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,

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

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.