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ÅÙ¼­ ÇØ¼® II-1. ÀÏ¹Ý ÁÂÇ¥°è ÅÙ¼­ÀÇ ¿¬»ê
    ±è°ü¼®  2019-06-03 21:39:43, Á¶È¸¼ö : 940
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II-1 ÀÏ¹Ý ÁÂÇ¥°è¿Í Tensor   

     a) 11.1 General coordinate(ÀÏ¹Ý ÁÂÇ¥°è)    <- ±×¸² 11.4 ÂüÁ¶
       ∘ Cartesian coordinates(ÁÂÇ¥°è): 𝐱1= x, 𝐱2= y, 𝐱3= z,  𝐮 = 𝑢1𝐞1 + 𝑢2𝐞2 + 𝑢3𝐞3
       ∘ Cylindrical coordinates(¿øÅë ÁÂÇ¥°è): 𝜉1= 𝑟, ¥î2= 𝜃,  𝜉3= 𝑧,   𝐱1= 𝑟 cos𝜃, 𝐱2 = 𝑟 sin𝜃,  x3 = 𝑧        
       ∘ Sphrical coordinates(±¸ ÁÂÇ¥°è): 𝜉1= 𝜌, 𝜉2= 𝜃,  𝜉3= 𝜙,   𝐱1= 𝜌 sin𝜃 cos𝜙, 𝐱2= 𝜌 sin𝜃 sin𝜙,  𝐱3= 𝜌 cos𝜃
       ∘ Curviinear coordinates(°î¼± ÁÂÇ¥°è) ¡æ general coordinate(ÀÏ¹Ý ÁÂÇ¥°è)·Î ÁöĪµÇ±âµµ ÇÔ. 
          ´Ù¸¥ ÁÂÇ¥°èµé°ú ´Þ¸® ƯÁ¤ÀÇ reference point(±âÁØÁ¡)ÀÌ ¾øÀ¸¸ç Cartesian ÁÂÇ¥°èÀÇ ¿øÁ¡¿¡ ´ëÇÑ »ó´ëÀûÀÎ À§Ä¡¿¡¼­ Á¤ÇØÁý´Ï´Ù.
          curvilinear ÁÂÇ¥°è´Â global coordinate(±¤¿ª ÁËÇ¥°è)ÀÎ Cartesian ÁÂÇ¥°è¿Í ´Þ¸® local coordinate(±¹¼Ò ÁÂÇ¥°è)¶ó°í ºÎ¸¨´Ï´Ù.
          À§ ±×¸²¿¡¼­ º¸µíÀÌ Cartesian ÁÂÇ¥°è¿¡¼­ ÀÓÀÇ °Å¸®¸¸Å­ ¶³¾îÁø ÇÑÁ¡±îÁö¸¦ ÀÕ´Â vector¸¦ À§Ä¡ vector 𝐱 ¶ó°í Ç¥±âÇÕ´Ï´Ù.
          ÀÌ Á¡¿¡¼­ curvilinear ÁÂÇ¥°è´Â ÁÂÇ¥°èÀÇ components(¼ººÐ)¿¡ µû¸¥ °î¸éÀ» °¡Áö°Ô µË´Ï´Ù.
     b) 11.2 ÀÏ¹Ý ÁÂÇ¥°è(°î¼± ÁÂÇ¥°è)¿¡¼­ÀÇ Ç¥±â¹ý 
       ∘  ÀÏ¹Ý ÁÂÇ¥°èÀÇ vector 𝐮 = 𝑢𝑖 𝐠𝑖 = 𝑢𝑖 𝐠𝑖  <- ¡Ø ÀÏ¹Ý tensorÀÇ Áߺ¹Áö¼ö´Â ¹Ýµå½Ã À§¿Í ¾Æ·¡·Î ±³Â÷Çϵµ·Ï ÇØ¾ß ÇÕ´Ï´Ù.   [11.2.2,5]
          𝐮 = 𝑢𝑖 𝐠𝑖 = 𝑢1𝐠𝑖 + 𝑢2𝐠2 + 𝑢3𝐠3  <-  𝑢𝑖: contravariant component(¹Ýº¯ ¼ººÐ),  𝐠𝑖: natural basis(ÀÚ¿¬ ±âÀú) vector   [11.2.3]    
          𝐮 = 𝑢𝑖 𝐠𝑖 = 𝑢1𝐠1 + 𝑢2𝐠2 + 𝑢3𝐠3  <-  𝑢𝑖: covariant component(°øº¯ ¼ººÐ),  𝐠𝑖: reciprocal basis(¿ª±âÀú) vector   [11.2.4]
       ∘  2Â÷ tensor Ç¥±â¹ý: ´ÙÀ½ÀÇ 4°¡Áö·Î Ç¥ÇöµÉ ¼ö ÀÖ½À´Ï´Ù.
