Hi guys! Let [itex]\left \{ B_{t} \right \}_{t\in \mathbb{R}}[/itex] be a one - parameter family of compact subsets of [itex]\mathbb{R}^{3}[/itex] with smooth (manifold) boundary (e.g. one - parameter family of closed balls). In my context, each [itex]B_{t}[/itex] belongs to a different constant time slice of Minkowski space - time. How does one compute [itex]\frac{\partial }{\partial t}\int_{B_{t}}f(t,\mathbf{x})dV[/itex]? Again in my context, [itex]f(t,\mathbf{x})[/itex] happens to be [itex]T_{00}(t,\mathbf{x})[/itex], the time - time component of the stress energy tensor and as such this component is assumed to be a smooth scalar field.(adsbygoogle = window.adsbygoogle || []).push({});

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# Differentiation under integral sign - one parameter family

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