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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)
Revision as of 17:42, 19 June 2007
, 17:42, 19 June 2007→Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>: wiki table
Figure 70-a shows a graphical way of picturing the tangent functor map d''F'' = ‹d''f'', d''g''› for the transformation ''F'' = ‹''f'', ''g''› = ›((u)(v)), ((u, v))›. This amounts to the same information about d''f'' and d''g'' that was given in the computation summary of Tables 66-i and 66-ii, the relevant rows of which are repeated here:
Figure 70-a shows a graphical way of picturing the tangent functor map d''F'' = ‹d''f'', d''g''› for the transformation ''F'' = ‹''f'', ''g''› = ›((u)(v)), ((u, v))›. This amounts to the same information about d''f'' and d''g'' that was given in the computation summary of Tables 66-i and 66-ii, the relevant rows of which are repeated here:
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<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
|
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) |
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
|
| dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) |
|-
| |
| d''f''
| = || ''uv'' || <math>\cdot</math> || 0
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| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
|
|-
| d''g''
| = || ''uv'' || <math>\cdot</math> || (d''u'', d''v'')
| + || ''u''(''v'') || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'', d''v'')
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
|
|}
|}
</font><br>
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