Timewise-average function

Definition

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

Let $u:\mathbb {R} \times M\to \mathbb {R}$ be any continuous function. Then the timewise-average function of $u$ is a map $f:\mathbb {R} \to \mathbb {R}$ 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.

Related notions

• Timewise-max function
• Timewise-min function