          𝐀 =  𝐴𝑖𝑗 𝐠𝑖 ⊗ 𝐠𝑗 = 𝐴𝑖𝑗 𝐠𝑖 ⊗ 𝐠𝑗 = 𝐴𝑖,𝑗 𝐠𝑖 ⊗ 𝐠𝑗 = 𝐴𝑖,𝑗 𝐠𝑖⊗ 𝐠𝑗  <- contravant(¹Ýº¯), covariant(°øº¯), mixed(È¥ÇÕ) ¼ººÐ   [11.2.6-8]
     c) 11.3 General(ÀϹÝ) tensorÀÇ ¼ºÁú
       ∘ Tanspose(ÀüÄ¡):
          symmetric tensor 𝐀T = (𝐴𝑖𝑗)T 𝐠𝑖 ⊗ 𝐠j = 𝐴𝑗𝑖 𝐠𝑖 ⊗ 𝐠j = 𝐴𝑖𝑗 𝐠j ⊗ 𝐠𝑖,  (𝐴𝑖𝑗)T = 𝐴𝑗𝑖,  (𝐴𝑖𝑗)T = 𝐴𝑗𝑖,  (𝐴𝑖,j)T = 𝐴j,𝑖   [11.3.1,2]
          symmetric tensor: If 𝐀T = 𝐀, then 𝐀: sym𝐀   <- square matrix                          
          skew-symetric(¹Ý´ëĪ) tensor: If 𝐀T = -𝐀, then 𝐀: skew𝐀   <- square matrix    [11.3.3]                                                
          𝐀 = sym𝐀 + skew𝐀,   sym𝐀 = 1/2 (𝐀 +  𝐀T),  skew𝐀 = 1/2 (𝐀 -  𝐀T)  <- Toeplitz decomposition 'Symmetric matrix'[link]
       ∘ µ¡¼À°ú »¬¼À: µ¿ÀÏÇÑ ±âÀú(basis)¸¦ °®´Â ¼ººÐ(component) °£¿¡ °¡´É
          ex1) 𝐀 - 𝐁 = (𝐴𝑖𝑗 - 𝐵𝑖𝑗) 𝐠𝑖 ⊗ 𝐠𝑗 =  𝑇𝑖𝑗 = 𝐓,  ex2) 𝐀 - 𝐁 = (𝐴𝑖,𝑗 - 𝐵𝑖,𝑗) 𝐠𝑖 ⊗ 𝐠𝑗 =  𝑇𝑖,𝑗 = 𝐓   [11.4.4,5]
     d) 11.5 General tensorÀÇ À¯¿ë¼º: ¡Ø ÁÂÇ¥°è¿¡ ¹«°üÇÑ ¼ö½Ä Ç¥ÇöÀÌ °¡´ÉÇϹǷÎ, Einstein¿¡ ÀÇÇØ ÀÏ¹Ý »ó´ë¼º ¿ø¸®(GR)¿¡ Ȱ¿ëµÇ¾úÀ½.  
