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1answer
181 views

Partial derivative of an integral operator functional

Suppose $f(x) \equiv f_0(x) + \epsilon t(x)$, where $x,\epsilon \in{\mathbb{R}}$. And let $\mathcal{L}[f(x)] \equiv \int_a^b f(x')dx'$. I want to differentiate $\mathcal{L}$ w.r.t. $\epsilon$. So I ...
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94 views

Am I allowed to move around an operator like this?

Can I take this product: $$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$ And factor out one of the $L$'s to get: $$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$ Where the operator $\frac{d}{dt}$ now ...