±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

ÅÙ¼­ ÇØ¼® I-1. Dyad¿Í ÅÙ¼­ÀÇ ¿¬»ê  🔵
    ±è°ü¼®  2019-05-27 13:44:06, Á¶È¸¼ö : 1,047
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ÀÏ¹Ý »ó´ë¼º ¿ø¸®(GR)´Â ÅÙ¼­ ¹æÁ¤½Ä(tensor equation)À̹ǷΠÀ̸¦ ÀÌÇØÇÏ·Á¸é tensor¸¦ ÇнÀÇÏ¿©¾ß¸¸ ÇÕ´Ï´Ù. ±× ±âÃʷμ­,
vector¶õ ¼öÇÐ/¹°¸®Çп¡¼­ Å©±â¿Í ¹æÇâÀ» °®´Â ±âÇÏÀû °´Ã¼(geometric object)À̸ç, scalar¶õ Å©±â¸¸À» °®´Â °´Ã¼(object)ÀÔ´Ï´Ù.
vector space R©ú À̶õ vectorµéÀÌ ¼­·Î ´õÇØÁö°Å³ª, scalar¿ÍÀÇ °ö¼ÀÀÌ °¡´ÉÇÑ ÁýÇÕÀ» °¡¸®Å°¸ç, À̶§ n°³ÀÇ ¼ººÐÀ» °®½À´Ï´Ù.

I-1 TensorÀÇ °³³ä

Tensor¶õ °³³äÀûÀ¸·Î´Â 'Á¤ÀÇµÈ ÁÂÇ¥°è(coordinate system)ÀÇ ¼ººÐÀ» °®µµ·Ï ÇÑ vectorÀÇ È®Àå'À̶ó°í ÇÒ ¼ö ÀÖ½À´Ï´Ù.
ÀÌ´Â tensor·Î Ç¥±âµÈ ¸ðµç ¹æÁ¤½Ä°ú ¹ýÄ¢µéÀÌ 'ÁÂÇ¥°è °£ÀÇ À̵¿°ú ȸÀü µîÀÇ °¢Á¾ º¯È¯¿¡ ´ëÇÑ ºÒº¯¼º'À» À¯ÀÚÇϱâ À§ÇÔÀÔ´Ï´Ù.
ƯÁ¤ tensor°¡ Á¤ÀǵǾî ÀÖ´Â ÁÂÇ¥°è´Â ¼±Çü µ¶¸³ÇÑ ±âÀú(linearly independent basis) tensorµé·Î½á ½Äº°ÇÒ ¼ö ÀÖ½À´Ï´Ù.

       * ¼±Çü µ¶¸³(linear independence): ÇϳªÀÇ vector space¿¡ ¼ÓÇÏ´Â vectorÀÇ ÁýÇÕ¿¡¼­ ¾î¶² vector¶óµµ ´Ù¸¥ vectorµéÀÇ
          linear combination(¼±Çü °áÇÕ)À¸·Î Á¤ÀÇÇÒ ¼ö ¾ø´Â °æ¿ì¸¦ ¸»ÇÕ´Ï´Ù.  ↦  𝐀: matrix of vectors,  det 𝐀 ¡Á 0         

I-2  (4.2) TensorÀÇ Â÷¼ö(order) 

      Tensor u = u1e1 + u2e2 + ... + unen  [en: unit baisis(´ÜÀ§ ±âÀú) tensor, ¡«en¡«= 1] * Cartesian ÁÂÇ¥°èÀÇ °æ¿ì
      NÂ÷ tensor´Â 3Â÷¿ø¿¡¼­´Â 3n°³ÀÇ, ÀϹÝÀû MÂ÷¿ø¿¡¼­´Â Mn°³ÀÇ basis tensor¿Í ¼ººÐ(component)À» °®½À´Ï´Ù..
      0Â÷ tensor´Â ÇѰ³ÀÇ ¼ººÐÀ» °®´Â scalarÀ̸ç, 1Â÷ tensor´Â vector·Î¼­ 3Â÷¿ø¿¡¼­´Â 31 = 3°³ÀÇ ¼ººÐÀ» °®À¸¸ç,
      2Â÷ tensor ¥ò𝑖𝑗´Â MÂ÷¿ø¿¡¼­ M2°³ÀÇ ¼ººÐÀ» °¡Áö¹Ç·Î 3Â÷¿ø¿¡¼­ 32 = 9°³, 4Â÷¿ø¿¡¼­ 42 = 16°³ÀÇ ¼ººÐÀ» °®½À´Ï´Ù. 
           Tensor ¥ò = [¥ò𝑖𝑗] =
              ⌈  ¥ò11  ¥ò12  ¥ò13
             ¦­ ¥ò21  ¥ò22  ¥ò23 ¦­  (𝑖,𝑗 = 1,2,3)   [4.2.6]
              ⌊  ¥ò31  ¥ò32  ¥ò33

