±âº» ÆäÀÌÁö Æ÷Æ®Æú¸®¿À ´ëÇѹα¹ÀÇ ÀüÅë°ÇÃà Áß±¹°ú ÀϺ»ÀÇ ÀüÅë°ÇÃà ¼­À¯·´°ú ¹Ì±¹ÀÇ °ÇÃà ±¹¿ª û¿À°æ Çö´ë ¿ìÁÖ·Ð ´ëÇѹα¹ÀÇ »êdz°æ ¹éµÎ´ë°£ Á¾ÁÖ»êÇà ³×ÆÈ È÷¸»¶ó¾ß Æ®·¹Å· ¸ùºí¶û Áö¿ª Æ®·¹Å· ¿ä¼¼¹ÌƼ ij³â µî Ƽº£Æ® ½ÇÅ©·Îµå ¾ß»ý »ý¹° ÆÄ³ë¶ó¸¶»çÁø °¶·¯¸® Ŭ·¡½Ä ·¹ÄÚµå °¶·¯¸® AT Æ÷·³ Æ®·¹Å· Á¤º¸ ¸µÅ©


 ·Î±×ÀÎ  È¸¿ø°¡ÀÔ

¹ÌºÐ±âÇÏÇÐ 1. °î¼±; Gauss °î·ü; °î¸é  🔵
    ±è°ü¼®  2019-06-16 16:50:51, Á¶È¸¼ö : 1,114
- Download #1 : dff_1_1.jpg (47.4 KB), Download : 5



  Áß·Â(gravity)Àº ½Ã°ø°£ÀÇ curvature * ÈÖ¾îÁüÀÇ ¹ßÇöÀ̶ó´Â ¾ÆÀν´Å¸ÀÎÀÇ »ó´ë¼ºÀÌ·ÐÀÇ ÀÌÇØ¸¦ À§Çؼ­ °ø°£°ú ½Ã°£À»
  ±âÇÏÇÐÀû ÀÌ·ÐÀ¸·Î¼­ ÆÄ¾ÇÇÏ´Â °ÍÀÌ ÀÌ Ã¥ÀÇ ±Ã±ØÀûÀÎ ¸ñÇ¥ÀÔ´Ï´Ù. ±× Áß¿¡¼­´Â ¹Ù·Î curvature°¡ ÁÖÃàÀÌ µÇ´Â °³³äÀÔ´Ï´Ù.
   curvatureÀÇ °³³äÀº 4Â÷¿ø À̻󿡼­´Â ´õ¿í º¹ÀâÇØÁö¹Ç·Î, Euclid 3Â÷¿ø °ø°£ E3¿¡¼­ÀÇ curvature¸¦ ¸ÕÀú ³íÀÇÇÕ´Ï´Ù.

1. Curves ÍØàÊ <- Figure I-1 ÂüÁ¶

     ∘  𝛂(t) = (x(t),y(t),z(t))  <- 𝛂: vector-valued fuction curve in 𝐄3, a ¡Â t ¡Â b, t: real variable parameter ØÚ˿ܨ⦠ 
     ∘  𝛂'(t) = (x'(t),y'(t),z'(t)) = lim𝛥¡æ0 [𝛂(t + 𝛥t) - 𝛂(t))]/𝛥t  <- 𝛂': derivative vector of curve 𝛂  
     ∘  s(t) = ¡òat¡«𝛂'(u)¡«du  <- s: arc length of curve 𝛂, t: time, ¡«𝛂'(u)¡«: instaneous speed áÜÕô,  𝛂': velocity vector
     ∘  𝐋 = ¡òab¡«𝛂'(u)¡«du  <- 𝐋: total length of the entire curve 𝛂(t), a ¡Â t ¡Â b, 𝐈: interval
     ∘  𝛂"(t) = (x"(t),y"(t),z"(t))  <- 𝛂": derivative of velocity vector 𝛂',  𝛂": acceleration vector 
     ∘  𝐓(s) = 𝛂'(s)  <-  t = s ¡æ  ds/dt = 1,  𝛂'(s) or 𝐓(s): unit tangent vector,  𝛂(s): unit speed curve
     ∘  𝐓(s) ∙ 𝐓(s) = 1, ¡Å 𝐓(s) ∙ 𝐓'(s) = 0  (<- by product rule) ¡Å 𝐓(s)¡Ñ𝐓'(s),  𝛂"(s) or 𝐓'(s): curvature vector

