 ∙ 𝐗2) + v'2(𝐗2 ∙ 𝐗2)  <- by eq.(3)
         𝐸 =  𝐗1 ∙ 𝐗1,  𝐹 =  𝐗1 ∙ 𝐗2,  𝐺 =  𝐗2 ∙ 𝐗2  <- following Gauss   [4-4]
         (ds/dt)2 = 𝐸(du/dt)2 + 2𝐹[(du/dt)(dv/dt)] + 𝐺(dv/dt)2   [4-5]
         ds2 = 𝐸 du2 + 2𝐹 dudv + 𝐺 dv2  <- the first fundamental form ð¯ìéÐñÜâû¡ãÒ or metric form ͪÕáû¡ãÒ   [4-6]      
    ∘  If 𝐯 = a𝐗1 + b𝐗2, 𝐰 = c𝐗1 + d𝐗2,  <- (a,b,c,d´Â ½Ç¼ö) 𝐌ÀÇ ÇÑÁ¡¿¡¼­ÀÇ tangent vectors
        then 𝐯 ∙ 𝐰 =  (a𝐗1 + b𝐗2) ∙ (c𝐗1 + d𝐗2) = 𝐸ac + 𝐹(ad + bc) + 𝐺bd.  
                          ⌈ 𝐸 𝐹 ⌉  ⌈ c ⌉
         𝐯 ∙ 𝐰 = (a,b) ⌊ 𝐹 𝐺 ⌋  ⌊ d ⌋  <- matrix product Ç¥±â ¹æ½Ä       [4-8]
    ∘  𝐔 = 𝐗1 ⨯ 𝐗2 /¡«𝐗1 ⨯ 𝐗2¡«  <- unit normal vector,  ¡ñ  𝐔¡Ñ𝐗1 and 𝐗2, ¡«𝐔¡«= 1, 𝐗1 ⨯ 𝐗2 ¡Á 0
        ¡«𝐗1 ⨯ 𝐗2¡«2 = (𝐗1 ∙ 𝐗1) (𝐗2 ∙ 𝐗2) - (𝐗1 ∙ 𝐗2)2 = 𝐸𝐺 - 𝐹2 <- the identity of Lagrange ¶ó±×¶ûÁÖ ùöÔõãÒ  [4-9]
        ¡ñ 𝐗1 ⨯  𝐗2 = 𝐗1 𝐗2 sin𝜃,  𝐗1 ∙ 𝐗2 = 𝐗1 𝐗2 cos𝜃,  ∣𝐗12 ∣𝐗22 sin2𝜃 = ∣𝐗12 ∣𝐗22 - ∣𝐗12 ∣𝐗22 cos2𝜃
    ∘  𝑔𝑖𝑗 =  𝐗𝑖 ∙ 𝐗𝑗 [¡Ø notaional change] ¡æ  𝑔11 = 𝐸, 𝑔21 = 𝐹, 𝑔22 = 𝐺
        ⌈ 𝑔11 𝑔12 ⌉      ⌈ 𝐸 𝐹 ⌉  
        ⌊ 𝑔21 𝑔22 ⌋  =   ⌊ 𝐹 𝐺 ⌋   <- the matrix of the geometric form
    ∘  𝑔 = ¡« 𝐗1 ⨯ 𝐗2¡«2  <- let  𝑔 = det (𝑔𝑖𝑗) = 𝐸𝐺 - 𝐹2 [¡Ø notaional change]   [4-10]
     ( the Einstein Summation Convention ¾ÆÀν´Å¸ÀÎ ùê Ð®å³ À» Àû¿ëÇϱ⠽ÃÀÛÇÕ´Ï´Ù .) <- Tensor Çؼ® I-1 ÂüÁ¶
    ∘  ds2 = 𝑔𝑖𝑗du𝑖du𝑗   [notaional change: ¢² 𝑖,𝑗 𝑔𝑖𝑗du𝑖du𝑗]  <- the first fundamental form or metric form  by eq.(4-6)
    ∘  𝐯 ∙ 𝐰 = 𝑔𝑖𝑗v𝑖w𝑗 [notaional change: ¢² 𝑖,𝑗 𝑔𝑖𝑗v𝑖]  <- by eq.(8);  𝐯¡Ñ𝐰, if and only if * 𝐯 ∙ 𝐰 = 0.   [4-11]
    ∘  𝛂' = u𝑖'𝐗𝑖 [notaional change: ¢² 𝑖 u𝑖'𝐗𝑖]  <- by eq.(3)
    ∘  𝑔𝑖𝑗: the components of the matrix inverse of (𝑔𝑖𝑗) ↦ (𝑔𝑖𝑗)(𝑔𝑖𝑗) = 𝐈  <- 𝐈: identity matrix
       𝑔𝑖𝑗𝑔𝑗𝑘 = 𝛿𝑘𝑖,  𝛿𝑘𝑖 = {(1 if 𝑖 = 𝑘), (0 if 𝑖 ¡Á 𝑘)} [notaional change: ¢² 𝑖,𝑗 𝑔𝑖𝑗𝑔𝑗𝑘]  [4-12]
        𝑔11 = 𝑔22/𝑔,  𝑔12 = 𝑔21 = -𝑔12/𝑔,   𝑔22 = 𝑔11/𝑔  <-  inverse matrix**,  𝑔 = det (𝑔𝑖𝑗); if and only if  𝑔 ¡Á 0
    ∘  If a surface 𝐗;𝐃 ¡æ E3, a region 𝛺 ¡ø 𝐃 on which 𝐗 is one-to-one, ¾î¶»°Ô 𝐗(𝛺) ¸éÀûÀ» ¹ß°ßÇÒ ¼ö ÀÖÀ»±î¿ä?  
