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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)
Revision as of 18:05, 5 June 2007
, 18:05, 5 June 2007→Differential of Conjunction: wiki table
Awaiting that determination, I proceed with what seems like the obvious course, and compute d''J'' according to the pattern in Table 45.
Awaiting that determination, I proceed with what seems like the obvious course, and compute d''J'' according to the pattern in Table 45.
<pre>
<font face="courier new">
Table 45. Computation of dJ
{| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
|+ Table 45. Computation of d''J''
| |
|
| DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| width="6%" | D''J''
| => |
| width="25%" | = ''u'' ''v'' ((d''u'')(d''v''))
| |
| width="23%" | + ''u'' (''v'')(d''u'') d''v''
| dj = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 |
| width="23%" | + (''u'') ''v'' d''u'' (d''v'')
| |
| width="23%" | + (''u'')(''v'') d''u'' d''v''
|-
</pre>
| width="6%" | ⇒
|-
| width="6%" | d''J''
| width="25%" | = ''u'' ''v'' (d''u'', d''v'')
| width="23%" | + ''u'' (''v'') d''v''
| width="23%" | + (''u'') ''v'' d''u''
| width="23%" | + (''u'')(''v'') <math>\cdot</math> 0
|}
|}
</font><br>
Figures 46-a through 46-d illustrate the proposition d''J'', rounded out in our usual array of prospects. This proposition of E''U''<sup> •</sup> is what we refer to as the (first order) differential of ''J'', and normally regard as ''the'' differential proposition corresponding to ''J''.
Figures 46-a through 46-d illustrate the proposition d''J'', rounded out in our usual array of prospects. This proposition of E''U''<sup> •</sup> is what we refer to as the (first order) differential of ''J'', and normally regard as ''the'' differential proposition corresponding to ''J''.