Let's say I have a function $f : \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$. Are there conditions under which the following holds
$$ \frac{\partial}{\partial\mathbf{x}^T}\int_\Omega f(\mathbf{x},t) \,{\rm d} t = \int_\Omega \frac{\partial}{\partial\mathbf{x}^T}f(\mathbf{x},t) \,{\rm d} t,$$
where $\Omega\subset\mathbb{R}$ is some open set? In this case, both the LHS and RHS quantities are in $\mathbb{R}^{n \times n}$.
For $n=1$ there are results that follow from Lebesgue Dominated Convergence, but I am unaware of results in higher dimensions. Intuitively, it seems like the extension shouldn't be too complicated.
The goal will be to reduce to 1D, so that we can apply that standard differentiation under the integral sign, which I will call DIS. See wikipedia for quick reference on this result.
Assume the following:
For almost all $t\in\Omega$, $\frac{\partial}{\partial\mathbf{x}^T}f(\mathbf{x},t)$ exists for all $\mathbf{x}\in\mathbb{R}^n$
$f(\mathbf{x},t)$ is integrable w.r.t $t$ for all $\mathbf{x}\in\mathbb{R}^n$
$||\frac{\partial}{\partial\mathbf{x}^T}f(\mathbf{x},t)||_2\leq g(t)$ for all $\mathbf{x}\in\mathbb{R}^n$ and almost all $t\in\Omega$, where $g$ is an integrable function
From 3, we can conclude the following for $i,j=1,\ldots,n$: $$\Big(\frac{\partial}{\partial\mathbf{x}^T}f(\mathbf{x},t)\Big)_{ij}\leq||\frac{\partial}{\partial\mathbf{x}^T}f(\mathbf{x},t)||_{max}\leq||\frac{\partial}{\partial\mathbf{x}^T}f(\mathbf{x},t)||_2\leq g(t)$$ for all $\mathbf{x}\in\mathbb{R}^n$ and almost all $t\in\Omega$. Now, apply DIS to yield the following: $$\Big(\frac{\partial}{\partial\mathbf{x}^T}\int_\Omega f(\mathbf{x},t)\ dt\Big)_{ij}=\Big(\int_\Omega \frac{\partial}{\partial\mathbf{x}^T}f(\mathbf{x},t)\ dt\Big)_{ij}$$
Then we can conclude that the two matrices in question are equal via equality of their elements.