I-3  (4.3) TensorÀÇ Áö¼ö(index)
 
      Tensor´Â Â÷¼ö¿Í °°Àº ¼ýÀÚÀÇ Áö¼ö(index)¸¦ °¡Áý´Ï´Ù. Áï, 2Â÷´Â 2°³, 3Â÷´Â 3°³, 4Â÷´Â 4°³ÀÇ Áö¼ö(ex: C𝑖𝑗𝑘𝑙)¸¦ °¡Áý´Ï´Ù.
      TensorÀÇ ¿¬»êÀº 1Â÷ tensorÀÎ vectorÀÇ °æ¿ì¿Í À¯»çÇÏ°Ô µÇ´Âµ¥, ´Ù¸¸ ±× Áö¼ö(index)´Â ²À ÀÏÄ¡ÇØ¾ß¸¸ ÇÕ´Ï´Ù. ex) C𝑗𝑘 = A𝑗𝑘 + B𝑗𝑘
      * Einstein's Summation Convention(ÇÕÀÇ ±Ô¾à) <- superscript(À§Ã·ÀÚ)¿Í subscript(¾Æ·¡Ã·ÀÚ)°£ Áߺ¹ °¡´É
       a) Tensor·Î Ç¥±âµÈ ½Ä¿¡¼­ Áߺ¹ Áö¼ö(dummy index)´Â ÇÕÀÇ ±âÈ£(¢²)¸¦ ´ëÄ¡ÇÕ´Ï´Ù. ex) A𝑘𝑘 =  ¢² A𝑘𝑘 (𝑘=1,2,3) = A11 + A22 + A33
       b) °°Àº Ç׿¡¼­ Áߺ¹ Áö¼ö´Â 2°³¸¦ ³ÑÀ» ¼ö ¾ø½À´Ï´Ù. ex) v =  ϵ𝑖𝑗𝑘 a𝑖b𝑗c𝑗 [𝑗°¡ 3°³¶ó Ʋ¸²] ¢¡ v =  ϵ𝑖𝑗𝑘 a𝑖b𝑗c𝑘       
      * Free index rule(ÀÚÀ¯ Áö¼ö ±ÔÄ¢)
       Tensor ¹æÁ¤½Ä¿¡¼­ ¾çÂÊ º¯ÀÇ ÀÚÀ¯ Áö¼ö(free index)´Â µ¿ÀÏÇØ¾ß ÇÕ´Ï´Ù. ex)  t𝑘 = ¥ò𝑗𝑘 n𝑘 [Ʋ¸²] ¢¡  t𝑗 = ¥ò𝑗𝑘n𝑘

