Preamble: I am looking to better understand Hadamard's Variational Formula $\frac{d\lambda}{d\tau} = -\int_{\partial U(\tau)}\left|\frac{\partial w}{\partial y}\right|^2v\cdot y dS$ for eigenvalues of domains under variation, so I got my copy of Evans' Partial Differential Equations (2nd edition). The variational formula appears as an exercise in the page 369 (chapter 6.6.) and to my knowledge beyond this problem, the formula is not discussed explicitly anywhere else in the book. This problem's hint is a reference to the appendix C.4, where the following theorem, which to my knowledge is the Raynolds' Transportation Theorem, is provided w/o a proof or a further discussion:
Theorem 6 (Differentiation formula for moving regions) If $f = f(x, \tau)$ is a smooth function, then $\frac{d}{d\tau}\int_{U(\tau)}fdx = \int_{\partial U(\tau)}fv\cdot ydS + \int_{U(\tau)}f_\tau dx$
Question: I am looking for a source which exemplifies the backgroud, the implications and the connection of Raynold's Transportation Theorem to Hadamard's Variational Formula with appropriate proofs for both.
An excellent book on Hadamard's variational formula is Hadamard’s Formula Inside and Out by P. Grinfeld. In it he discusses in detail the calculus of moving domains and the differentiation formula for moving regions (Reynold's transportation formula).
Moreover, he uses the formula repeatedly to derive many results in the context of variational spectral problems including the formula in Evan's. He also looks at some applications.