Hodge dual: Difference between revisions

Line 92:

With a basis of dimension ''n'', suppose we have a ''k''-form '''V''' and wish to find its dual '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure:

# Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order

# Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order. '''C''' will have the same length as '''V''' and '''M'''.

# '''C''' will have the same length as '''V''' and '''M'''

# For each combination <math>C_i</math>, compute <math>S_i = \sum C_i - \frac{k (k+1)}{2}</math>.

# ~~Sum the numbers in~~ each combination ~~'''C'''~~<~~sub~~>~~''i''~~</~~sub~~> ~~with ceil~~(''k~~''/~~2~~) to find '''S'''<sub>''i''~~</~~sub~~>

# Multiply the ''i''-th element of '''V''' by <math>(-1)^{S_i}</math>

# Multiply the ''i''th element of ''V'' by −1<~~sup>'''S'''<sub>''i''</sub~~></~~sup~~>

# Reverse the elements of '''V'''.

~~and then reverse~~ the elements of '''V'''.

To find an unknown '''V''' from a known '''M''', first reverse '''M''' and then adjust the signs.

@@ Line 92: / Line 92: @@
 With a basis of dimension ''n'', suppose we have a ''k''-form '''V''' and wish to find its dual '''M'''. The elements of '''V''' are associated with ''k''-combinations, and of '''M''' with {{nowrap|(''n'' − ''k'')}}-combinations, of the basis elements. Because of the symmetry of binomial coefficients, '''V''' and '''M''' will have the same length. To find '''M''' we adjust the signs of '''V''' with the following procedure:
-# Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order
+# Let '''C''' be the ''k''-combinations of the numbers 1 through ''n'' in lexicographic order. '''C''' will have the same length as '''V''' and '''M'''.
-# '''C''' will have the same length as '''V''' and '''M'''
+# For each combination <math>C_i</math>, compute <math>S_i = \sum C_i - \frac{k (k+1)}{2}</math>.
-# Sum the numbers in each combination '''C'''<sub>''i''</sub> with ceil(''k''/2) to find '''S'''<sub>''i''</sub>
+# Multiply the ''i''-th element of '''V''' by <math>(-1)^{S_i}</math>
-# Multiply the ''i''th element of ''V'' by −1<sup>'''S'''<sub>''i''</sub></sup>
+# Reverse the elements of '''V'''.
-and then reverse the elements of '''V'''.
 To find an unknown '''V''' from a known '''M''', first reverse '''M''' and then adjust the signs.