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2008.10.18
7.9 Differential forms and Hodge theory
7.9.1 Invariant volume elements
An orientable manifold M is endowed with a metric g.
So we can define the invariant volume element.
This section give the invariant volume element both in the coordinate basis and non-coordinate basis.
7.9.2 Duality transformations (Hodge star)
If a manifold M is endowed with a metric g, Hodge * operation is an isomorphism mapping.
The totally anti-symmetric tensor \epsilon is very important.
7.9.3 Inner products of r-forms
The inner products give us a "volume".
7.9.4 Adjoints of exterior derivatives
We have an important relationship between d and d^{\dagger}.
The operator d is nilpotent(幂零), so does d^{\dagger}.
d and d^{\dagger} are similar to an operator in bra and ket of quantum mechanics.
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