I-3  (4.5) Æ¯º°ÇÑ Á¾·ùÀÇ Tensor
  
      a) Kronecker delta: ¥ä𝑖𝑗 = 0 (𝑖¡Á𝑗), 1 (𝑖=𝑗)   [4.5.1]  <- ³ªÁß¿¡ ³ª¿À´Â unit (´ÜÀ§) tensorÀÇ Áö¼ö Ç¥±âÀÓ.
               ¥ä11 = 1       ¥ä12 = 0       ¥ä13 = 0
               ¥ä21 = 0       ¥ä22 = 1       ¥ä23 = 0    [4.5.2]
               ¥ä31 = 0       ¥ä32 = 0       ¥ä33 = 1
          Kronecker delta´Â °áÇÕµÈ ´Ù¸¥ tensorÀÇ µ¿ÀÏÇÑ Áö¼ö¸¦ ÀÚ½ÅÀÇ ´Ù¸¥ Áö¼ö·Î ´ëÄ¡ÇÏ´Â Áß¿äÇÑ ¿ªÇÒÀÇ ¼ºÁúÀ» °®½À´Ï´Ù.
              ex1) ¥ä𝑗𝑘 x𝑗 = x𝑘     ex2) ¥ä𝑖𝑚 ¥ä𝑗𝑛 T𝑚𝑛 = T𝑖𝑗
      b) Permutation symbol(¼øÈ¯ ±âÈ£):  𝑒𝑖𝑗𝑘 = {1 (123, 231, 312), -1 (321, 213, 132), 0 (ÀÌ¿ÜÀÇ °æ¿ì)}   [4.5.6]
              ex) 𝑒𝑗𝑘𝑘 = e𝑗11 + 𝑒𝑗22 + 𝑒𝑗33 = 0 + 0 + 0 = 0. ¡ñ °¢±â ¸ðµÎ°¡ 'ÀÌ¿ÜÀÇ °æ¿ì'À̹ǷÎ
      c) Tensor¿Í Cartesian ÁÂÇ¥°è:       
              ex) Carttesian ÁÂÇ¥°è 3Â÷¿ø 2Â÷ tensor *
                   Tensor ¥ò = [¥ò𝑖𝑗] =
                     ⌈ ¥ò11  ¥ò12  ¥ò13 ⌉            
                    ¦­¥ò21   ¥ò22  ¥ò23¦­  (𝑖,𝑗 = 1,2,3)   [4.5.12]
                     ⌊ ¥ò31   ¥ò32  ¥ò33 ⌋    

I-4 TensorÀÇ ±âº» ¿¬»ê

      a) 5.1 Dot product (³»Àû)
               ab = (a𝑖 𝐞𝑖) ∙ (b𝑗 𝐞𝑗)  = a𝑖b𝑗 𝐞𝑖 ∙ 𝐞𝑗 = a𝑖b𝑗 ¥ä𝑖𝑗 = a𝑖b𝑖;  ¥ä𝑖𝑗 ¡Õ 𝐞𝑖 ∙ 𝐞𝑗 <- kronecker delta Á¤ÀÇ   [5.1.1-4]
      b) 5.2 Cross product (¿ÜÀû)
               ab = (a𝑖 𝐞𝑗) ⨯ (b𝑗 𝐞𝑗)  = a𝑖b𝑗 𝐞𝑖 ⨯ 𝐞𝑗   [5.2.1]
               ab =
                 ∣ 𝐞1 𝐞2 𝐞3
                 ∣ a1 a2 a3 ∣ = 𝑒𝑖𝑗𝑘 a𝑖b𝑗 𝐞𝑘  <-  𝐞𝑖 ⨯ 𝐞𝑗 = 𝑒𝑖𝑗𝑘 𝐞𝑘   [5.2.2,3]
                 ∣ b1 b2 b3
      c) 5.3 Triple dot product (»ïÁß ³»Àû)
               (ab) ∙ c =
                 ∣ a1 a2 a3 ∣     ∣ a1 b1 c1
                 ∣ b1 b2 b3 ∣ = ∣ a2 b2 c2 ∣  =  𝑒𝑖𝑗𝑘 a𝑖b𝑗c𝑘;  𝑒𝑖𝑗𝑘 ¡Õ (𝐞𝑖 ⨯ 𝐞𝑗) ∙ 𝐞𝑘 <- permutation symbol Á¤ÀÇ   [5.3.1,2]
                 ∣ c1 c2 c3 ∣     ∣ a3 b3 c3

