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$$K = \Bigg|\frac{dT}{ds}\Bigg| = \Bigg|\frac{\frac{dT}{dt}}{\frac{ds}{dt}}\Bigg|$$

where $T$ is the unit tangent vector to the curve and $s$ is arc lenght.

I don't know why that relationship is True in infinitesimals, in algebra I know the ratio $\frac{1}{dt}$ both in the numerator and the denominator is one but I am not sure how that works in infinitesimal and with an absolute value

Sebastiano
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samsamradas
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1 Answers1

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You have $s=s(t)$ and $T=T(s)=T(s(t))$.
Applying the chain rule you get $\frac{dT}{dt}=\frac{dT}{ds}\frac{ds}{dt}$ and then divide by $\frac{ds}{dt}$ to obtain your equality.

Temoi
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  • is my approach in the comments also correct? – samsamradas Jan 03 '24 at 18:55
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    No, because $\frac{ds}{dt}$ (and similarly the others) are not fractions but just a notation for the derivative of the function $s$ with respect to the variable $t$ (see also: https://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio#:~:text=%24%5Cboldsymbol%7B%5Cdfrac%7Bdy,notation%20of%20the%20derivative%20(c. ) – Temoi Jan 03 '24 at 18:58