Changes

To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from ''U''^ • = [''u'', ''v''] to ''X''^ • = [''x'', ''y''] that is defined by the following system of equations:

To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from ''U''^ • = [''u'', ''v''] to ''X''^ • = [''x'', ''y''] that is defined by the following system of equations:

<pre>

<br><font face="courier new">

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

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

|                                                           |

|         x  =   f(u, v)  =   ((u)(v))                   |

{| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"

|                                                           |

| &nbsp;

|         y   =   g(u, v)  =   ((u, v))                   |

| ''x''

|                                                           |

| =

| ''f''‹''u'', ''v''›

</pre>

| =

| ((''u'')(''v''))

| &nbsp;

|-

| &nbsp;

| ''y''

| =

| ''g''‹''u'', ''v''›

| =

| ((''u'', ''v''))

| &nbsp;

|}

|}

</font><br>

The component notation ''F''&nbsp;=&nbsp;‹''F''₁,&nbsp;''F''₂›&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull; allows us to give a name and a type to this transformation, and permits us to define it by means of the compact description that follows:

The component notation ''F''&nbsp;=&nbsp;‹''F''₁,&nbsp;''F''₂›&nbsp;=&nbsp;‹''f'',&nbsp;''g''›&nbsp;:&nbsp;''U''^&nbsp;&bull;&nbsp;&rarr;&nbsp;''X''^&nbsp;&bull; allows us to give a name and a type to this transformation, and permits us to define it by means of the compact description that follows: