April 15, 2018 / by Matthew Kvalheim / In news

Differentiation under the integral sign

In calculus, Leibniz’s rule for differentiation under the integral sign states that, modulo precise regularity assumptions,

There is a nice generalization of this result to the case of integrating a time-dependent differential $k$-form over a time-varying $k$-submanifold with boundary.

Let $M_t\subset Q$ be a smoothly time-dependent k-dimensional submanifold with boundary of an n-dimensional boundary-less manifold $Q$, and let $\omega_t$ be a smooth time-dependent differential $k$-form on $Q$, and let $V_t \in \Gamma$$(TQ\mid_{M_t})$ be the field of velocities of points on $M_t$ (where $\Gamma(\cdot)$ denotes the space of smooth sections of whatever bundle is in parentheses). Then:

Here $d_x\omega_t$ is the exterior derivative with respect to spatial directions only. Also, e.g. $V_t \lrcorner \omega$ denotes the contraction of $V_t$ with $\omega$ (the interior product), and $\dot{\omega_t} = \frac{\partial}{\partial t} \omega_t$.

For a proof of this fact, see the very nice paper Differentiation Under the Integral Sign by H. Flanders.