          Cartesian ÁÂÇ¥°è¿¡¼­:  ex) velocity 𝐯 = 𝑣𝑖𝐞𝑖,  𝑣𝑖 = 𝑎𝑖𝑡,  𝑣𝑖𝐞𝑖 = 𝑎𝑖𝐞𝑖𝑡  ¡Å  𝐯 = 𝐚𝑡   <- 𝐚: acceleration, 𝑡: time   [11.5.1,4]
          General ÁÂÇ¥°è¿¡¼­:   ex) velocity 𝐯 = 𝑣𝑖𝐠𝑖, 𝑣𝑖 = 𝑎𝑖𝑡,   𝑣𝑖𝐠𝑖 = 𝑎𝑖𝐠𝑖𝑡  ¡Å  𝐯 = 𝐚𝑡   <- 𝐚: acceleration, 𝑡: time   [11.5.2,5]

II-2 ÀÏ¹Ý ÁÂÇ¥°è¿¡¼­ÀÇ ¿¬»ê

     a) 12.1 Inner product(³»Àû)
       ∘ Metric tensor:  𝑔𝑖𝑗 ¡Õ 𝐠𝑖 ∙ 𝐠𝑗;  If 𝐚 ∙ 𝐛 = 𝑎𝑖𝑏𝑗 𝐠𝑖 ∙ 𝐠𝑗, then 𝐚 ∙ 𝐛 = 𝑎𝑖𝑏𝑗 𝑔𝑖𝑗   [12.1.6]
           metric tensor [𝑔𝑖𝑗] =
                                 ⌈  𝑔11  𝑔12  𝑔13
                                ¦­ 𝑔21  𝑔22  𝑔23¦­  (𝑖,𝑗 = 1,2,3)   [12.1.7]
                                 ⌊  𝑔31  𝑔32  𝑔33
       ∘ Kronecker delta:  𝛿𝑖𝑗 ¡Õ 𝐠𝑖 ∙ 𝐠𝑗 = {0 (𝑖¡Á𝑗);  1 (𝑖=𝑗)};  If 𝐚 ∙ 𝐛 =  𝑎𝑖𝑏𝑗 𝐠𝑖 ∙ 𝐠j, then 𝐚 ∙ 𝐛 = 𝑎𝑖𝑏𝑗 𝛿𝑖𝑗 =  𝑎𝑖𝑏i  <- index ±³È¯ (j¡æi of 𝑏)   [12.1.11,12]
       ∘ Reciprocal(¿ª) metric tensor:  𝑔𝑖𝑗 ¡Õ 𝐠𝑖 ∙ 𝐠𝑗;  If 𝐚 ∙ 𝐛 = 𝑎𝑖𝑏𝑗 𝐠𝑖 ∙ 𝐠𝑗, then 𝐚 ∙ 𝐛 = 𝑎𝑖𝑏j 𝑔𝑖𝑗   [12.1.13,14]
       ∘ Identity(´ÜÀ§) tensor:  𝐈 = 𝛿𝑖𝑗 𝐠𝑖 ⊗ 𝐠𝑗 = 𝐠𝑖 ⊗ 𝐠𝑖,   𝐈 =  𝐈T = 𝛿𝑗𝑖 𝐠𝑖 ⊗ 𝐠𝑗 = 𝐠𝑗 ⊗ 𝐠𝑗  ¡Å  𝐈 = 𝐠𝑖 ⊗ 𝐠𝑖 = 𝐠𝑖 ⊗ 𝐠𝑖   [12.1.15-17]
       ∘ Index ±âÈ£∙À§Ä¡ ¹Ù²Ù±â: ¡Ø metric tensor/reciprocal metric tensor´Â ±ÙÁ¢ÇØ ÀÖ´Â indexÀÇ ±âÈ£¿Í À§Ä¡¸¦ µ¿½Ã¿¡ ¹Ù²Þ. 
