When reading the proof of one dimensional Euler-Lagrange equation, e.g. on Wikipedia, I stuck myself at one point: $\frac{\text{d}}{\text{d}\varepsilon}\int_a^b F_\varepsilon \text{d}x = \int_a^b \frac{\text{d}F_\varepsilon}{\text{d}\varepsilon} \text{d}x$. Why is it possible to change the order of derivative and integration?
2026-03-28 21:06:54.1774732014
change of order of derivative and integral in the proof of Euler-Lagrange equation
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It is a consequence of the Leibniz Integral Rule when the limits of integration are constants.