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Suppose I have a moving curve $\alpha:I\times[0,T] \to \mathbb{R}^n.$ Its length is $\int_\alpha ds = \int_I |\alpha_x|dx.$ If I want to find the time derivative of this, I guess I differentiate under the integral sign.

If I have a function like $\int_I{|\alpha_{ss}|ds}$, how do I find its time derivative? Can I just differentiate the $|\alpha_{ss}|$ part inside the integral or do I need to do something with $ds$ too?

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    Nope. I guess take it as smooth as needed.2012-06-02

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