          𝑎𝑖𝑏𝑗 𝐠𝑖 ∙ 𝐠𝑗 = 𝑎𝑖𝑏𝑗 𝑔𝑖𝑗 = 𝑎𝑖𝑏i,  𝑎𝑖𝑏𝑗 𝐠𝑖 ∙ 𝐠𝑗 = 𝑎𝑖𝑏𝑗 𝛿𝑗𝑖 = 𝑎𝑖𝑏i,  𝑎𝑖𝑏𝑗 𝐠𝑖 ∙ 𝐠𝑗 = 𝑎𝑖𝑏𝑗 𝛿𝑖𝑗 = 𝑎𝑖𝑏𝑖,  𝑎𝑖𝑏𝑗 𝐠𝑖 ∙ 𝐠𝑗 = 𝑎𝑖𝑏𝑗 𝑔𝑖𝑗 = 𝑎𝑖𝑏𝑖,   𝐚 ∙ 𝐛 = 𝑎𝑖𝑏𝑗 = 𝑎𝑖𝑏𝑗   [12.1.24-28]
     b) 12.2 Inner product(³»Àû) ÀÀ¿ë
       ∘ Basis(±âÀú) vectorÀÇ º¯È¯: 𝐠𝑖 = 𝑔𝑖𝑗 𝐠𝑗,  𝐠𝑖 = 𝑔𝑖𝑗 𝐠𝑗   [12.2.1,2]
       ∘ Metric tensor °ü°è½Ä: [𝑔𝑖𝑗] = [𝑔𝑖𝑗]-1  ¡ñ 𝑔𝑖𝑗 ∙ 𝑔𝑖𝑗 =  𝐠𝑖 ∙ 𝐠𝑗 ∙ 𝐠𝑖 ∙ 𝐠𝑗 = 𝐠𝑖 ∙ 𝐠𝑗 ∙ 𝐠𝑗 ∙ 𝐠𝑖 =  𝛿𝑖𝑗 𝛿𝑗𝑖 = 𝐈   [12.2.6-8]     
Trace(´ë°¢ÇÕ): tr𝐀 = 𝐀 : 𝐈 ¡Õ 𝐴𝑖,𝑖,   tr(𝐀2) = tr(𝐀 ∙ 𝐀) = 𝐀 : 𝐀 = 𝐴𝑖,𝑗𝐴j,𝑖,  (tr𝐀)2 = tr(𝐀) tr(𝐀) =  𝐴𝑖,𝑖 𝐴𝑗,𝑗   [12.2.12-14]
       ∘ VectorÀÇ Å©±â:  𝐧 = 𝑛𝑖 𝐠𝑖ÀÇ Å©±â ¡æ ¡«𝐧¡«= ¡î (𝐧 ∙ 𝐧) = ¡î (𝑛𝑖 𝐠𝑖 ∙ 𝑛𝑗 𝐠𝑗) = ¡î (𝑛𝑖𝑛𝑗𝑔𝑖𝑗) = ¡î (𝑛i𝑛𝑗),  ¡Å ¡«𝐧¡«= ¡î (𝐧 ∙ 𝐧) = ¡î (𝑛i𝑛i)   [12.2.16-18]
       ∘ Basis(±âÀú) vectorÀÇ Å©±â: ¡«𝐠𝑖¡«= ¡î (𝐠𝑖̄ ∙ 𝐠𝑖̄) = ¡î 𝑔𝑖̄𝑖̄, ¡«𝐠𝑖¡«= ¡î (𝐠𝑖̄ ∙ 𝐠𝑖̄) = ¡î 𝑔𝑖̄𝑖̄  <- ¡Ø 𝑖̄ : Áߺ¹Áö¼ö(dummy index) ¹ÌÀû¿ë Ç¥±âÀÓ.   [12.2.19-24]
          ex) ¿øÅë ÁÂÇ¥°èÀÇ ÀÚ¿¬ ±âÀú vector Å©±â¡«𝐠𝑖¡«¸¦ ±¸ÇϽÿÀ. ◂  
               ¡«𝐠1¡«= ¡î (𝐠1 ∙ 𝐠1) = ¡î 𝑔11, ¡«𝐠2¡«= ¡î (𝐠2 ∙ 𝐠2) = ¡î 𝑔22, ¡«𝐠3¡«= ¡î (𝐠3 ∙ 𝐠3) = ¡î 𝑔33  ▮   [12.2.25]
     c) 12.3 Tensor component(¼ººÐ) º¯È¯
       ∘ Tensor ¼ººÐ ±¸Çϱâ: ¿øÇÏ´Â ¹æÇâÀÇ basis vector¸¦ tensor¿¡ dot product¸¦ ÇÔ. <- 1Â÷ tensor: 1ȸ, 2Â÷ tensor: 2ȸ          
          ex1) 𝐮 ∙ 𝐠𝑗 = 𝑢𝑖 𝐠𝑖 ∙ 𝐠𝑗 = 𝑢𝑖𝛿𝑖𝑗 = 𝑢𝑗,  ex2)  𝐠𝑘 ∙ 𝐀 ∙ 𝐠𝑙 = 𝐠𝑘 ∙ (𝐴𝑖𝑗 𝐠𝑖 ⊗ 𝐠𝑗) ∙ 𝐠𝑙 = 𝐴𝑖𝑗 (𝐠𝑘 ∙ 𝐠𝑖)(𝐠𝑗 ∙ 𝐠𝑙) = 𝐴𝑖𝑗 𝑔𝑘𝑖 𝑔𝑗𝑙 = 𝐴𝑘𝑙   [12.3.1,3]
       ∘ Tensor ±¸¼ºÇϱâ: tensorÀÇ ¼ººÐ¾Æ ¾Ë·ÁÁ® ÀÖÀ» ¶§´Â basis vector¸¦ °öÇÏ¿© tensor¸¦ ȸº¹ÇÒ ¼ö ÀÖ½À´Ï´Ù.   