       ¸ÕÀú 𝛺¸¦ u1, u2ÃàÀ» µû¶ó °¢º¯ÀÌ 𝛥u1, 𝛥u2ÀÎ ÀÛÀº Á÷»ç°¢ÇüÀ¸·Î ³ª´« ÈÄ¿¡ À̸¦ 𝐗(𝛺)·Î mappingÇØ º¾´Ï´Ù.
       𝐗(𝛺)¿¡¼­ 𝛺¿¡¼­ÀÇ ÀÛÀº »ç°¢ÇüµéÀÌ À¯»ç ÆòÇà»çº¯ÇüÀ¸·Î µÇ¸ç °¢º¯Àº ¡«𝐗1¡«𝛥u1,¡«𝐗2¡«𝛥u2°¡ µË´Ï´Ù.
       ¡Å 𝛥𝐴 ≈¡«𝐗1¡«¡«𝐗2¡«sin 𝜃 𝛥u1𝛥u2 = ¡«𝐗1 ⨯ 𝐗2¡«𝛥u1𝛥u2 = (¡î 𝑔) 𝛥u1𝛥u2
       𝐀 = ¡ó𝛺¡«𝐗1 ⨯ 𝐗2¡«du1du2 = ¡ó𝛺 (¡î 𝑔) du1du2  <- function 𝐗 to 𝛺: one-to-one and regular   [4-13]

5. The Second Fundamental Form <- Figure I-25 ÂüÁ¶

      ∘  ÀÌÈķδ the Einstein Summation Convention ¾ÆÀν´Å¸ÀÎ ùê Ð®å³ À» °è¼Ó Àû¿ëÇÕ´Ï´Ù.  
      ∘  The Gauss Formulas : a curve 𝛂(s) = 𝐗(u1(s), u2(s)) on 𝐌, s: arc length,  
         unit tangent vector 𝐓 = 𝛂' = u𝑖'𝐗𝑖;  acceleration or curvature vector 𝐓' = 𝛂";  curvature k(s) = ¡«𝐓'¡«  
         𝛂" = 𝛂"tan + 𝛂"nor  <-  tangent and normal to the surface; in order to study the surface 𝐌   [5-14]
         𝛂" = u𝑖"𝐗𝑖 + u𝑖'd𝐗𝑖/ds  <- by product rule   [5-15]
         𝐗𝑖 ¡Õ ¡Ó𝐗/¡Óu𝑖 = ( ¡Óx/¡Óu𝑖, ¡Óy/¡Óu𝑖, ¡Óz/¡Óu𝑖),  𝑖 = 1, 2
         𝐗𝑖𝑗 ¡Õ ¡Ó2𝐗/¡Óu𝑗¡Óu𝑖 =  ¡Ó𝐗𝑖/¡Óu𝑗 = ( ¡Ó2x/¡Óu𝑗¡Óu𝑖, ¡Ó2y/¡Óu𝑗¡Óu𝑖, ¡Ó2z/¡Óu𝑗¡Óu𝑖),  𝑖,𝑗 = 1, 2
         𝐗𝑖 = 𝐗𝑖(u1(s), u2(s)) ¡æ d𝐗𝑖/ds = (¡Ó𝐗𝑖/¡Óu1) u1' + (¡Ó𝐗𝑖/¡Óu2) u2' = (¡Ó𝐗𝑖/¡Óu𝑗)u𝑗' = u𝑗'𝐗𝑖𝑗  <- by chain rule
         𝛂" = u𝑟"𝐗𝑟 + u𝑖'u𝑗'𝐗𝑖𝑗   [5-16]
         𝐗𝑖𝑗 = 𝛤𝑟𝑖𝑗𝐗𝑟 + 𝐿𝑖𝑗𝐔,  𝑖,𝑗 = 1,2  <- the Gauss Formulas ; cf) Tensor Çؼ® II-4 b) Christoffel ±âÈ£   [5-17]
         𝛂" = 𝛂"tan + 𝛂"nor = (u𝑟" + 𝛤𝑟𝑖𝑗 u𝑖'u𝑗')𝐗𝑟 + (𝐿𝑖𝑗 u𝑖'u𝑗')𝐔   [5-18]
