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ONT Re: Differential Logic


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DLOG.  Note D68

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Operator Maps:  Areal Views

                              o
                             /X\
                            /XXX\
                           oXXXXXo
                          /X\XXX/X\
                         /XXX\X/XXX\
                        oXXXXXoXXXXXo
                       / \XXX/X\XXX/ \
                      /   \X/XXX\X/   \
                     o     oXXXXXo     o
                    / \   / \XXX/ \   / \
                   /   \ /   \X/   \ /   \
                  o     o     o     o     o
                 =|\   / \   / \   / \   /|=
                = | \ /   \ /   \ /   \ / | =
               =  |  o     o     o     o  |  =
              =   |  |\   / \   / \   /|  |   =
             =    |u | \ /   \ /   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       /\\
          ////\      | du \ /   \ / dv |      /\\\\
         o/////o     o-----o     o-----o     o\\\\\o
        //\/////\           \   /           /\\\\\/\\
       ////\/////\           \ /           /\\\\\/\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         / \\\\/\\\\\/ \
    /   \/////\//   \       =   =       /   \\/\\\\\/   \
   o     o/////o     o     =     =     o     o\\\\\o     o
  / \   / \//// \   / \   =       =   / \   / \\\\/ \   / \
 /   \ /   \//   \ /   \ =         = /   \ /   \\/   \ /   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   / \   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /   \ /   \ / |
|  o     o     o     o  |           |  o     o     o     o  |
|  |\   / \   / \   /|  |           |  |\   / \   / \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \ /   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\          !h!J
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       /
      x = uv               \     /             dx = uv
                            \   /
                             \ /
                              o
Figure 56-a1.  Radius Map of the Conjunction J = uv

                              o
                             /X\
                            /XXX\
                           oXXXXXo
                          //\XXX//\
                         ////\X////\
                        o/////o/////o
                       /\\/////\////\\
                      /\\\\/////\//\\\\
                     o\\\\\o/////o\\\\\o
                    / \\\\/ \//// \\\\/ \
                   /   \\/   \//   \\/   \
                  o     o     o     o     o
                 =|\   / \   /\\   / \   /|=
                = | \ /   \ /\\\\ /   \ / | =
               =  |  o     o\\\\\o     o  |  =
              =   |  |\   / \\\\/ \   /|  |   =
             =    |u | \ /   \\/   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       /\\
          ////\      | du \ /   \ / dv |      /\\\\
         o/////o     o-----o     o-----o     o\\\\\o
        //\/////\           \   /           / \\\\/ \
       ////\/////\           \ /           /   \\/   \
      o/////o/////o           o           o     o     o
     / \/////\//// \         = =         /\\   / \   /\\
    /   \/////\//   \       =   =       /\\\\ /   \ /\\\\
   o     o/////o     o     =     =     o\\\\\o     o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/ \   / \\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/   \ /   \\/   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   /\\   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /\\\\ /   \ / |
|  o     o     o     o  |           |  o     o\\\\\o     o  |
|  |\   / \   / \   /|  |           |  |\   / \\\\/ \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \\/   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           EJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |. 
                  o-----o           o-----o 
                         \         /
                          \       / dx = (u, du)(v, dv)
      x = uv               \     /
                            \   /   dx = uv + u dv + v du + du dv
                             \ /
                              o
Figure 56-a2.  Secant Map of the Conjunction J = uv