I-5  (5.4,5) DyadÀÇ °³³ä°ú ¿¬»ê

      a) Dyad¿Í dyad product(°ö) **
          dyad¶õ µÎ vectorÀÇ °öÀ¸·Î ÀÌ·ç¾îÁø 2Â÷ tensor¸¦ ¸»Çϸç, ÀÌ °öÀ» dyad product¶ó ÇÕ´Ï´Ù. ¡æ ab or  a b   [5.4.1,2]
          dyad¶õ 2Â÷ tensor·Î¼­ components¿Í basis dyad·Î ±¸¼ºµË´Ï´Ù. ¡æ ab = a𝑖b𝑗 𝐞𝑖 ⊗ 𝐞𝑗 = T = T𝑖𝑗 𝐞𝑖 ⊗ 𝐞𝑗   [5.5.2]
      b) Basis(±âÀú) dyad                                                                                                         
              °¢°¢ÀÇ identity basis(´ÜÀ§ ±âÀú) vectorÀÇ °öÀ̹ǷÎ,
              ex) 𝐞1 ⊗ 𝐞2
                     ⌈ 1 ⌉                    
                    ¦­0¦­  [ 0  1  0 ]  =  (°á°ú: ¾Æ·¡ ÂüÁ¶)   [5.4.6]              
                     ⌊ 0 ⌋                         
              ÀüºÎ¸¦ °è»êÇßÀ» ¶§ °á°ú´Â...  [5.4.7]
                𝐞1 ⊗ 𝐞1 =                   𝐞1 ⊗ 𝐞2 =                 𝐞1 ⊗ 𝐞3 =                                                                                     
                  ⌈ 1  0  0 ⌉                   ⌈ 0  1  0 ⌉                  ⌈ 0  0  1 ⌉
                 ¦­0  0  0¦­                  ¦­0  0  0¦­                 ¦­0  0  0¦­
                  ⌊ 0  0  0 ⌋                   ⌊ 0  0  0 ⌋                  ⌊ 0  0  0 ⌋ 
                𝐞2 ⊗ 𝐞1 =                   𝐞2 ⊗ 𝐞2 =                 𝐞2 ⊗ 𝐞3 =
                  ⌈ 0  0  0 ⌉                   ⌈ 0  0  0 ⌉                   ⌈ 0  0  0 ⌉                          
                 ¦­1  0  0¦­                  ¦­0  1  0¦­                  ¦­0  0  1¦­
                  ⌊ 0  0  0 ⌋                   ⌊ 0  0  0 ⌋                   ⌊ 0  0  0 ⌋
                𝐞3 ⊗ 𝐞1 =                   𝐞3 ⊗ 𝐞2 =                 𝐞3 ⊗ 𝐞3 =
                  ⌈ 0  0  0 ⌉                   ⌈ 0  0  0 ⌉                   ⌈ 0  0  0 ⌉
                 ¦­0  0  0¦­                  ¦­0  0  0¦­                  ¦­0  0  0¦­
                  ⌊ 1  0  0 ⌋                   ⌊ 0  1  0 ⌋                   ⌊ 0  0  1 ⌋
      c) Identity(´ÜÀ§) dyad
                 I  =
                   ⌈ 1  0  0 ⌉                  
                  ¦­0  1  0¦­ =  ¥ä𝑖𝑗 𝐞𝑖 ⊗ 𝐞𝑗 =  𝐞𝑖 ⊗ 𝐞𝑖   [5.5.28]
                   ⌊ 0  0  1 ⌋     
      d) 5.6 Dyad ¿¬»ê <- [Wikipedia] 'Dyadics' 'Tensor product' [link] <- ¡Ø ¾ÕÀ¸·Î ÀÚÁÖ ¾²ÀÓ.
          ∘ Dyad¿Í vector dot product: (ab) ∙ c = a (bc),  c ∙ (ab) = (ca) b,  a ∙ (bc) ∙ d = (ab)(cd)    [5.6.3]
          ∘ Dyad¿Í vector cross product: (ab) ⨯ c = a (bc),  c ⨯ (ab) = (ca) b
          ∘ Dyad¿Í dyad dot product: (ab) ∙ (cd) = (bc) (ad)    [5.6.8]
          ∘ DyadÀÇ double dot product(ÀÌÁß Á¡°ö): (ab) : (cd) = (ac) (bd)    [5.6.16]

