Changes

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.

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<font face="courier new">

Table 45.  Computation of dJ

o-------------------------------------------------------------------------------o

|+ 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''

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| width="6%" | &rArr;

|-

| 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

|}

|}

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Figures&nbsp;46-a through 46-d illustrate the proposition d''J'', rounded out in our usual array of prospects.  This proposition of E''U''^&nbsp;&bull; is what we refer to as the (first order) differential of ''J'', and normally regard as ''the'' differential proposition corresponding to ''J''.

Figures&nbsp;46-a through 46-d illustrate the proposition d''J'', rounded out in our usual array of prospects.  This proposition of E''U''^&nbsp;&bull; is what we refer to as the (first order) differential of ''J'', and normally regard as ''the'' differential proposition corresponding to ''J''.