Research proof index

Swap the time derivative $\partial_t$ with the integral $\int$.

Swap the spatial derivative $\partial_x$ with the integral $\int$.

Swap the second spatial derivative $\partial_{xx}$ with the integral $\int$.

Uniqueness of weak multi-derivatives on U: any two candidates must agree almost everywhere on U. The proof reduces to the du Bois-Reymond lemma via the defining identity.

A Cauchy sequence in W^{k,p}(U) induces a Cauchy sequence in Lp for each weak derivative component.

Triangle inequality for a single weak-derivative eLpNorm.