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I'm working through some problems in Differential Geometry but I always get stuk at this expression. One of the problems for instance asks to calculate the arclength paramater of the following function:

$\epsilon: J\rightarrow \mathbb{R}^3: s \rightarrow \alpha(s) + \frac{1}{\kappa}\underline{n} $

where $\alpha$ is a path. When calculating the arclength paramater you will have to solve the following integral:

$\int_0^t\sqrt{{(\frac{1}{\kappa(s)})'^2)+(\frac{\tau(s)}{\kappa(s)}})^2}ds$

I have no idea how to simplify the second expression. I've come across it multiple times and always get stuck. Would appreciate any help.

Othman El Hammouchi
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    First, this is incorrect. You have to integrate the square root of $(\tau/\kappa)^2 + ((1/\kappa)')^2$. But you never need to actually do these computations. You use the chain rule to compute the derivatives with respect to the new arclength. Please edit your question to give the actual question at hand. (And of course you can't simplify $\tau/\kappa$ in general; the two quantities are in general totally independent.) – Ted Shifrin Oct 18 '20 at 22:43
  • I don't understand your correction, could you clarify? How do you use the chain rule here? – Othman El Hammouchi Oct 19 '20 at 07:49

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