Changes

Alternatively, the difference map D''J'' can be expanded over the cells of [d''u'', d''v''] to arrive at the formulation shown in Table 42.  The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns of the middle portion of the Table.

Alternatively, the difference map D''J'' can be expanded over the cells of [d''u'', d''v''] to arrive at the formulation shown in Table 42.  The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns of the middle portion of the Table.

<pre>

<font face="courier new">

Table 42.  Computation of DJ (Method 2)

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

|+ Table 42.  Computation of D''J'' (Method 2)

|                                                                               |

| DJ =  !e!J           + EJ                                                  |

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

|                                                                               |

| width="8%"  | D''J''

|     = J<u, v>        + J<u + du, v + dv>                                  |

| width="4%" | =

|                                                                               |

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

| width="4%" | +

|                                                                               |

| colspan="5" | E''J''

|     =  0               +  u dv            + v du            +  du dv        |

|                                                                               |

| width="8%" | &nbsp;

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

| width="4%"  | =

|                                                                               |

| width="16%" | ''J''‹''u'', ''v''›

| width="4%"  | +

</pre>

| colspan="5" | ''J''‹''u'' + d''u'', ''v'' + d''v''›

|-

| width="8%"  | &nbsp;

| width="4%" | =

| width="16%" | ''u'' ''v''

| width="4%"  | +

| colspan="5" | (''u'', d''u'')(''v'', d''v'')

|-

| width="8%"  | &nbsp;

| width="4%" | =

| width="16%" | 0

| width="4%"  | +

| width="4%" | +

| width="16%" | ''v'' d''u''

| width="4%"  | +

| width="28%" | d''u'' d''v''

| width="8%"  | D''J''

| width="4%"  | =

| width="16%" | 0

| width="4%"  | +

| width="16%" | ''u'' (d''u'') d''v''

| width="4%"  | +

| width="16%" | ''v'' d''u'' (d''v'')

| width="4%"  | +

| width="28%"  | ((''u'', ''v'')) d''u'' d''v''

|}

|}

</font><br>

Even more simply, the same result is reached by matching up the propositional coefficients of <math>\epsilon</math>''J'' and E''J'' along the cells of [d''u'',&nbsp;d''v''] and adding the pairs under boolean sums (that is, "mod 2", where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0), as shown in Table&nbsp;43.

Even more simply, the same result is reached by matching up the propositional coefficients of <math>\epsilon</math>''J'' and E''J'' along the cells of [d''u'',&nbsp;d''v''] and adding the pairs under boolean sums (that is, "mod 2", where 1&nbsp;+&nbsp;1&nbsp;=&nbsp;0), as shown in Table&nbsp;43.