Differentiation of a smooth family of finite-chain integrals

By [F2], every smooth pullback coefficient on the standard simplex is integrable; [F3] realizes its integral by zero extension to a bounded nondegenerate rectangle $Q$. The Darboux definitions [F4] give linearity: for integrable $u,w$, choose grid partitions whose upper-minus lower sums for each are arbitrarily small, take a common refinement, and use $\inf (u+w)\ge \inf u+\inf w$ and $\sup (u+w)\le \sup u+\sup w$ on each cell. The upper-minus-lower gap for $u+w$ is therefore bounded by the sum of the chosen gaps, so $u+w$ is integrable. Its lower and upper integrals are squeezed between the sums of the lower integrals of $u,w$ and the sums of their upper integrals; letting the gaps tend to zero gives ${\int }_Q(u+w)={\int }_{Q}u+{\int }_{Q}w$. For a scalar $c>0$, upper and lower sums scale by $c$; for $c<0$ they exchange after scaling, giving ${\int }_Q(cu)=c{\int }_{Q}u$, and $c=0$ gives zero. Also, if $\mathrm{\mid }u\mathrm{\mid }\le \epsilon$ on $Q$, each lower sum is at least $-\epsilon \mathrm{vol}(Q)$ and each upper sum at most $\epsilon \mathrm{vol}(Q)$, since cell volumes add to $\mathrm{vol}(Q)$ by [F5]. Thus $\mathrm{\mid }{\int }_{\sigma }u\mathrm{\mid }\le \epsilon \mathrm{vol}(Q)$ for an integrable function on the simplex bounded by $\epsilon$.

Fix a parameter chart and a point $p$. For one simplex $\sigma$ write ${\sigma }^{\ast }{\alpha }_p=f(p,z)d{z}_1\wedge d{z}_2\wedge d{z}_3$ on $T_3$. Choose a closed parameter rectangle $R$ around $p$ inside the chart. The displayed inequalities defining $T_3$ make it closed and bounded, hence compact by [F6]; [F2] identifies it as the integration domain. The product $R\times T_3$ is a closed bounded Euclidean set, hence compact by [F6], so ${\mathrm{\partial }}_{a}f$ is uniformly continuous there by [F7]. For a small nonzero increment $h$ in coordinate $p^a$, [F8] gives $\frac{f(p+h{e}_a,z)-f(p,z)}{h}-{\mathrm{\partial }}_{a}f(p,z)=\frac{1}{h}{\int }_0^h({\mathrm{\partial }}_{a}f(p+s{e}_a,z)-{\mathrm{\partial }}_{a}f(p,z))ds\mathrm{.}$ Uniform continuity makes the right side tend uniformly to zero on $T_3$ as $h\to 0$. By step 1.1, the integral of this error tends to zero. Linearity therefore gives ${\mathrm{\partial }}_a{\int }_{\sigma }f(p,z)dz={\int }_{\sigma }{\mathrm{\partial }}_{a}f(p,z)dz\mathrm{.}$ Applying the same argument to every parameter derivative of $f$ proves that all parameter partial derivatives of the simplex integral exist and are integrals of the corresponding derivatives. They are continuous: on a compact parameter rectangle, uniform continuity of each next derivative and step 1.1 bound the change in its integral by the uniform coefficient change times $\mathrm{vol}(Q)$. Hence this simplex integral is smooth.

By [F1], $\mathrm{\Sigma }$ is a finite linear combination of such simplices. Linearity from step 1.1 makes $I$ the corresponding finite linear combination of their smooth integrals and gives the asserted derivative formula for every parameter coordinate. Linearity in the tangent vector then gives the formula for every $v\in T_{p}P$.