     ∘  Definition I-1
        k(s) = ¡«𝐓'(s)¡«= ¡«𝛂"(s)¡«
        k(s)·Î ÁöĪµÇ´Â °î¼± 𝛂ÀÇ 𝛂(s)¿¡¼­ÀÇ curvature ÍØáãÀº 𝐓'(s)ÀÇ ±æÀÌÀÌ´Ù.

     ∘  𝐍(s) = 𝐓'(s)/¡«𝐓'(s)¡«= 𝐓'(s)/k(s). ¡Å 𝐓'(s) = k(s)𝐍(s)   <- 𝐍(s): principal normal vector
     ∘  c(s) = 𝛂(s) + [1/k(s)]/𝐍(s)  <- 𝛂(s): °î¼± 𝛂¿¡ÀÇ tangent point, c(s): center of curvature
        osculating circle ïÈõºê­: ¹ÝÁö¸§ 1/k(s), Á᫐ c(s)ÀÇ ¿ø,  osculating plane: 𝐓(s)¿Í 𝐍(s)ÀÇ Æò¸é
     ∘  𝐁 = 𝐓 X 𝐍   <- binormal vector of the unit length: osculating plane¿¡ ¼öÁ÷ÀÎ vector
        𝐁' = (𝐓 X 𝐍)' = 𝐓' X 𝐍 + 𝐓 X 𝐍' = 0 + 𝐓 X 𝐍'= 𝐓 X 𝐍' (¡ñ 𝐓'= k𝐍)
        𝐍 = 𝐁 X 𝐓, 𝐓 = 𝐍 X 𝐁,  𝐍¡Ñ𝐁, 𝐍¡Ñ𝐍',  𝐁'¡Ñ𝐁, 𝐁'¡Ñ𝐍'  ¡Å 𝐁' ¡ð𝐍
     ∘  𝐁'  = - 𝜏𝐍  <- 𝜏 = 𝜏(s): torsion ºñƲ¸²(²¿ÀÓ) ÇÔ¼ö·Î¼­ osculating planeÀÇ turning rate(üÞï® Ýïáã)¸¦ ÃøÁ¤ÇÔ.
     ∘  𝐓' = k𝐍,  𝐍' = - k𝐓 + 𝜏𝐁,  𝐁'  = - 𝜏𝐍  <- the Formulas of Frenet

         ¾ÆÁÖ °£·«ÇÏ°Ô ¿ä¾àÇÑ ÀÌ»óÀÇ °ü°è½ÄÀ» Á¦´ë·Î ÀÌÇØÇÑ´Ù¸é ¿ì¸®´Â 3Â÷¿ø¿¡¼­ÀÇ ÀÓÀÇÀÇ °î¼±À» ÇØ¼®/ÀçÇöÇÒ ¼ö ÀÖ½À´Ï´Ù.

