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

 

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|+ style="height:30px" | <math>\text{Table 47.} ~~ \text{Computation of}~ \mathrm{r}J\!</math>

|+ Table 47. Computation of r''J''

|

|

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

<math>\begin{array}{*{5}{l}}

| width="6%"  | r''J''

\mathrm{r}J & = & \mathrm{D}J & + & \mathrm{d}J

| width="5%"  | =

\end{array}\!</math>

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

| width="3%"  | +

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

|}

|-

|-

|

|

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

<math>\begin{array}{*{9}{l}}

| width="6%"  | D''J''

\mathrm{D}J

| width="25%" | = ''u'' ''v'' ((d''u'')(d''v''))

& = & u \!\cdot\! v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}

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

& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

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

& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

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

& = & u \!\cdot\! v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}

& + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v

& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u

& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot 0

|-

|-

| width="6%"  | d''J''

|

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

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

\mathrm{r}J ~

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

& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~~~~~~

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

& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,

& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,

& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v

\end{array}\!</math>

|}

|}

|-

 

 

 

 

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"

|

|

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

<math>\begin{array}{*{7}{l}}

| width="6%"  | r''J''

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

& = & \boldsymbol\varepsilon J

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

& + & \mathrm{D}J

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

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

& = & \boldsymbol\varepsilon J

& + & \mathrm{d}J

& = & \mathrm{d}^0 J

& + & \mathrm{d}^1 J

& + & \mathrm{d}^2 J

|}

|}

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:

 

which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.

 

Figures&nbsp;48-a through 48-d illustrate the proposition <math>\mathrm{r}J = \mathrm{d}^2 J,\!</math> which forms the remainder map of <math>J\!</math> and also, in this instance, the second order differential of <math>J.\!</math>

:{| cellpadding=2

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

| [[Image:Diff Log Dyn Sys -- Figure 48-a -- Remainder of J.gif|center]]

| =

| <math>\epsilon</math>''J''

| +

|-

|-

| &nbsp;

| height="20px" valign="top" | <math>\text{Figure 48-a.} ~~ \text{Remainder of}~ J ~\text{(Areal)}\!</math>

| =

| <math>\epsilon</math>''J''

|-

| d⁰''J''

| d¹''J''

| d²''J''

|}

|}

Figures&nbsp;48-a through 48-d illustrate the proposition r''J''&nbsp;=&nbsp;d²''J'', which forms the remainder map of ''J'' and also, in this instance, the second order differential of ''J''.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

| [[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]

| height="20px" valign="top" | <math>\text{Figure 48-b.} ~~ \text{Remainder of}~ J ~\text{(Bundle)}\!</math>

<br>

<br>

<p>[[Image:Diff Log Dyn Sys -- Figure 48-b -- Remainder of J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 48-c -- Remainder of J.gif|center]]

<p><center><font size="+1">'''Figure 48-b. Remainder of ''J''&nbsp;&nbsp;(Bundle)'''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 48-c.} ~~ \text{Remainder of}~ J ~\text{(Compact)}\!</math>

<br>

<br>

<p>[[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]</p>

| [[Image:Diff Log Dyn Sys -- Figure 48-d -- Remainder of J.gif|center]]

<p><center><font size="+1">'''Figure 48-d. Remainder of ''J''&nbsp;&nbsp;(Digraph)'''</font></center></p>

| height="20px" valign="top" | <math>\text{Figure 48-d.} ~~ \text{Remainder of}~ J ~\text{(Digraph)}\!</math>

=====Summary of Conjunction=====

=====Summary of Conjunction=====