          ex1) 𝑢𝑗 (= 𝐮 ∙ 𝐠𝑗)¸¦ ¾Ë¸é,  𝑢j 𝐠𝑗 = 𝐮;  ex2) 𝑢𝑗 (= 𝐮 ∙ 𝐠𝑗)¸¦ ¾Ë¸é, 𝑢𝑗 𝐠𝑗 = 𝐮   [12.3.15,16]. 
       ∘ 12.4 Tensor ¼ººÐ º¯È¯ ÀýÂ÷: 𝐚 = 𝑢𝑖 𝐠𝑖  <- contavariant ¼ººÐ°ú ÀÚ¿¬ ±âÀú vector°¡ ÁÖ¾îÁú °æ¿ì  
          1) metric tensor ±¸Çϱâ:  ; 𝑔𝑖𝑗 = 𝐠𝑖 ∙ 𝐠𝑗
             ex) ¿øÅë ÁÂÇ¥°è (𝑟, 𝜃, 𝑧)¿¡¼­ x = 𝑟 cos 𝜃, y = 𝑟 sin 𝜃, z = 𝑧 ÀÏ ¶§ ÇØ´ç metric tensor 𝑔𝑖𝑗¸¦ ±¸ÇϽÿÀ.  ◂
                  ⌈ 𝐠1 ⌉       ⌈  cos𝜃       sin𝜃       0   ⌉     ⌈ 𝐞1 ⌉    
                 ¦­𝐠2¦­ =   ¦­ -𝑟 sin 𝜃   𝑟 cos𝜃   0  ¦­    ¦­𝐞2¦­  [12.1.8]
                  ⌊ 𝐠3 ⌋       ⌊    0            0          1   ⌋     ⌊ 𝐞3 ⌋  
                ¡Å  𝐠1 = cos𝜃 𝐞1 + sin𝜃 𝐞2,  𝐠2 = -𝑟 sin𝜃 𝐞1 + 𝑟 cos𝜃 𝐞2,  𝐠3 = 𝐞3 
                𝑔11 =  𝐠1 ∙ 𝐠1 =  (cos𝜃 𝐞1 + sin𝜃 𝐞2) ∙ (cos𝜃 𝐞1 + sin𝜃 𝐞2) = 1,  °°Àº ¹æ½ÄÀ¸·Î °è»êÀ» °è¼ÓÇÑ °á°ú´Â,
               metric tensor [𝑔𝑖𝑗] =
                                     ⌈ 1    0    0 ⌉                  
                                    ¦­ 0    𝑟  0¦­  ▮   [12.1.9]
                                     ⌊ 0    0    1 ⌋  
          2) reciproca(¿ª) metric tensor ±¸Çϱâ: 𝑔𝑖𝑗  <- [𝑔𝑖𝑗] = [𝑔𝑖𝑗]-1 °ü°è½ÄÀ¸·Î ºÎÅÍ
             ex) ¿øÅë ÁÂÇ¥°è (𝑟, 𝜃, 𝑧)ÀÇ °æ¿ì¿¡ ¿ª metric tensor ±¸ÇϽÿÀ.  ◂  
               reciprocal metric tensor [𝑔𝑖𝑗] =
                                                      ⌈ 1    0    0 ⌉ -1      ⌈ 1     0     0 ⌉              
                                                     ¦­0    𝑟2   0¦­    =   ¦­0   1/𝑟2   0¦­   ▮   [12.2.9]
                                                      ⌊ 0    0    1 ⌋          ⌊ 0     0     1 ⌋    
          3) reciproca(¿ª) ±âÀú vector ±¸Çϱâ: 𝐠𝑖 = 𝑔𝑖𝑗 𝐠𝑗
             ex) ¿øÅë ÁÂÇ¥°è (𝑟, 𝜃, 𝑧)ÀÇ °æ¿ì¿¡ ¿ª±âÀú vector 𝐠𝑖¸¦ À¯µµÇϽÿÀ.  ◂
                 𝐠1 = 𝑔11𝐠1 + 𝑔12𝐠2 + 𝑔13𝐠3 = 𝐠1 + 0 + 0 = 𝐠1  ¡Å  𝐠1 = cos𝜃 𝐞1 + sin𝜃 𝐞2
                 𝐠2 = 𝑔21𝐠1 + 𝑔22𝐠2 + 𝑔23𝐠3 = 0 + (1/𝑟2 ) 𝐠2 + 0 + 0 = (1/𝑟2 ) 𝐠2  ¡Å  𝐠2 = -(1/𝑟) sin𝜃 𝐞1 + (1/𝑟) cos𝜃 𝐞2
                 𝐠3 = 𝑔31𝐠1 + 𝑔32𝐠2 + 𝑔33𝐠3 = 0 + 0 + 𝐠3 = 𝐠3  ¡Å  𝐠3 = 𝐞3  ▮   [12.2.4,5]        
          4) º¯È¯µÈ ¼ººÐ ±¸Çϱâ: 𝑢𝑖 = 𝑔𝑖𝑗 𝑢𝑗
          5) º¯È¯µÈ tensor ±¸¼ºÇÔ: 𝐮 =  𝑢𝑖 𝐠𝑖  <- covariant ¡æ contravariant ÀÇ °æ¿ì¿¡µµ À¯»ç ¹æ½ÄÀ¸·Î ÁøÇàµË´Ï´Ù.
     d) 12.5 Outer product(¿ÜÀû)
       ∘ 𝐠𝑖 ⨯ 𝐠𝑗 ¡Õ 𝝐𝑖𝑗𝑘 𝐠𝑘,   𝐚 ⨯ 𝐛 = 𝑎𝑖𝑏𝑗 (𝐠𝑖 ⨯ 𝐠𝑗) = 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗 𝐠𝑘   <- 𝝐𝑖𝑗𝑘: ÀÏ¹Ý ÁÂÇ¥°èÀÇ ¼øÈ¯ ±âÈ£   [12.5.2,3]
       ∘ 𝐠𝑖 ⨯ 𝐠 ¡Õ 𝝐𝑖𝑗𝑘 𝐠𝑘,   𝐚 ⨯ 𝐛 = 𝑎𝑖𝑏𝑗 (𝐠𝑖 ⨯ 𝐠) = 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗 𝐠k  ¡Å ÀÏ¹Ý ÁÂÇ¥°èÀÇ vectorÀÇ ¿ÜÀû: 𝐚 ⨯ 𝐛 = 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗 𝐠k or 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗 𝐠k   [12.5.5,6]     
     e) 12.6 Triple inner product(»ïÁß ³»Àû)  
         (𝐚 ⨯ 𝐛) ∙ 𝐜 = 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗 𝐠𝑘 ∙ 𝑐𝑙 𝐠𝑙 = 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗𝑐𝑙 𝐠𝑘 ∙ 𝐠𝑙 = 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗𝑐𝑙 𝛿𝑘𝑙 = 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗𝑐𝑘  ¡Å (𝐚 ⨯ 𝐛) ∙ 𝐜 = 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗𝑐𝑘 or 𝝐𝑖𝑗𝑘 𝑎𝑖𝑏𝑗𝑐𝑘   [12.6.1-3]
       ∘ Permutation symbol(¼øÈ¯ ±âÈ£) Á¤ÀÇ:  𝝐𝑖𝑗𝑘 ¡Õ (𝐠𝑖 ⨯ 𝐠𝑗) ∙ 𝐠𝑘  <- 𝐄 = 𝝐𝑖𝑗𝑘 𝐠𝑖 ⊗ 𝐠𝑗 ⊗ 𝐠𝑘ÀÇ covariant ¼ººÐ   [12.6.5]
           𝝐𝑖𝑗𝑘 =
               ⌈   𝑉𝑔    (123, 231, 312)
              ¦­ -𝑉𝑔    (321, 132, 213)  <-  𝑉𝑔: 3°³ ±âÀú vectorµé·Î ÀÌ·ç¾îÁø ÆòÇàÀ°¸éüÀÇ Ã¼Àû   [12.6.8]