I-6 TensorÀÇ Æ¯¼º°ú ¿¬»ê
  
      a) 6.1 Tensor dot product
         ∘ Contraction(Ãà¾à):  index(Áö¼ö)¸¦ Áߺ¹Çؼ­ ÇÕÇϸé order(Â÷¼ö)°¡ 2Â÷ ³·¾ÆÁü. ex) 𝐀 - 2Â÷ tensor, 𝐀𝑘𝑘- 0Â÷ tensor, scalar
                                          2Â÷ tensorÀÇ dot product -> 2Â÷ tensor, 2Â÷ tensorÀÇ double product -> 0Â÷ tensor  
         ∘ TensorÀÇ ¼ººÐ ±¸Çϱâ: ±× basis vector¸¦ dot productÇÔ. ex) 𝐀 ∙ 𝐞𝑖 = A𝑖𝑗 𝐞𝑖 ⊗ 𝐞𝑗∙ 𝐞𝑖 =  A𝑖𝑗 𝐞𝑖 ¥ä𝑗𝑖 =  A𝑖1+ A𝑖2+ A𝑖3   [6.1.22-4]
         ∘ 2Â÷ tensorÀÇ dot product: 𝐀2 = 𝐀 ∙ 𝐀,   𝐀3 = 𝐀 ∙ 𝐀 ∙ 𝐀    [6.1.33,34]
         ∘ double dot product:  𝐀 : 𝐁 = A𝑖𝑗 𝐞𝑖 ⊗ 𝐞𝑗 : B𝑘𝑙 𝐞𝑘 ⊗ 𝐞𝑙 = A𝑖𝑗B𝑘𝑙 (𝐞𝑖 ⊗ 𝐞𝑗 : 𝐞𝑘 ⊗ 𝐞𝑙) = A𝑖𝑗B𝑘𝑙 (𝐞𝑖 ∙ 𝐞𝑘)(𝐞𝑗 ∙ 𝐞𝑙) = A𝑖𝑗B𝑘𝑙 ¥ä𝑖𝑘 ¥ä𝑗𝑙 = A𝑖𝑗B𝑖𝑗  
             <- ¡ñ  (ab) : (cd) = (ac) (bd);  µÎ¹øÀÇ Ãà¾à ¼öÇà ÈÄ 0Â÷ tensor, scalar°¡ µÊ.   [6.1.35-40]
      b) 6.2 Identity(´ÜÀ§) tensor
         ∘ 2Â÷ identity tensor:  𝐈 = ¥ä𝑖𝑗 𝐞𝑖 ⊗ 𝐞𝑗,  𝐈 ∙ 𝐚 = 𝐚   [6.2.6,10]
         ∘ Trace(´ë°¢ÇÕ):  𝐀 : 𝐈 = tr(𝐀) ¡Õ 𝐀𝑘𝑘 = 𝐀11 + 𝐀22 + 𝐀33   [6.2.11]
                 tr(𝐀) = tr(𝐀T) = A𝑖𝑖,  𝐀 : 𝐁 = A𝑖𝑗B𝑖𝑗,  tr(𝐀2) = tr(𝐀 ∙ 𝐀) = 𝐀 ∶ 𝐀 = A𝑖𝑗A𝑖𝑗   [6.2.12-15]
         ∘ TensorÀÇ Å©±â: ¡«𝐀¡«= ¡î (𝐀 : 𝐀) = ¡î tr(𝐀2) = ¡î A𝑖𝑗A𝑖𝑗   [6.2.17,18]
      c) 6.3  TensorÀÇ inverse(¿ª)
         ∘ Tensor¿Í matrix(Çà·Ä):  tensorÀÇ inverse 𝐀-1 or [A𝑖𝑗]-1,  𝐀 ∙ 𝐀-1 = 𝐈  <- 2Â÷ identity(´ÜÀ§) tensor   [6.3.1,2]
         ∘ Determinant: det 𝐀,  minor determinant: 𝑀𝑖𝑗,  cofactor: Ȃ = (-1)𝑖+𝑗 𝑀𝑖𝑗   [6.3.5,6]
             det 𝐀T = det 𝐀,  det (𝐀 𝐁) = (det 𝐀) (det 𝐁),  det 𝐀-1 = 1 / det 𝐀   [6.3.12-14]  
         ∘ Inverse matrix(¿ªÇà·Ä):  𝐀-1 = (Ȃ)T/ det 𝐀  <- ¡Ø Áß¿äÇÔ;   (𝐀-1)-1 = 𝐀,  (𝐀T)-1 = (𝐀-1)T   [6.3.12]
         ∘ Orthogonal(Á÷±³) tensor :  𝐀-1 = 𝐀T,  𝐀 ∙ 𝐀-1 = 𝐀 ∙ 𝐀T =  𝐈   [6.3.25,26]
      d) 6.4 Eigenvalue(°íÀµ°ª)ÀÇ ¹®Á¦>
             (𝐀 - ¥ë𝐈) ∙ 𝐱 = 0     ¥ë: eigenvalue,   𝐱: eigenvector ,  𝐈: identity vector   [6.4.1]
            det (𝐀 - ¥ë𝐈) = 0   [6.4.2]    
            ¥ë3 - 𝐼 ¥ë2 +  𝐼𝐼 ¥ë2 - 𝐼𝐼𝐼 = 0   [6.4.3]  
            𝐼 = tr𝐀 = A 𝑖𝑖 ,   𝐼𝐼 = 1/2[(tr𝐀)2 - (tr(𝐀2)] = 1/2[(A𝑖𝑖 A𝑗𝑗 - A𝑖𝑗 A𝑖𝑗)],   𝐼𝐼𝐼 = det 𝐀 = 𝑒𝑖𝑗𝑘 A1𝑖 A2𝑗 A3𝑘   [6.4.5,6] ***
      e) 6.5 Determinant¿Í permutaion symbol(¼øÈ¯ ±âÈ£)
         ∘  det 𝐀 = det[A𝑖𝑗] = 𝑒𝑖𝑗𝑘 A1𝑖 A2𝑗A3𝑘 = 𝑒𝑖𝑗𝑘  A𝑖1A𝑗2A𝑘3   [6.5.2]
         ∘  (𝐚 ⨯ 𝐛 ∙ 𝐜)(𝐝 ⨯ 𝐞 ∙ 𝐟) =   <-  ¡Ø II-6 f)¿¡¼­ »ç¿ë;  det 𝐀T = det 𝐀,  det (𝐀 𝐁) = (det 𝐀) (det 𝐁) Ȱ¿ë   [6.5.25]
              ∣ 𝑎1  𝑎2  𝑎3 ∣  ∣ 𝑑1  𝑒1  𝑓1 ∣      ∣ 𝐚 ∙ 𝐝    𝐚 ∙ 𝐞    𝐚 ∙ 𝐟 ∣
              ∣ 𝑏1  𝑏2  𝑏3 ∣  ∣ 𝑑2  𝑒2  𝑓2 ∣  =  ∣ 𝐛 ∙ 𝐝    𝐛 ∙ 𝐞    𝐛 ∙ 𝐟 ∣                                  
              ∣ 𝑐1  𝑐2   𝑐3 ∣  ∣ 𝑑3  𝑒3  𝑓3 ∣      ∣ 𝐜 ∙ 𝐝     𝐜 ∙ 𝐞    𝐜 ∙ 𝐟 ∣

