Line 3,276: |
Line 3,276: |
| {| 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> and <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> and <math>\operatorname{d}Y = \{ \texttt{(} \operatorname{d}y \texttt{)}, \operatorname{d}y \}.</math> |
| |} | | |} |
| | | |
− | <pre>
| |
| 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. | |
| | | |
| + | <pre> |
| o---------------------------------------o | | o---------------------------------------o |
| | | | | | | |