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DLOG. Note D60
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Remainder of Conjunction (concl.)
Figures 48-a through 48-d illustrate the proposition rJ = d^2.J,
which forms the remainder map of J and also, in this instance,
the second order differential of J.
o---------------------------------------o
| |
| o |
| / \ |
| / \ |
| / \ |
| o o |
| / \ / \ |
| / \ / \ |
| / \ / \ |
| o o o |
| / \ /%\ / \ |
| / \ /%%%\ / \ |
| / \ /%%%%%\ / \ |
| o o%%%%%%%o o |
| / \ /%\%%%%%/%\ / \ |
| / \ /%%%\%%%/%%%\ / \ |
| / \ /%%%%%\%/%%%%%\ / \ |
| o o%%%%%%%o%%%%%%%o o |
| |\ / \%%%%%/%\%%%%%/ \ /| |
| | \ / \%%%/%%%\%%%/ \ / | |
| | \ / \%/%%%%%\%/ \ / | |
| | o o%%%%%%%o o | |
| | |\ / \%%%%%/ \ /| | |
| | | \ / \%%%/ \ / | | |
| | u | \ / \%/ \ / | v | |
| o---+---o o o---+---o |
| | \ / \ / | |
| | \ / \ / | |
| | du \ / \ / dv | |
| o-------o o-------o |
| \ / |
| \ / |
| \ / |
| o |
| |
o---------------------------------------o
Figure 48-a. Remainder of J (Areal)
o-----------------------------o
| |
| o-----o 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 |
| / /`\ @------\-----------@ | du |%%%| dv | |
| / /```\ \ | | o o%%%o o |
| / /`````\ \ | | \ \%/ / |
| / /```````\ \ | | \ o / |
| o o`````````o o | | \ / \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````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 |
@ | du |%%%| dv | |
| o o%%%o o |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-----o o-----o |
| |
o-----------------------------o
Figure 48-b. Remainder of J (Bundle)
o---------------------------------------------------------------------o
| |
| |
| o-------------------o o-------------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | | du dv | | |
| | u @<------------------------->@ v | |
| | | | | |
| | | | | |
| | | | | |
| o o @ o o |
| \ \ ^ / / |
| \ \ | / / |
| \ \ | / / |
| \ \ | / / |
| \ \|/ / |
| \ du | dv / |
| \ /|\ / |
| o-------------------o | o-------------------o |
| | |
| | |
| v |
| @ |
| |
o---------------------------------------------------------------------o
Figure 48-c. Remainder of J (Compact)
o-----------------------------------------------------------o
| |
| u v |
| @ |
| ^ |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| du | dv |
| u (v) @<----------|---------->@ (u) v |
| du | dv |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| v |
| @ |
| (u) (v) |
| |
o-----------------------------------------------------------o
Figure 48-d. Remainder of J (Digraph)
Jon Awbrey
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