p.s.  ÃÖ´ö±â Àú <ÅÙ¼­ ÇØ¼® °³·Ð> (¹üÇѼ­Àû 2003)À» ÅØ½ºÆ®·Î ÆíÁýÇϰí, ÀϺΠ¿À·ù/¿ÀŸµéÀ» ¼öÁ¤ÇßÀ½.
        ¸î°¡Áö¸¦ °ËÅäÇÑ ³¡¿¡ ¼±ÅÃÇÑ ÀÌ Ã¥ÀÌ ´Ù¸¥ ¿µ¹®Ã¥/¹ø¿ªÆÇµéº¸´Ù ÀÌÇØ°¡ ½¬¿ì¸é¼­µµ ½Ç¿ëÀûÀÎ µíÇßÀ½.
        ¿¹Á¦ Ãß°¡¿Í Ipad À§ÁÖÀÇ Çà·Ä°ú determinant Ç¥±â ¼öÁ¤ µî Àü¹ÝÀûÀ¸·Î ¾÷µ¥ÀÌÆ®ÇÔ. [u. 10/2019]
*, ** »ó´ë¼º ÀÌ·ÐÀº ½Ã°£ Â÷¿øÀ» Æ÷ÇÔÇÑ 4Â÷¿ø 2Â÷ tensorÀÎ 4-vector dyad¸¦ »ç¿ëÇÔ.
***  ¿À·§µ¿¾È ¹æÄ¡Çß´ø ¿ÀŸ¸¦ ¼öÁ¤ÇßÀ½. [u. 5/2020]