           𝛂" = [u1" +  𝛤111(u1')2 +  2𝛤112u1'u2' + 𝛤122u2')2]𝐗1 + [u2" +  𝛤211(u1')2 +  2𝛤212u1'u2' + 𝛤222(u2')2]𝐗2
                 + [𝐿11(u1')2 + 2𝐿12u1'u2' + 𝐿22(u2')2]𝐔  <- full equation by the Einstein summation convention
     ∘  𝐿𝑖𝑗 u𝑖'u𝑗'  <-   how does it curve relative that space? extrinsic(èâî¤îÜ) geometry  [5-19]
        𝐿𝑖𝑗 =  𝐗𝑖𝑗 ∙ 𝐔,  𝑖,𝑗 = 1,2  <- the second fundamental form of the surfaces  [5-20]
            ex) Let 𝐗(u,v) = (u, v, f(u,v)), (u,v) ∊ D, then 𝐗1 = (1, 0, fu), 𝐗2 = (0, 1, fv).
               𝐸 = 1 + fu2,  𝐹 = fuv,  𝐺 = 1 + fv2,  𝑔 = 𝐸𝐺 - 𝐹2 = 1 + fu2 + fv2,  𝐗1 ⨯ 𝐗2 = (-fu, -fv, 1),  𝐔 = (-fu, -fv)/¡î 𝑔
               𝐗11= (0, 0, fuu), 𝐗12= (0, 0, fuv), 𝐗11= (0, 0, fvv) ¡æ 𝐗𝑖𝑗 = (0, 0, f𝑖𝑗); ¡Å 𝐿𝑖𝑗 = f𝑖𝑗 / ¡î 𝑔,  𝑖,𝑗 = 1, 2

     ∘  Definition I-3
         𝐯 = v𝑖 𝐗𝑖°¡ 𝐏¿¡¼­ 𝐌¿¡ Á¢ÇÏ´Â a unit vector¶ó¸é, the normal curvature ÛöÍØáã in the v direction k´Â,
         𝑘n(𝐯) = 𝐿𝑖𝑗 v𝑖v𝑗 ·Î Á¤ÀǵȴÙ.   [5-21]

     ∘  𝛂°¡ 𝐌¿¡ ÀÖ´Â ÀÓÀÇÀÇ °î¼±À¸·Î 𝛂(s0) = 𝐏, 𝛂(s0)' = 𝐯À̸é, 𝛂(s0)'= u𝑖'(s0)𝐗𝑖(u1(s0), u2(s0)), v𝑖 = u𝑖'(s0)ÀÌ µË´Ï´Ù.
         𝑘n(𝐯) = 𝐿𝑖𝑗 u𝑖'u𝑗' = 𝛂" ∙ 𝐔 = ∓¡«𝛂"¡« <- by eq.16 ¿¡¼­ ur" 𝐗r ∙ 𝐔 = 0; eq. 20   [5-22]
         '0'ÀÌ¾Æ´Ñ tangent vector¿¡ ´ëÇؼ­ normal curvatureÀÇ °³³äÀ» ´ÙÀ½°ú °°ÀÌ È®Àå½Ãų ¼ö ÀÖ½À´Ï´Ù.
         𝑘n(𝐯) =  𝐿𝑖𝑗 v𝑖v𝑗/¡«𝐯¡«2 = 𝐿𝑖𝑗 v𝑖v𝑗/ 𝑔mnvmvn  <-  by eq. 21; ¿ìº¯ÀÇ v¸¦ v/¡«v¡«·Î ±³Ã¼ÇÔ.
          
p.s.  °î¸é 𝐌 À§¿¡ ÀÖ´Â °î¼± ¹æÁ¤½ÄÀÇ 1Â÷ 2Â÷ ¹ÌºÐ °á°ú¸¦ ÀÌ¿ëÇؼ­ °î¸é ÀÚüÀÇ curvature¸¦ ¹ß°ßÇÏ´Â °ÍÀÌ ÇÙ½ÉÀÓ.