                              o
                             //\
                            ////\
                           o/////o
                          /X\////X\
                         /XXX\//XXX\
                        oXXXXXoXXXXXo
                       /\\XXX/X\XXX/\\
                      /\\\\X/XXX\X/\\\\
                     o\\\\\oXXXXXo\\\\\o
                    / \\\\/ \XXX/ \\\\/ \
                   /   \\/   \X/   \\/   \
                  o     o     o     o     o
                 =|\   / \   /\\   / \   /|=
                = | \ /   \ /\\\\ /   \ / | =
               =  |  o     o\\\\\o     o  |  =
              =   |  |\   / \\\\/ \   /|  |   =
             =    |u | \ /   \\/   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       / \
          ////\      | du \ /   \ / dv |      /   \
         o/////o     o-----o     o-----o     o     o
        //\/////\           \   /           /\\   /\\
       ////\/////\           \ /           /\\\\ /\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         /\\\\\/\\\\\/\\
    /   \/////\//   \       =   =       /\\\\\/\\\\\/\\\\
   o     o/////o     o     =     =     o\\\\\o\\\\\o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/ \\\\/ \\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/   \\/   \\/   \
o     o     o     o     o           o     o     o     o     o
|\   / \   / \   / \   /|           |\   / \   /\\   / \   /|
| \ /   \ /   \ /   \ / |           | \ /   \ /\\\\ /   \ / |
|  o     o     o     o  |           |  o     o\\\\\o     o  |
|  |\   / \   / \   /|  |           |  |\   / \\\\/ \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \\/   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           DJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       / dx = (u, du)(v, dv) - uv
      x = uv               \     /
                            \   /   dx = u dv + v du + du dv
                             \ /
                              o
Figure 56-a3.  Chord Map of the Conjunction J = uv

                              o
                             //\
                            ////\
                           o/////o
                          /X\////X\
                         /XXX\//XXX\
                        oXXXXXoXXXXXo
                       /\\XXX//\XXX/\\
                      /\\\\X////\X/\\\\
                     o\\\\\o/////o\\\\\o
                    / \\\\/\\////\\\\\/ \
                   /   \\/\\\\//\\\\\/   \
                  o     o\\\\\o\\\\\o     o
                 =|\   / \\\\/ \\\\/ \   /|=
                = | \ /   \\/   \\/   \ / | =
               =  |  o     o     o     o  |  =
              =   |  |\   / \   / \   /|  |   =
             =    |u | \ /   \ /   \ / | v|    =
            o     o--+--o     o     o--+--o     o
           //\       |   \   / \   /   |       / \
          ////\      | du \ /   \ / dv |      /   \
         o/////o     o-----o     o-----o     o     o
        //\/////\           \   /           /\\   /\\
       ////\/////\           \ /           /\\\\ /\\\\
      o/////o/////o           o           o\\\\\o\\\\\o
     / \/////\//// \         = =         /\\\\\/ \\\\/\\
    /   \/////\//   \       =   =       /\\\\\/   \\/\\\\
   o     o/////o     o     =     =     o\\\\\o     o\\\\\o
  / \   / \//// \   / \   =       =   / \\\\/\\   /\\\\\/ \
 /   \ /   \//   \ /   \ =         = /   \\/\\\\ /\\\\\/   \
o     o     o     o     o           o     o\\\\\o\\\\\o     o
|\   / \   / \   / \   /|           |\   / \\\\/ \\\\/ \   /|
| \ /   \ /   \ /   \ / |           | \ /   \\/   \\/   \ / |
|  o     o     o     o  |           |  o     o     o     o  |
|  |\   / \   / \   /|  |           |  |\   / \   / \   /|  |
|u | \ /   \ /   \ / | v|           |u | \ /   \ /   \ / | v|
o--+--o     o     o--+--o     o     o--+--o     o     o--+--o
 . |   \   / \   /   |       /X\       |   \   / \   /   | .
  .| du \ /   \ / dv |      /XXX\      | du \ /   \ / dv |.
   o-----o     o-----o     /XXXXX\     o-----o     o-----o
    .     \   /           /XXXXXXX\           \   /     .
     .     \ /           /XXXXXXXXX\           \ /     .
      .     o           oXXXXXXXXXXXo           o     .
       .               //\XXXXXXXXX/\\               .
        .             ////\XXXXXXX/\\\\             .
      !e!J           //////\XXXXX/\\\\\\           dJ
          .         ////////\XXX/\\\\\\\\         .
           .       //////////\X/\\\\\\\\\\       .
            .     o///////////o\\\\\\\\\\\o     .
             .    |\////////// \\\\\\\\\\/|    .
              .   | \////////   \\\\\\\\/ |   .
               .  |  \//////     \\\\\\/  |  .
                . |   \////       \\\\/   | .
                 .| x  \//         \\/ dx |.
                  o-----o           o-----o
                         \         /
                          \       /
      x = uv               \     /  dx = u dv + v du
                            \   /
                             \ /
                              o
Figure 56-a4.  Tangent Map of the Conjunction J = uv

Jon Awbrey

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