Intuitive

Concrete

This tool gives a nice formula to the derivative of a one dimensional integral with dependencies to the varied quantity every where, with $f, a$ and $b$ having the right conditions

$$ \left.\frac{\partial}{\partial \varepsilon} \int_{a(\varepsilon)}^{b(\varepsilon)}f(\varepsilon, x) d x \right|_{\varepsilon=0} = \\ \int_{a}^{b} \left.\frac{\partial}{\partial \varepsilon} f(\varepsilon, x)\right|_{\varepsilon=0} d x + f(0,b)\left.\frac{\partial}{\partial \varepsilon} b(\varepsilon)\right|_{\varepsilon=0} - f(0,a)\left.\frac{\partial}{\partial \varepsilon} a(\varepsilon)\right|_{\varepsilon=0} $$

Abstract

Why is it interesting?

This formula is really nice in one dimenisonal Variational Calculus and with $a(0) = a$, $b(0) = b$, and for the particle mechanics Noether's Theorem. For integrals over arbitrary manifolds, it generalizes to the Lie derivative of a volume element.