        ƯÈ÷ °î¸é À§¿¡¼­ÀÇ Christoffel symbols¸¦ Æ÷ÇÔÇÑ Gauss Formulas-'Tensor Çؼ® II-2'°ú ºñ±³µÊ-¿¡ ÁÖ¸ñ ¹Ù¶÷.
       * 'if and only if ' ȤÀº ÁÙ¿©¼­ 'iff '¶ó°íµµ ¾²´Âµ¥ µ¿Ä¡(ÔÒö·) ¶Ç´Â ÇÊ¿äÃæºÐÁ¶°ÇÀ» °¡¸®Å´.
       **  inverse matrix(æ½ú¼Öª) 𝐀-1 = (Ȃ𝑖𝑗)T/det 𝐀  <- cofactor(æ®ì×í­) Ȃ𝑖𝑗 = (-1)𝑖+𝑗𝑀𝑖𝑗,  𝑀𝑖𝑗: the 𝑖,𝑗 minor of 𝐀


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°øÁö  'Çö´ë ¿ìÁÖ·Ð'¿¡ °üÇÑ Å½±¸ÀÇ Àå    °ü¸®ÀÚ 1 2017-08-15
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156    Appendix  Ab. Einstein Equation etc    ±è°ü¼® 3 2025-02-07
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155      Supplement  Sa. Chpater 2. Problems  ✍🏻    ±è°ü¼® 3 2025-02-07
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154  Baumann's Cosmology  8a. Quantum Conditions    ±è°ü¼® 4 2025-01-08
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153    Cosmology  8b. Quantum Fluctuations    ±è°ü¼® 4 2025-01-08
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152      Cosmology  8c. Primordial Power Spectra    ±è°ü¼® 4 2025-01-08
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151        Cosmology  8d. Obs. Constraints; 9 Outlook    ±è°ü¼® 4 2025-01-08
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150  Baumann's Cosmology  7a. CMB Physics  ✅    ±è°ü¼® 5 2024-12-13
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149    Cosmology  7b. Primordial Sound Waves    ±è°ü¼® 5 2024-12-13
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148      Cosmology  7c. CMB Power Spectrum    ±è°ü¼® 5 2024-12-13
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147        Cosmology  7d. Glimpse at CMB Polarization    ±è°ü¼® 5 2024-12-13
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146          Cosmology  7e. Summary and Problems    ±è°ü¼® 5 2024-12-13
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145  Baumann's Cosmology  6a. Relativistic Perturbation    ±è°ü¼® 4 2024-11-08
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144    Cosmology  6b. Conservation Eqs; Initial Conditions    ±è°ü¼® 4 2024-11-08
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143      Cosmology  6c. Growth of Matter Perturbations    ±è°ü¼® 4 2024-11-08
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142        Cosmology  6d. Summary and Problems    ±è°ü¼® 4 2024-11-08
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141  Baumann's Cosmology  4a. Cosmological Inflation    ±è°ü¼® 5 2024-10-21
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140    Cosmology  4b. Physics of Inflation    ±è°ü¼® 5 2024-10-21
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139      Cosmology  5a. Newtonian Perturbation    ±è°ü¼® 5 2024-10-21
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138        Cosmology  5b. Statistical Properties    ±è°ü¼® 5 2024-10-21
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137          Cosmology  5c. Summary and Problems    ±è°ü¼® 5 2024-10-21
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136  Baumann's Cosmology  3a. Hot Big Bang  ✅     ±è°ü¼® 4 2024-09-22
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135    Cosmology  3b. Thermal Equilibrium    ±è°ü¼® 4 2024-09-22
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134      Cosmology  3c. Boltzmann Equation    ±è°ü¼® 4 2024-09-22
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133        Cosmology  3d. Beyond Equilibrium    ±è°ü¼® 4 2024-09-22
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132  Baumann's Cosmology  1. Introduction  ⚫  [1]  ±è°ü¼® 5 2024-09-01
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130      Cosmology  2b. Dynamics      ±è°ü¼® 5 2024-09-01
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129        Cosmology  2c. Friedmann Equations    ±è°ü¼® 5 2024-09-01
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128          Cosmology  2d. Our Universe    ±è°ü¼® 5 2024-09-01
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127  Waves  1. Wave fundamentals  ✅    ±è°ü¼® 8 2024-05-07
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126    Waves  2. The wave equation    ±è°ü¼® 8 2024-05-07
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125      Waves  3a. General solution; Boundary conditions    ±è°ü¼® 8 2024-05-07
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124        Waves  3b. Fourier theory    ±è°ü¼® 8 2024-05-07
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