               ⌊   0      (223, 331 ±âŸ)
       ∘ Reciprocal basis(¿ª±âÀú) vector   <- ±×¸² 12.1 ÂüÁ¶
          𝐠1 ∙ 𝐠1 = 1,  𝐠1 ∙ 𝐠2 = 0,  𝐠1 ∙ 𝐠3 = 0,  (<- Kronecker delta)  ¡Å 𝐠1 = 𝜆(𝐠2 ⨯ 𝐠3),  𝐠1 ∙ 𝐠1 = 𝜆(𝐠2 ⨯ 𝐠3) ∙ 𝐠1 = 𝜆𝑉𝑔 = 1,  𝜆 = 1/𝑉𝑔.  
         ¡Å  𝐠1 = (𝐠2 ⨯ 𝐠3)/𝑉𝑔,   𝐠2 = (𝐠3 ⨯ 𝐠1)/𝑉𝑔,   𝐠3 = (𝐠1 ⨯ 𝐠2)/𝑉𝑔   [12.6.15-17]
       ∘ Reciprocal permutation symbol(¿ª¼øÈ¯ ±âÈ£): 𝝐𝑖𝑗𝑘 ¡Õ (𝐠𝑖 ⨯ 𝐠𝑗) ∙ 𝐠𝑘  <- 𝐅 = 𝝐𝑖𝑗𝑘 𝐠𝑖 ⊗ 𝐠𝑗 ⊗ 𝐠𝑘ÀÇ contravariant ¼ººÐ   [12.6.26]
           𝝐𝑖𝑗𝑘 =
               ⌈   1/𝑉𝑔   (123, 231, 312)
              ¦­-1/𝑉𝑔   (321, 132, 213)  <-  𝑉 𝑔: 3°³ ±âÀú vectorµé·Î ÀÌ·ç¾îÁø ÆòÇàÀ°¸éüÀÇ Ã¼Àû   [12.6.30]
               ⌊   0       (223, 331 ±âŸ)
     f) 12.7 ¼öÇÐÀû À¯µµ °úÁ¤
       ∘ Permutation symbol(¼øÈ¯ ±âÈ£) °ü°è½Ä:  𝝐𝑖𝑗𝑘 =  𝑉𝑔 𝑒𝑖𝑗𝑘  <- Cartesian ÁÂÇ¥°èÀÇ ¼øÈ¯ ±âÈ£ÀÇ »ç¿ë °¡´É   [12.7.1] 
       ∘ Triple inner product(»ïÁß ³»Àû) °ªÀÇ À¯µµ:  (𝐠𝑖 ⨯ 𝐠𝑗 ∙ 𝐠𝑘)(𝐠p ⨯ 𝐠q ∙ 𝐠r) = 𝝐𝑖𝑗𝑘 𝝐𝑝𝑞𝑟  <- ¡Ø [6.5.25] ÂüÁ¶   [12.7.2]
           𝝐𝑖𝑗𝑘 𝝐𝑝𝑞𝑟 =
              ∣ 𝐠𝑖 ∙ 𝐠𝑝    𝐠𝑖 ∙ 𝐠𝑞    𝐠𝑖 ∙ 𝐠𝑟 ∣      ∣ 𝑔𝑖𝑝  𝑔𝑖𝑞  𝑔𝑖𝑟
              ∣ 𝐠𝑗 ∙ 𝐠𝑞    𝐠𝑗 ∙ 𝐠𝑞    𝐠j ∙ 𝐠r ∣  =  ∣ 𝑔𝑗𝑝  𝑔𝑗𝑞  𝑔𝑗𝑟 ∣  =  det [𝐠𝑖𝑗],   𝑔 ¡Õ det [𝐠𝑖𝑗]   [12.7.4,5]
              ∣ 𝐠𝑘 ∙ 𝐠𝑝   𝐠𝑘 ∙ 𝐠𝑞    𝐠𝑘 ∙ 𝐠𝑟 ∣     ∣ 𝑔𝑘𝑝  𝑔𝑘𝑞 𝑔𝑘𝑟
         basis vector  (𝝐123)2 = det [𝐠] (𝑖,𝑗 = 1,2,3) = 𝑔,  𝝐123 = ¡î 𝑔 =  𝑉𝑔  ¡Å  𝝐𝑖𝑗𝑘 =  𝑉𝑔 𝑒𝑖𝑗𝑘 = ¡î 𝑔 𝑒𝑖𝑗𝑘   [12.7.6-8]
       ∘ Reciprocal permutation symbol(¿ª¼øÈ¯ ±âÈ£):  𝝐𝑖𝑗𝑘 =  𝐠𝑖 ⨯ 𝐠𝑗 ∙ 𝐠𝑘,  𝝐𝑖𝑗𝑘 =  (1/¡î 𝑔) 𝑒𝑖𝑗𝑘  <- ¿ª±âÀú vector ÂüÁ¶   [12.7.13-19]
     g) 12.8 Eigenvalue(°íÀ¯°ª) °ü·Ã
          (𝐀 - ¥ë𝐈) ∙ 𝐱 = 0     ¥ë: eigenvalue(°íÀ¯°ª),   𝐱: eigenvector(°íÀ¯ vector) ,  𝐈: identity vector(´ÜÀ§ vector)   [12.8.1]
