Changes

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{| align="center" cellpadding="10" width="90%"

{| align="center" cellpadding="10"

| align="center" |

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<pre>

<math>\begin{array}{rcccccc}

& = & x  & \cdot & y

|  f    =     x  y                     |

| Df    =     x  y  · ((dx)(dy))       |

& = &  x & \cdot & y  & \cdot & ((\operatorname{d}x)(\operatorname{d}y))

|      +     x (y) · (dx) dy          |

& + &  x & \cdot & (y) & \cdot & (\operatorname{d}x)~\operatorname{d}y

|      +   (x) y  ·   dx (dy)         |

& + & (x) & \cdot &  y  & \cdot &   \operatorname{d}x~(\operatorname{d}y)

|      +   (x)(y) ·   dx  dy          |

& + & (x) & \cdot & (y) & \cdot &   \operatorname{d}x~~\operatorname{d}y

\end{array}</math>

</pre>

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Any proposition worth its salt, as they say, has many equivalent ways to look at it, any of which may reveal some unsuspected aspect of its meaning.  We will encounter more and more of these alternative readings as we go.

Any proposition worth its salt affords many equivalent ways to look at it, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning.  We will encounter more and more of these alternative readings as we go.

==Note 5==

==Note 5==