Changes

Let us now recapitulate the story so far.  In effect, we have been carrying out a decomposition of the enlarged proposition E''J'' in a series of stages.  First, we considered the equation E''J''&nbsp;=&nbsp;<math>\epsilon</math>''J''&nbsp;+&nbsp;D''J'', which was involved in the definition of D''J'' as the difference E''J''&nbsp;&ndash;&nbsp;<math>\epsilon</math>''J''.  Next, we contemplated the equation D''J''&nbsp;=&nbsp;d''J''&nbsp;+&nbsp;r''J'', which expresses D''J'' in terms of two components, the differential d''J'' that was just extracted and the residual component r''J''&nbsp;=&nbsp;D''J''&nbsp;&ndash;&nbsp;d''J''.  This remaining proposition r''J'' can be computed as shown in Table&nbsp;47.

Let us now recapitulate the story so far.  In effect, we have been carrying out a decomposition of the enlarged proposition E''J'' in a series of stages.  First, we considered the equation E''J''&nbsp;=&nbsp;<math>\epsilon</math>''J''&nbsp;+&nbsp;D''J'', which was involved in the definition of D''J'' as the difference E''J''&nbsp;&ndash;&nbsp;<math>\epsilon</math>''J''.  Next, we contemplated the equation D''J''&nbsp;=&nbsp;d''J''&nbsp;+&nbsp;r''J'', which expresses D''J'' in terms of two components, the differential d''J'' that was just extracted and the residual component r''J''&nbsp;=&nbsp;D''J''&nbsp;&ndash;&nbsp;d''J''.  This remaining proposition r''J'' can be computed as shown in Table&nbsp;47.

<pre>

<font face="courier new">

Table 47.  Computation of rJ

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

|+ Table 47.  Computation of r''J''

|                                                                               |

| rJ =       DJ        +       dJ                                            |

{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

|                                                                               |

| width="6%"  | r''J''

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

| width="5%" | =

|                                                                               |

| DJ  = u v ((du)(dv)) +   u (v)(du) dv  + (u) v du (dv) + (u)(v) du dv  |

| width="3%"  | +

|                                                                               |

| align="center" width="20%" | d''J''

| dJ  = u v (du, dv)   +   u (v) dv      + (u) v du      + (u)(v) . 0   |

| width="46%" | &nbsp;

|                                                                               |

|}

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

|-

|                                                                               |

| rJ  = 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''))

</pre>

| width="23%" | + ''u'' (''v'')(d''u'') d''v''

| width="23%" | + (''u'') ''v'' d''u'' (d''v'')

| width="23%" | + (''u'')(''v'') d''u'' d''v''

|-

| width="6%"  | d''J''

| width="25%" | = ''u'' ''v''&nbsp;&nbsp;(d''u'', d''v'')

| width="23%" | + ''u'' (''v'') d''v''

| width="23%" | + (''u'') ''v'' d''u''

| width="23%" | + (''u'')(''v'') <math>\cdot</math> 0

|}

|-

|

{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"

| width="6%"  | r''J''

| width="25%" | = ''u'' ''v''&nbsp;&nbsp;&nbsp;d''u'' d''v''

| width="23%" | + ''u'' (''v'') d''u'' d''v''

| width="23%" | + (''u'') ''v'' d''u'' d''v''

| width="23%" | + (''u'')(''v'') d''u'' d''v''

|}

|}

</font><br>

As it happens, the remainder r''J'' falls under the description of a second order differential r''J''&nbsp;=&nbsp;d²''J''.  This means that the expansion of E''J'' in the form:

As it happens, the remainder r''J'' falls under the description of a second order differential r''J''&nbsp;=&nbsp;d²''J''.  This means that the expansion of E''J'' in the form: