| Line 4,654: |
Line 4,654: |
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| − | |}
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| − | <br>
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| − |
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| − | ==Detail of Calculation for the Difference Map==
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| − |
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| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
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| − | |+ '''Detail of Calculation for <math>\operatorname{D}f = \operatorname{E}f + f</math>'''
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| − | |- style="background:ghostwhite; height:60px"
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| − | |
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| − | |
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| − | <math>\begin{array}{cr}
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| − | & \operatorname{E}f|_{\operatorname{d}x\ \operatorname{d}y} \\
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| − | + & f|_{\operatorname{d}x\ \operatorname{d}y} \\
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| − | = & \operatorname{D}f|_{\operatorname{d}x\ \operatorname{d}y} \\
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| − | \end{array}</math>
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| − | |
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| − | <math>\begin{array}{cr}
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| − | & \operatorname{E}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
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| − | + & f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
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| − | = & \operatorname{D}f|_{\operatorname{d}x\ (\operatorname{d}y)} \\
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| − | \end{array}</math>
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| − | |
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| − | <math>\begin{array}{cr}
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| − | & \operatorname{E}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
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| − | + & f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
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| − | = & \operatorname{D}f|_{(\operatorname{d}x)\ \operatorname{d}y} \\
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| − | \end{array}</math>
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| − | |
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| − | <math>\begin{array}{cr}
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| − | & \operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
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| − | + & f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
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| − | = & \operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)} \\
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| − | \end{array}</math>
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| − | |- style="height:40px"
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| − | | <math>f_{0}\!</math>
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| − | | <math>0 + 0 = 0\!</math>
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| − | | <math>0 + 0 = 0\!</math>
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| − | | <math>0 + 0 = 0\!</math>
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| − | | <math>0 + 0 = 0\!</math>
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| − | |-
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" | <math>f_{1}\!</math>
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| − | |-
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| − | | height="60px" | <math>f_{2}\!</math>
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| − | |-
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| − | | height="60px" | <math>f_{4}\!</math>
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| − | |-
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| − | | height="60px" | <math>f_{8}\!</math>
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| − | |}
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & y & \operatorname{d}x & \operatorname{d}y \\
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| − | + & (x) & (y) & \operatorname{d}x & \operatorname{d}y \\
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| − | = & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & (y) & \operatorname{d}x & \operatorname{d}y \\
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| − | + & (x) & y & \operatorname{d}x & \operatorname{d}y \\
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| − | = & (x, & y) & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & y & \operatorname{d}x & \operatorname{d}y \\
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| − | + & x & (y) & \operatorname{d}x & \operatorname{d}y \\
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| − | = & (x, & y) & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & (y) & \operatorname{d}x & \operatorname{d}y \\
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| − | + & x & y & \operatorname{d}x & \operatorname{d}y \\
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| − | = & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & (y) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | + & (x) & (y) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & y & \operatorname{d}x & (\operatorname{d}y) \\
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| − | + & (x) & y & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & & y & \operatorname{d}x & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & (y) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | + & x & (y) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & y & \operatorname{d}x & (\operatorname{d}y) \\
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| − | + & x & y & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & & y & \operatorname{d}x & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & y & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & (x) & (y) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & (y) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & (x) & y & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & y & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & x & (y) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & x & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & (y) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & x & y & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & x & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | + & (x) (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | + & (x)\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | + & x\ (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | + & x\ y & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |-
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" | <math>f_{3}\!</math>
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| − | |-
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| − | | height="60px" | <math>f_{12}\!</math>
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| − | |}
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & & \operatorname{d}x & \operatorname{d}y \\
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| − | + & (x) & & \operatorname{d}x & \operatorname{d}y \\
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| − | = & 1 & & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & & \operatorname{d}x & \operatorname{d}y \\
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| − | + & x & & \operatorname{d}x & \operatorname{d}y \\
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| − | = & 1 & & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & & \operatorname{d}x & (\operatorname{d}y) \\
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| − | + & (x) & & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & 1 & & \operatorname{d}x & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & & \operatorname{d}x & (\operatorname{d}y) \\
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| − | + & x & & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & 1 & & \operatorname{d}x & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |
| |
| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & 0 & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & x & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & 0 & & (\operatorname{d}x) & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |
| |
| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x) & & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | + & (x) & & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & 0 & & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & x & & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | + & x & & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & 0 & & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |-
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" | <math>f_{6}\!</math>
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| − | |-
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| − | | height="60px" | <math>f_{9}\!</math>
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| − | |}
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x , y) & \operatorname{d}x & \operatorname{d}y \\
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| − | + & (x , y) & \operatorname{d}x & \operatorname{d}y \\
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| − | = & 0 & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & ((x , y)) & \operatorname{d}x & \operatorname{d}y \\
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| − | + & ((x , y)) & \operatorname{d}x & \operatorname{d}y \\
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| − | = & 0 & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |
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| − | {| align="center"
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & ((x , y)) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | + & (x , y) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & 1 & \operatorname{d}x & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
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| − | <math>\begin{smallmatrix}
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| − | & (x , y) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | + & ((x , y)) & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & 1 & \operatorname{d}x & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
| |
| − | |}
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| − | |
| |
| − | {| align="center"
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| − | |-
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| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & ((x , y)) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & (x , y) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & 1 & (\operatorname{d}x) & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & (x , y) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & ((x , y)) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & 1 & (\operatorname{d}x) & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |
| |
| − | {| align="center"
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| − | |-
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| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & (x , y) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | + & (x , y) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & ((x , y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | + & ((x , y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
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| − | |}
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| − | |-
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| − | |
| |
| − | {| align="center"
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| − | |-
| |
| − | | height="60px" | <math>f_{5}\!</math>
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| − | |-
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| − | | height="60px" | <math>f_{10}\!</math>
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| − | |}
| |
| − | |
| |
| − | {| align="center"
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| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & & y & \operatorname{d}x & \operatorname{d}y \\
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| − | + & & (y) & \operatorname{d}x & \operatorname{d}y \\
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| − | = & & 1 & \operatorname{d}x & \operatorname{d}y \\
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| − | \end{smallmatrix}</math>
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| − | |-
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| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & & (y) & \operatorname{d}x & \operatorname{d}y \\
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| − | + & & y & \operatorname{d}x & \operatorname{d}y \\
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| − | = & & 1 & \operatorname{d}x & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
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| − | |-
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| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | + & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | = & & 0 & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & & y & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | + & & y & \operatorname{d}x & (\operatorname{d}y) \\
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| − | = & & 0 & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
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| − | |-
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| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & & y & (\operatorname{d}x) & \operatorname{d}y \\
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| − | + & & (y) & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & & 1 & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
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| − | & & (y) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | + & & y & (\operatorname{d}x) & \operatorname{d}y \\
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| − | = & & 1 & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | + & & (y) & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & & y & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | + & & y & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | = & & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
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| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |-
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| − | | height="60px" | <math>f_{7}\!</math>
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| − | |-
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| − | | height="60px" | <math>f_{11}\!</math>
| |
| − | |-
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| − | | height="60px" | <math>f_{13}\!</math>
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| − | |-
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| − | | height="60px" | <math>f_{14}\!</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & ((x) & (y)) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | + & (x & y) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | = & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & ((x) & y) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | + & (x & (y)) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | = & (x, & y) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & (x & (y)) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | + & ((x) & y) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | = & (x, & y) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & (x & y) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | + & ((x) & (y)) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | = & ((x, & y)) & \operatorname{d}x & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & ((x) & y) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | + & (x & y) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | = & & y & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & ((x) & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | + & (x & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | = & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & (x & y) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | + & ((x) & y) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | = & & y & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & (x & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | + & ((x) & (y)) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | = & & (y) & \operatorname{d}x & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & (x & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | + & (x & y) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | = & x & & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & (x & y) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | + & (x & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | = & x & & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & ((x) & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | + & ((x) & y) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | = & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & ((x) & y) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | + & ((x) & (y)) & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | = & (x) & & (\operatorname{d}x) & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | + & (x\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | + & (x\ (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | + & ((x)\ y) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |-
| |
| − | | height="60px" |
| |
| − | <math>\begin{smallmatrix}
| |
| − | & ((x) (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | + & ((x) (y)) & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | = & 0 & (\operatorname{d}x) & (\operatorname{d}y) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |- style="height:40px"
| |
| − | | <math>f_{15}\!</math>
| |
| − | | <math>1 + 1 = 0\!</math>
| |
| − | | <math>1 + 1 = 0\!</math>
| |
| − | | <math>1 + 1 = 0\!</math>
| |
| − | | <math>1 + 1 = 0\!</math>
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ==Differential Forms==
| |
| − |
| |
| − | ===Expanded on a Logical Basis===
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Differential Forms Expanded on a Logical Basis'''
| |
| − | |- style="background:ghostwhite; height:36px"
| |
| − | |
| |
| − | | <math>f\!</math>
| |
| − | | <math>\operatorname{D}f</math>
| |
| − | | <math>\operatorname{d}f</math>
| |
| − | |- style="height:36px"
| |
| − | | <math>f_{0}\!</math>
| |
| − | | <math>(~)\!</math>
| |
| − | | <math>0\!</math>
| |
| − | | <math>0\!</math>
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{1} \\
| |
| − | f_{2} \\
| |
| − | f_{4} \\
| |
| − | f_{8} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (x) & (y) \\
| |
| − | (x) & y \\
| |
| − | x & (y) \\
| |
| − | x & y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (y) & \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | (x) & (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | ((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
| |
| − | y & \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | (x) & (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | (x, y) & \operatorname{d}x\ \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | x & (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | (x, y) & \operatorname{d}x\ \operatorname{d}y \\
| |
| − | y & \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | x & (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | ((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (y) & \partial x & + & (x) & \partial y \\
| |
| − | y & \partial x & + & (x) & \partial y \\
| |
| − | (y) & \partial x & + & x & \partial y \\
| |
| − | y & \partial x & + & x & \partial y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{3} \\
| |
| − | f_{12} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (x) \\
| |
| − | x \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \partial x \\
| |
| − | \partial x \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{6} \\
| |
| − | f_{9} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (x, & y) \\
| |
| − | ((x, & y)) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | (\operatorname{d}x)\ \operatorname{d}y \\
| |
| − | \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | (\operatorname{d}x)\ \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \partial x & + & \partial y \\
| |
| − | \partial x & + & \partial y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{5} \\
| |
| − | f_{10} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (y) \\
| |
| − | y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \partial y \\
| |
| − | \partial y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{7} \\
| |
| − | f_{11} \\
| |
| − | f_{13} \\
| |
| − | f_{14} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (x & y) \\
| |
| − | (x & (y)) \\
| |
| − | ((x) & y) \\
| |
| − | ((x) & (y)) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | y & \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | x & (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | ((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | x & (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | (x, y) & \operatorname{d}x\ \operatorname{d}y \\
| |
| − | y & \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | (x) & (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | (x, y) & \operatorname{d}x\ \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x\ (\operatorname{d}y) & + &
| |
| − | (x) & (\operatorname{d}x)\ \operatorname{d}y & + &
| |
| − | ((x, y)) & \operatorname{d}x\ \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | y & \partial x & + & x & \partial y \\
| |
| − | (y) & \partial x & + & x & \partial y \\
| |
| − | y & \partial x & + & (x) & \partial y \\
| |
| − | (y) & \partial x & + & (x) & \partial y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |- style="height:36px"
| |
| − | | <math>f_{15}\!</math>
| |
| − | | <math>((~))\!</math>
| |
| − | | <math>0\!</math>
| |
| − | | <math>0\!</math>
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Expanded on an Algebraic Basis===
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Differential Forms Expanded on an Algebraic Basis'''
| |
| − | |- style="background:ghostwhite; height:36px"
| |
| − | |
| |
| − | | <math>f\!</math>
| |
| − | | <math>\operatorname{D}f</math>
| |
| − | | <math>\operatorname{d}f</math>
| |
| − | |- style="height:36px"
| |
| − | | <math>f_{0}\!</math>
| |
| − | | <math>(~)\!</math>
| |
| − | | <math>0\!</math>
| |
| − | | <math>0\!</math>
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{1} \\
| |
| − | f_{2} \\
| |
| − | f_{4} \\
| |
| − | f_{8} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (x) & (y) \\
| |
| − | (x) & y \\
| |
| − | x & (y) \\
| |
| − | x & y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (y) & \operatorname{d}x & + &
| |
| − | (x) & \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | y & \operatorname{d}x & + &
| |
| − | (x) & \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x & + &
| |
| − | x & \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | y & \operatorname{d}x & + &
| |
| − | x & \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
| |
| − | y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x & + & x & \operatorname{d}y \\
| |
| − | y & \operatorname{d}x & + & x & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{3} \\
| |
| − | f_{12} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (x) \\
| |
| − | x \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \operatorname{d}x \\
| |
| − | \operatorname{d}x \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \operatorname{d}x \\
| |
| − | \operatorname{d}x \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{6} \\
| |
| − | f_{9} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (x, & y) \\
| |
| − | ((x, & y)) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \operatorname{d}x & + & \operatorname{d}y \\
| |
| − | \operatorname{d}x & + & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \operatorname{d}x & + & \operatorname{d}y \\
| |
| − | \operatorname{d}x & + & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{5} \\
| |
| − | f_{10} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (y) \\
| |
| − | y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \operatorname{d}y \\
| |
| − | \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | \operatorname{d}y \\
| |
| − | \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | f_{7} \\
| |
| − | f_{11} \\
| |
| − | f_{13} \\
| |
| − | f_{14} \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | (x & y) \\
| |
| − | (x & (y)) \\
| |
| − | ((x) & y) \\
| |
| − | ((x) & (y)) \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | y & \operatorname{d}x & + &
| |
| − | x & \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x & + &
| |
| − | x & \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | y & \operatorname{d}x & + &
| |
| − | (x) & \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x & + &
| |
| − | (x) & \operatorname{d}y & + &
| |
| − | \operatorname{d}x\ \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center"
| |
| − | |
| |
| − | <math>\begin{smallmatrix}
| |
| − | y & \operatorname{d}x & + & x & \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x & + & x & \operatorname{d}y \\
| |
| − | y & \operatorname{d}x & + & (x) & \operatorname{d}y \\
| |
| − | (y) & \operatorname{d}x & + & (x) & \operatorname{d}y \\
| |
| − | \end{smallmatrix}</math>
| |
| − | |}
| |
| − | |- style="height:36px"
| |
| − | | <math>f_{15}\!</math>
| |
| − | | <math>((~))\!</math>
| |
| − | | <math>0\!</math>
| |
| − | | <math>0\!</math>
| |
| | |} | | |} |
| | <br> | | <br> |