Changes

{| align="center" cellpadding="6" width="90%"

{| align="center" cellpadding="6" width="90%"

| <math>\operatorname{E}U ~=~ U \times \operatorname{d}U ~=~ X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math>

| <math>\operatorname{E}U ~=~ U \times \operatorname{d}U ~=~ X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math>

|-

|}

| with

 

|-

with

 

{| align="center" cellpadding="6" width="90%"

| <math>\operatorname{d}X = \{ \texttt{(} \operatorname{d}x \texttt{)}, \operatorname{d}x \}</math> &nbsp;and&nbsp; <math>\operatorname{d}Y = \{ \texttt{(} \operatorname{d}y \texttt{)}, \operatorname{d}y \}.</math>

| <math>\operatorname{d}X = \{ \texttt{(} \operatorname{d}x \texttt{)}, \operatorname{d}x \}</math> &nbsp;and&nbsp; <math>\operatorname{d}Y = \{ \texttt{(} \operatorname{d}y \texttt{)}, \operatorname{d}y \}.</math>

|}

|}

The interpretations of these new symbols can be diverse, but the easiest

The interpretations of these new symbols can be diverse, but the easiest

for now is just to say that dx means "change x" and dy means "change y".

option for now is just to say that <math>\operatorname{d}x</math> means "change <math>x\!</math>" and <math>\operatorname{d}y</math> means "change <math>y\!</math>". To draw the differential extension <math>\operatorname{E}U</math> of our present universe <math>U = X \times Y</math> as a venn diagram, it would take us four logical dimensions <math>X, Y, \operatorname{d}X, \operatorname{d}Y,</math> but we can project a suggestion of what it's about on the universe <math>X \times Y</math> by drawing arrows that cross designated borders, labeling the arrows as

To draw the differential extension EU of our present universe U = X x Y

<math>\operatorname{d}x</math> when crossing the border between <math>x\!</math> and <math>\texttt{(} x \texttt{)}</math> and as <math>\operatorname{d}y</math> when crossing the border between <math>y\!</math> and <math>\texttt{(} y \texttt{)},</math> in either direction, in either case.

as a venn diagram, it would take us four logical dimensions X, Y, dX, dY,

but we can project a suggestion of what it's about on the universe X x Y

by drawing arrows that cross designated borders, labeling the arrows as

dx when crossing the border between x and (x) and as dy when crossing

the border between y and (y), in either direction, in either case.

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