To each path we can associate a dual 1-form given by
.
Here are two examples: The following path passes through one edge twice:
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The corresponding dual 1-form has evaluates to 2 on this edge.
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The second example contains the edges and .
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In the corresponding dual 1-form these two edges cancel:
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Let us calculate the derivative of the dual 1-form associated to a path :
If is closed, this will be zero for all vertices . This is because a closed path enters any vertex as often as it leaves it. On the other hand, if is not closed, the evaluates to on the vertex to on and to zero on all other vertices.
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In particular this implies that for a closed path the dual 1-form is also closed, i.e. .