Changes between Version 6 and Version 7 of u/johannjc/scratchpad5


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Timestamp:
10/01/15 11:16:31 (9 years ago)
Author:
Jonathan
Comment:

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  • u/johannjc/scratchpad5

    v6 v7  
    7474$A \partial_i  B_{ij} \left [ C_\lambda  \partial_j T^{\lambda + 1} + D_\lambda  \partial_j T^{\lambda} T' \right ]$
    7575
    76 where $ C_\lambda= \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )$ and $ D_\lambda = \phi \left ( \lambda + 1 \right )$
     76where $ C= \left ( 1 - \phi \left ( \lambda + 1 \right ) \right )$ and $ D = \phi \left ( \lambda + 1 \right )$
    7777
    7878Now we can expand the derivatives and get
    … …  
    8888We can also write this as
    8989
    90 $\partial_t T = \displaystyle \sum_{\parallel,\perp}{E \partial_j T^{\lambda + 1} + F \partial_j T^{\lambda} T' + G  \partial_i \partial_j T^{\lambda + 1} + H \partial_i \partial_j T^{\lambda} T'} $
     90$\partial_t T = \displaystyle \sum_{\parallel,\perp}{CE \partial_j T^{\lambda + 1} + DE \partial_j T^{\lambda} T' + CF  \partial_i \partial_j T^{\lambda + 1} + DF \partial_i \partial_j T^{\lambda} T'} $
    9191
    9292where
    9393
    94 $E = A \left ( \partial_i  B_{ij} \right ) C$
     94$E = A \left ( \partial_i  B_{ij} \right )$
    9595
    96 $F =  A \left ( \partial_i  B_{ij} \right ) D$
    97 
    98 $G = A B_{ij} C$
    99 
    100 $H = A B_{ij} D$
    101 
     96$F = A B_{ij}$
    10297
    10398
    … …  
    118113
    119114
    120 $-T_0 -  \phi' \left [ \alpha_0 T^{\lambda+1}_0 -  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} - \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
    121 $-T'_0 + \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}}$
     115$T_0 -  \displaystyle \sum_{\parallel, \perp} C \left [ \alpha_0 T^{\lambda+1}_0 -  \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda+1}_{\pm \hat{j}} - \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda+1}_{\pm \hat{i} \pm \hat{j}} \right ] = $
     116$T'_0 + \displaystyle \sum_{\parallel, \perp} D \left [  \alpha_0 T^{\lambda}_0T'_0 + \displaystyle \sum_{\pm j} \alpha_{\pm j} T^{\lambda}_{\pm \hat{j}} T'_{\pm \hat{j}} + \sum_{\pm i, \pm j,|i| \ne |j|} \alpha_{\pm i, \pm j} T^{\lambda}_{\pm \hat{i} \pm \hat{j}} T'_{\pm \hat{i} \pm \hat{j}} \right ]$
    122117
    123118where