Physicists usually do not worry about the order of integration and differentiation, since in most cases the functions involved are at least \( C^1 \)-continuous. However, one cannot always take this for granted. For instance, I recently encountered the following time-domain wave boundary integral equation [Langer and Schanz 2008, eq. (18.25)]:

4\pi c(x) u(x,t) = \int_\Gamma h(y, t-r) \frac{1}{r} \text{d}\Gamma_y + \int_\Gamma u(y, t-r)\frac{1}{r^2}\frac{\partial r}{\partial n}\text{d} \Gamma_y + \int_\Gamma \dot{u}(y,t-r)\frac{1}{r}\frac{\partial r}{\partial n}\text{d}\Gamma_y,

where \( c(x) \) is the solid angle, \( u \) is the unknown solution, \( \Gamma \) is the boundary of the domain of interest \( \Omega \), \( h \) is the Neumann boundary condition, and \( r = \|y-x\| \). This function describes a wave problem with homogeneous initial conditions and zero sources.

We are interested in the temporal derivative of \( u \), i.e., \( \frac{\partial u}{\partial t} \), so we take the derivative on both sides of the equation above:

4\pi c(x) \partial_tu(x,t) = \partial_t \int_\Gamma h(y, t-r) \frac{1}{r} \text{d}\Gamma_y + \partial_t \int_\Gamma u(y, t-r)\frac{1}{r^2}\frac{\partial r}{\partial n}\text{d} \Gamma_y + \partial_t \int_\Gamma \dot{u}(y,t-r)\frac{1}{r}\frac{\partial r}{\partial n}\text{d}\Gamma_y.

Let us take the first term on the right hand side as an example. Note that the integration domain is slightly misleading, and one may simply interchange the integral and derivative symbols, which seems formally correct. However, when \( t \) is small enough, the true integration domain of the first integral will not cover the whole \( \Gamma \). We could identify that, the true integration domain should be:

\Sigma = \Sigma(t) = \{y\in\Gamma : \|y-x\| \leq t\}.

So the differentiation of the first term should follow the Reynolds transport theorem (we refer to Zhang et al. [2020]):

\partial_t \int_\Gamma h(y, t-r) \frac{1}{r} \text{d}\Gamma_y = \int_\Sigma \partial_t h(y, t-r) \frac{1}{r} \text{d}\Sigma_y + \int_{\partial \Sigma} n \cdot \partial_t y \, \Delta (h/r) \text{d}(\partial\Sigma_y),

where

\Delta(h/r) = \lim_{\varepsilon\to0^-}(h/r)(y+\varepsilon n)-\lim_{\varepsilon\to0^+}(h/r)(y+\varepsilon n), \forall y \in\partial \Sigma.

References

• Langer and Schanz 2008, Time Domain Boundary Element Method.
• Zhang et al. 2020, A Differentiable Theory of Radiative Transfer.