Differentiation under the integral sign

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

\[\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t)\, dt = f(x,b(x))b'(x) - f(x,a(x))a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}f(x,t)\, dt.\]

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:

\[\frac{d}{dt}\int_{M_t}\omega_t = \int_{M_t}V_t\lrcorner d_x\omega_t + \int_{\partial M_t}V_t \lrcorner \omega_t + \int_{M_t}\dot{\omega}_t.\]

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.