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The Leibniz integral rule, in its multivariate form, deals with differentiation of the following sort: $$ \frac{\partial}{\partial t} \int_{D(t)} F({\bf x}, t) \, d{\bf x} \, , \qquad D(t)\in \mathbb{R}^d \, .$$

I am looking for a fully rigorous formulation of this theorem, as well as a proper proof. So far, I could only find:

  1. The one-dimensional case (see e.g., Courant calculus book).
  2. Physics-flavored books, where the normal speed $v_n ({\bf x})$ is not defined (see e.g., here). While insightful, this is not what I need.
  3. Very abstract, measure theoretical versions of it.

My question: Could you please refer me to a multivariate-calculus statement and proof of this theorem?

Update: I searched most of the textbooks here and here, but still no luck. Any ideas?

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