Hodge dual: Difference between revisions

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The Hodge dual \( \star(v_1 \wedge v_2) \) directly gives the generator of <math> \ker M </math>.

== ~~Computing the~~ Hodge dual ==

== Computation ==

~~Again 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

The Hodge dual can be computed quickly by realizing that if we write the basis in lexographical order, we only have to reverse the coefficients and change some signs.

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

@@ Line 87: / Line 87: @@
 The Hodge dual \( \star(v_1 \wedge v_2) \) directly gives the generator of <math> \ker M </math>.
-== Computing the Hodge dual ==
+== Computation ==
-Again 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
+The Hodge dual can be computed quickly by realizing that if we write the basis in lexographical order, we only have to reverse the coefficients and change some signs.
+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