2. Gauss Curvature I <- Figure I-10 ÂüÁ¶   

     ∘  ¸Å²ô·¯¿î(smooth) °î¸é(surface) 𝐌 ¡ø 𝐄3¿¡ ÀÓÀÇÀÇ Á¡ 𝐏¿¡ unit normal vector¸¦ 𝐔¶ó°í ÇÕ´Ï´Ù. À̶§ 𝐯°¡ Á¡ 𝐏¿¡¼­
         𝐌¿¡ Á¢Çϴ unit vector¶ó¸é, ±× 𝐯¿¡ ÀÇÁ¸ÇÏ´Â 𝛂𝐯 °î¼±ÀÌ ÀÖ´Â 𝐌°ú 𝐏¿¡¼­ ±³Â÷ÇÏ´Â Æò¸éÀº 𝐯¿Í 𝐔¿¡ ÀÇÇØ °áÁ¤µË´Ï´Ù.
         ±×·¡¼­ ¿ì¸®´Â 𝛂𝐯¸¦ °î¸é 𝐌ÀÇ Á¡ 𝐏¿¡¼­ÀÇ normal section in the 𝐯 direction À̶ó°í ºÎ¸¨´Ï´Ù.
     ∘  𝐤n(𝐯) = ∓1/ 𝐑(𝐯)  <- 𝐑(𝐯): osculating circle ¹Ý°æ,  𝐤n(𝐯): 𝐌ÀÇ 𝐏¿¡¼­ normal curvature in the 𝐯 direction         
         𝛂𝐯ÀÇ principal normal vector¿Í surface normal vectorÀÎ 𝐔ÀÇ ¹æÇâÀÌ µ¿ÀÏÇϸé +, ¹Ý´ëÀ̸é -, curvature°¡ 0À̸é 0.
     ∘  Gauss curvature 𝐊 (𝐌ÀÇ 𝐏¿¡¼­)¶õ?
         𝐊(𝐏) = 𝐤1 𝐤2  <-  𝐯1: tangent vector in a principal direction  𝐤1: maxinmum value of 𝐤n(𝐯),
                                 𝐯2: tangent vector in a principal direction  𝐤2: minimum value of 𝐤n(𝐯), 𝐯1 and  𝐯1: Ç×»ó Á÷±³ÇÔ.
         ex1)  𝐊  in a cylinder: (¹Ý°æ RÀÎ °æ¿ì) 𝐤1= 1/R, 𝐤2= 0  ¡Å 𝐊 = 0
         ex2)  𝐊  in a sphere: (¹Ý°æ RÀÎ °æ¿ì) 𝐤n(𝐯) = ∓1/R2
         ex3)  𝐊  in a sddle-shaped surface: z = 1/2(y2 - x2),  𝐊 < 0  
         ex4)  𝐊  in a torus:  𝐊 > 0 on outside surface,  𝐊 > 0 on inside surface, 𝐊 = 0 on two circle of top and bottom
  
         CurvatureÀÇ intrinsic Ò®î¤îÜ Ç¥±â ´öºÐ¿¡ ÀϹݻó´ë¼º¿ø¸®ÀÇ 4Â÷¿ø ½Ã°ø°£ curvature¸¦ Á¤ÀÇÇÒ ¼ö ÀÖ½À´Ï´Ù. 
         GaussÀÇ Therema Egregium (»©¾î³­ ïÒ×â)·Î ÀýÁ¤À» ÀÌ·ç´Â °î¸é ÀÌ·Ð ´öºÐ¿¡ intrinsic Ç¥±â°¡ °¡´ÉÇØÁ³½À´Ï´Ù.
         intrinsicÇÑ curvatureÀÇ Àç°ø½ÄÈ­´Â °íÂ÷¿øÀÇ manifold ÒýåÆô÷¿Í 'curved space'¸¦ ÀϹÝÈ­ÇÒ ¼ö ÀÖ°Ô ÇÕ´Ï´Ù.