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°øÁö  'Çö´ë ¿ìÁÖ·Ð'¿¡ °üÇÑ Å½±¸ÀÇ Àå    °ü¸®ÀÚ 1 2017-08-15
11:36:55
1315
°øÁö  À§Å°¹é°ú ¾÷µ¥ÀÌÆ®: º¼Ã÷¸¸ ¹æÁ¤½Ä, ¿£Æ®·ÎÇÇ   ✅   [1]  ±è°ü¼® 1 2021-09-28
06:56:21
2453
161  Palmer's The Primacy of Doubt <Ä«¿À½º ¿¡ºê¸®¿þ¾î>  ✍🏻    ±è°ü¼® 2 2025-05-11
08:43:54
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160    Gleick's CHAOS <Ä«¿À½º: »õ·Î¿î °úÇÐÀÇ ÃâÇö>  ✍🏻    ±è°ü¼® 2 2025-05-11
08:43:54
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159  Supplement  Chapter 2e. Problems    ±è°ü¼® 3 2025-03-24
15:13:37
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158    Supplement  Chapter 3e. Problems    ±è°ü¼® 3 2025-03-24
15:13:37
87
157      Supplement  Chapter 4c. Problems    [1]  ±è°ü¼® 3 2025-03-24
15:13:37
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156  Appendix  Aa. Elements of GR    ±è°ü¼® 2 2025-01-28
20:37:19
104
155    Appendix  Ab. Einstein Equation    ±è°ü¼® 2 2025-01-28
20:37:19
104
154  Baumann's Cosmology  8a. Quantum Conditions    ±è°ü¼® 4 2025-01-08
22:13:54
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153    Cosmology  8b. Quantum Fluctuations    ±è°ü¼® 4 2025-01-08
22:13:54
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152      Cosmology  8c. Primordial Power Spectra    ±è°ü¼® 4 2025-01-08
22:13:54
200
151        Cosmology  8d. Obs. Constraints; 9 Outlook    ±è°ü¼® 4 2025-01-08
22:13:54
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150  Baumann's Cosmology  7a. CMB Physics  ✅    ±è°ü¼® 5 2024-12-13
19:16:42
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149    Cosmology  7b. Primordial Sound Waves    ±è°ü¼® 5 2024-12-13
19:16:42
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148      Cosmology  7c. CMB Power Spectrum    ±è°ü¼® 5 2024-12-13
19:16:42
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147        Cosmology  7d. Glimpse at CMB Polarization    ±è°ü¼® 5 2024-12-13
19:16:42
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146          Cosmology  7e. Summary and Problems    ±è°ü¼® 5 2024-12-13
19:16:42
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145  Baumann's Cosmology  6a. Relativistic Perturbation    ±è°ü¼® 4 2024-11-08
17:16:07
414
144    Cosmology  6b. Conservation Eqs; Initial Conditions    ±è°ü¼® 4 2024-11-08
17:16:07
414
143      Cosmology  6c. Growth of Matter Perturbations    ±è°ü¼® 4 2024-11-08
17:16:07
414
142        Cosmology  6d. Summary and Problems    ±è°ü¼® 4 2024-11-08
17:16:07
414
141  Baumann's Cosmology  4a. Cosmological Inflation    ±è°ü¼® 5 2024-10-21
22:17:39
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140    Cosmology  4b. Physics of Inflation    ±è°ü¼® 5 2024-10-21
22:17:39
483
139      Cosmology  5a. Newtonian Perturbation    ±è°ü¼® 5 2024-10-21
22:17:39
483
138        Cosmology  5b. Statistical Properties    ±è°ü¼® 5 2024-10-21
22:17:39
483
137          Cosmology  5c. Summary and Problems    ±è°ü¼® 5 2024-10-21
22:17:39
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136  Baumann's Cosmology  3a. Hot Big Bang  ✅     ±è°ü¼® 4 2024-09-22
23:39:47
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135    Cosmology  3b. Thermal Equilibrium    ±è°ü¼® 4 2024-09-22
23:39:47
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134      Cosmology  3c. Boltzmann Equation    ±è°ü¼® 4 2024-09-22
23:39:47
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133        Cosmology  3d. Beyond Equilibrium    ±è°ü¼® 4 2024-09-22
23:39:47
1671
132  Baumann's Cosmology  1. Introduction  ⚫  [1]  ±è°ü¼® 5 2024-09-01
12:43:52
7220
131    Cosmology  2a. Expanding Universe    ±è°ü¼® 5 2024-09-01
12:43:52
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130      Cosmology  2b. Dynamics      ±è°ü¼® 5 2024-09-01
12:43:52
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129        Cosmology  2c. Friedmann Equations    ±è°ü¼® 5 2024-09-01
12:43:52
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128          Cosmology  2d. Our Universe    ±è°ü¼® 5 2024-09-01
12:43:52
7220

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