          det (𝐀 - ¥ë𝐈) = 0     [12.8.2]
          ¥ë3 - 𝐼 ¥ë2 +  𝐼𝐼 ¥ë2 - 𝐼𝐼𝐼 = 0    <-  𝐼, 𝐼𝐼, 𝐼𝐼𝐼: invariant scalars   [12.8.3]
          𝐼 = tr𝐀 = 𝐴𝑖.𝑖    𝐼𝐼 = 1/2{(tr𝐀 )2 - (tr(𝐀 2)} = 1/2 {(𝐴𝑖.𝑖 𝐴𝑗.𝑗 - 𝐴𝑖.𝑗 𝐴𝑖.𝑗)}    𝐼𝐼𝐼 = det 𝐀 = 𝑒𝑖𝑗𝑘 𝐴𝑖.1 𝐴𝑗.2 𝐴𝑘.3   [12.8.4-6]
      


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°øÁö  'Çö´ë ¿ìÁÖ·Ð'¿¡ °üÇÑ Å½±¸ÀÇ Àå    °ü¸®ÀÚ 1 2017-08-15
11:36:55
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°øÁö  À§Å°¹é°ú ¾÷µ¥ÀÌÆ®: º¼Ã÷¸¸ ¹æÁ¤½Ä, ¿£Æ®·ÎÇÇ   ✅   [1]  ±è°ü¼® 1 2021-09-28
06:56:21
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161  Palmer's The Primacy of Doubt <Ä«¿À½º ¿¡ºê¸®¿þ¾î>  ✍🏻    ±è°ü¼® 2 2025-05-11
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160    Gleick's CHAOS <Ä«¿À½º: »õ·Î¿î °úÇÐÀÇ ÃâÇö>  ✍🏻    ±è°ü¼® 2 2025-05-11
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159  Supplement  Chapter 2e. Problems    ±è°ü¼® 3 2025-03-24
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158    Supplement  Chapter 3e. Problems    ±è°ü¼® 3 2025-03-24
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157      Supplement  Chapter 4c. Problems    [1]  ±è°ü¼® 3 2025-03-24
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156  Appendix  Aa. Elements of GR    ±è°ü¼® 2 2025-01-28
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155    Appendix  Ab. Einstein Equation    ±è°ü¼® 2 2025-01-28
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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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131    Cosmology  2a. Expanding Universe    ±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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