3. Surfaces ÍØØü in E3 <- Figure I-20 ÂüÁ¶  

     ∘  °î¸é 𝐌Àº 𝓡2ÀÇ open subset 𝐃¿¡ Á¤ÀÇµÈ µÎ°³ º¯¼öÀÇ vector ÇÔ¼ö 𝐗:𝐃 ¡æ E3ÀÇ image·Î locally(ÏÑá¶îÜÀ¸·Î) Ç¥±â °¡´É                  
           𝐗 = 𝐗(u,v) = (x(u,v), y(u,v), z(u,v))   [3-2]
     ∘  ÆíÀÇ»ó ¿ì¸®´Â ÀÌ ÇÔ¼ö¸¦ 𝐃¿¡¼­ ÃÖ¼Ò »ïÂ÷±îÁö ¹ÌºÐÇÒ ¼ö ÀÖÀ¸¸ç, 𝐗°¡ regular ïáöÎîÜÀÏ ¶§´Â ´ÙÀ½ÀÇ °æ¿ì
           𝐗1(u, v) = ¡Ó𝐗/¡Óu = (¡Óx/¡Óu, ¡Óy/¡Óu, ¡Óz/¡Óu)
           𝐗2(u, v) = ¡Ó𝐗/¡Óv = (¡Óx/¡Óv, ¡Óy/¡Óv, ¡Óz/¡Óv)
         »ó±â µÎ vector´Â 𝐃ÀÇ °¢ (u,v)¿¡ ´ëÇØ linearlly independnt(àÊû¡ Լء)ÇÏ¿©¼­, 𝐗1 X 𝐗2 ¡Á 0 in 𝐃.  
     ∘  ÀϹÝÀûÀ¸·Î regularity(ïáöÎàõ)Àº 𝐃ÀÇ °¢ Á¡µéÀÌ neighborhood(ÐÎÛ¨) 𝛺¸¦ °®À¸¸ç, ÀÌ¿¡ ´ëÇØ ÇÔ¼ö 𝐗°¡
         ¿¬¼ÓÀûÀÎ inverse function(æ½ùÞâ¦) 𝐗(𝛺) ¡æ 𝛺 ¸¦ °®´Â one-to-one(ìéÓßìé)ÀÇ °ü°è¸¦ º¸ÀåÇÕ´Ï´Ù. ±×·¡¼­
         ÀÌ·¯ÇÑ ´ëÀÀÀ» system of curvillinear coordinate(ÍØàÊ ñ¨øöͧ) u, v on 𝐗(𝛺) ¡ø 𝐌À¸·Î »ý°¢ÇÒ ¼ö ÀÖ½À´Ï´Ù.
     ∘  𝛂(t) = 𝐗 (u(t), v(t))  <- 𝐃¿¡¼­ u(t), v(t)°¡ smooth °î¼±À̸é,  𝐗¿¡ ÀÇÇÑ imageµµ 𝐌¿¡¼­ 𝛂µµ smooth °î¼±
            𝛂'(t) = (¡Ó𝐗/¡Óu)(du/dt) +  (¡Ó𝐗/¡Óv)(dv/dt)  <- by chain rule
            𝛂'(t) = u'𝐗1 + v'𝐗2  <- in abbriviated notation   [3-3]

     ∘  Definition I-2
         𝐌¿¡¼­ 𝐏¸¦ Åë°úÇÏ´Â °î¼±ÀÇ velociy vector 𝐯°¡ 𝐏¿¡ ÀÖÀ¸¸é, vector 𝐯¸¦ 𝐏¿¡¼­ 𝐌À¸·ÎÀÇ tangent vector ¶ó ºÎ¸¥´Ù.
         𝐏¿¡¼­ 𝐌À¸·ÎÀÇ ¸ðµç tangent vectorÀÇ ÁýÇÕÀº 𝐏¿¡¼­ 𝐌ÀÇ tangent plane ¶ó°í ºÎ¸£°í, 𝐓p𝐌À¸·Î Ç¥±âÇÑ´Ù.

