Timewise-average function

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Definition

Let M be a differential manifold equipped with a measure over which we can integrate.

Let u:\R \times M \to \R be any continuous function. Then the timewise-average function of u is a map f:\R \to \R defined by:

f(t) = \frac{\int_M u(t,x) dx}{\int_M dx}

For instance, the above notion makes sense if we embed the manifold in Euclidean space and look at the induced measure, or in a Riemannian manifold where we take the naturally induced measure.

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