Volume Form - Divergence

Divergence

Given a volume form ω on M, one can define the divergence of a vector field X as the unique scalar-valued function, denoted by div X, satisfying

where LX denotes the Lie derivative along X. If X is a compactly supported vector field and M is a manifold with boundary, then Stokes' theorem implies

which is a generalization of the divergence theorem.

The solenoidal vector fields are those with div X = 0. It follows from the definition of the Lie derivative that the volume form is preserved under the flow of a solenoidal vector field. Thus solenoidal vector fields are precisely those that have volume-preserving flows. This fact is well-known, for instance, in fluid mechanics where the divergence of a velocity field measures the compressibility of a fluid, which in turn represents the extent to which volume is preserved along flows of the fluid.

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