     ∘  𝐏¿¡¼­ÀÇ ¾î¶² tangent vector 𝐗(u0, v0)´Â  𝐗1(u0, v0)°ú 𝐗1(u0, vo)ÀÇ ÇÑ linear combinationÀÔ´Ï´Ù.
         °Å²Ù·Î ½Ç¼ö a, b¿ÍÀÇ linear combination 𝐯 = (a𝐗1 + b𝐗2)(u0, v0)´Â 𝐌¿¡ ÀÖ´Â ÇÑ °î¼±ÀÇ velocity vectorÀÔ´Ï´Ù.
         ¿¹¸¦ µé¸é 𝛂(t) = 𝐗(u0+ at, v0+ bt)´Â   𝛂'(0) = 𝐯 ¸¦ °®½À´Ï´Ù. ±×·¡¼­ regurality(ïáöÎàõ) of 𝐗¶õ 𝐓p𝐌°¡
         𝐗1°ú  𝐗2¸¦ basis(±âÀú)·Î ÇÏ´Â °¢ Á¡ 𝐏¿¡¼­ÀÇ 2-dimensional vector space¸¦ ÀǹÌÇÏ´Â °ÍÀÔ´Ï´Ù.
         À̶§ 𝐗1(u0, v0)°ú 𝐗2(u0, v0)´Â ±×µé ÀÚ½ÅÀÌ ±× °î¼±ÀÇ velocity vectors
            u¡æ 𝐗 (u, v0)     (v = v0, fixed)   <- u-parameter curve
            v¡æ 𝐗 (u0, v)     (u = u0, fixed)   <- v-parameter curve

p.s. Richard L. Faber Differential Geometry and Relativity Theory (Marcel Deckker 1983) Chapter I.
       ¹ÌºÐ±âÇÏÇаú »ó´ë¼º ¿ø¸®¿¡ °ü½ÉÀ» µÐ ¿µ¾î±Ç ´ëÇб³ ¼öÇаú/¹°¸®Çаú °íÇгâ»ýµéÀ» ´ë»óÀ¸·Î ÁýÇʵǾú´Ù ÇÔ.
       ¿©·¯ GRÃ¥µéÀ» »ìÆì º¸´Ù°¡ ¼öÇÐÀû rigor¿Í ¹ÙÅÁÀÌ ÇÊ¿äÇÑ Á¦°Ô ¾ÆÁÖ ÀûÇÕÇÑ Ã¥À¸·Î »ý°¢µÇ¾î ¼±Á¤ÇÏ¿´À½.
       * curvature´Â ¹®¸Æ¿¡ µû¶ó ÈÖ¾îÁü ÀÚü¸¦ ÀǹÌÇϰųª °î·üÀ» ÁöĪ


Name
Spamfree

     ¿©±â¸¦ Ŭ¸¯ÇØ ÁÖ¼¼¿ä.

Password
Comment

  ´ä±Û¾²±â   ¸ñ·Ïº¸±â
¹øÈ£ Á¦               ¸ñ À̸§ ¿¬°ü ³¯Â¥ Á¶È¸
°øÁö  'Çö´ë ¿ìÁÖ·Ð'¿¡ °üÇÑ Å½±¸ÀÇ Àå    °ü¸®ÀÚ 1 2017-08-15
11:36:55
1312
°øÁö  À§Å°¹é°ú ¾÷µ¥ÀÌÆ®: º¼Ã÷¸¸ ¹æÁ¤½Ä, ¿£Æ®·ÎÇÇ   ✅   [1]  ±è°ü¼® 1 2021-09-28
06:56:21
2452
161  Palmer's The Primacy of Doubt <Ä«¿À½º ¿¡ºê¸®¿þ¾î>  ✍🏻    ±è°ü¼® 2 2025-05-11
08:43:54
56
160    Gleick's CHAOS <Ä«¿À½º: »õ·Î¿î °úÇÐÀÇ ÃâÇö>  ✍🏻    ±è°ü¼® 2 2025-05-11
08:43:54
56
159  Supplement  Chapter 2e. Problems    ±è°ü¼® 3 2025-03-24
15:13:37
81
158    Supplement  Chapter 3e. Problems    ±è°ü¼® 3 2025-03-24
15:13:37
81
157      Supplement  Chapter 4c. Problems    [1]  ±è°ü¼® 3 2025-03-24
15:13:37
81
156  Appendix  Aa. Elements of GR    ±è°ü¼® 2 2025-01-28
20:37:19
102
155    Appendix  Ab. Einstein Equation    ±è°ü¼® 2 2025-01-28
20:37:19
102
154  Baumann's Cosmology  8a. Quantum Conditions    ±è°ü¼® 4 2025-01-08
22:13:54
191
153    Cosmology  8b. Quantum Fluctuations    ±è°ü¼® 4 2025-01-08
22:13:54
191
152      Cosmology  8c. Primordial Power Spectra    ±è°ü¼® 4 2025-01-08
22:13:54
191
151        Cosmology  8d. Obs. Constraints; 9 Outlook    ±è°ü¼® 4 2025-01-08
22:13:54
191
150  Baumann's Cosmology  7a. CMB Physics  ✅    ±è°ü¼® 5 2024-12-13
19:16:42
1775
149    Cosmology  7b. Primordial Sound Waves    ±è°ü¼® 5 2024-12-13
19:16:42
1775
148      Cosmology  7c. CMB Power Spectrum    ±è°ü¼® 5 2024-12-13
19:16:42
1775
147        Cosmology  7d. Glimpse at CMB Polarization    ±è°ü¼® 5 2024-12-13
19:16:42
1775
146          Cosmology  7e. Summary and Problems    ±è°ü¼® 5 2024-12-13
19:16:42
1775
145  Baumann's Cosmology  6a. Relativistic Perturbation    ±è°ü¼® 4 2024-11-08
17:16:07
405
144    Cosmology  6b. Conservation Eqs; Initial Conditions    ±è°ü¼® 4 2024-11-08
17:16:07
405
143      Cosmology  6c. Growth of Matter Perturbations    ±è°ü¼® 4 2024-11-08
17:16:07
405
142        Cosmology  6d. Summary and Problems    ±è°ü¼® 4 2024-11-08
17:16:07
405
141  Baumann's Cosmology  4a. Cosmological Inflation    ±è°ü¼® 5 2024-10-21
22:17:39
475
140    Cosmology  4b. Physics of Inflation    ±è°ü¼® 5 2024-10-21
22:17:39
475
139      Cosmology  5a. Newtonian Perturbation    ±è°ü¼® 5 2024-10-21
22:17:39
475
138        Cosmology  5b. Statistical Properties    ±è°ü¼® 5 2024-10-21
22:17:39
475
137          Cosmology  5c. Summary and Problems    ±è°ü¼® 5 2024-10-21
22:17:39
475
136  Baumann's Cosmology  3a. Hot Big Bang  ✅     ±è°ü¼® 4 2024-09-22
23:39:47
1663
135    Cosmology  3b. Thermal Equilibrium    ±è°ü¼® 4 2024-09-22
23:39:47
1663
134      Cosmology  3c. Boltzmann Equation    ±è°ü¼® 4 2024-09-22
23:39:47
1663
133        Cosmology  3d. Beyond Equilibrium    ±è°ü¼® 4 2024-09-22
23:39:47
1663
132  Baumann's Cosmology  1. Introduction  ⚫  [1]  ±è°ü¼® 5 2024-09-01
12:43:52
7217
131    Cosmology  2a. Expanding Universe    ±è°ü¼® 5 2024-09-01
12:43:52
7217
130      Cosmology  2b. Dynamics      ±è°ü¼® 5 2024-09-01
12:43:52
7217
129        Cosmology  2c. Friedmann Equations    ±è°ü¼® 5 2024-09-01
12:43:52
7217
128          Cosmology  2d. Our Universe    ±è°ü¼® 5 2024-09-01
12:43:52
7217

    ¸ñ·Ïº¸±â   ´ÙÀ½ÆäÀÌÁö     ±Û¾²±â 1 [2][3][4][5]
    

Copyright 1999-2025 Zeroboard / skin by zero & Artech