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The curvature of a curve (rate of change of the unit tangent vector with respect to arc length) is defined as $\kappa = \frac{|\underline{r}'(t) \times \underline{r}''(t)|}{|\underline{r}'(t)|^3}$ The proof of this is shown in my textbook, but I don't understand one step.

They say: $\underline{r}'(t) = \underline{T}(t) |\underline{r}'(t)| = \underline{T}(t) s'(t)$ and then $ \underline{r}''(t) = \underline{T}'(t) s'(t) + \underline{T}(t)s''(t).$

First question: Would $s''(t)$ not equal $0$? My reasoning being since $s'(t) = |\underline{r}'(t)|,$ a constant and then subsequently differentiate again to get $0$?

After the above step, they say $\underline{r}'(t) \times \underline{r}''(t) = [s'(t)]^2 \underline{T}(t) \times \underline{T}'(t)$. I am not sure how they get this. The expression I got when I simplified was:$ \underline{T}(t)s'(t) \times (\underline{T}'(t)s'(t) + \underline{T}(t)s''(t)) = \underline{T}(t)s'(t) \times \underline{T}'(t)s'(t) + \underline{T}(t)s'(t) \times \underline{T}(t)s''(t)$ And the latter term disappears because $\underline{T}(t) \times \underline{T}(t) = \underline{0}$ Many thanks.

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    I see. So we have $(a\underline{b}) \times (a\underline{c}) = a^2(\underline{b}) \times \underline{c}$2012-12-13

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First question: Would $s′′(t)$ not equal 0? My reasoning being since $s'(t) = |\underline{r}'(t)|$, a constant and then subsequently differentiate again to get 0?

But why must the magnitude of $\underline{r}'(t)$ be constant? It is written as a function of $t$, so as $t$ changes, $\underline{r}'(t)$ can change, and thus its magnitude can change.

After the above step, they say $\underline{r}'(t) \times \underline{r}''(t) = [s'(t)]^2 \underline{T}(t) \times \underline{T}'(t)$. I am not sure how they get this.

You just need to realize how scalar multiplication works across cross products. The key is $(k\,\mathbf{a})\times \mathbf{b}=\mathbf{a}\times(k\,\mathbf{b})=k\,(\mathbf{a}\times\mathbf{b}),$ paying special attention to the last equality.

Then, using that last equality twice and the fact that $\underline{T}\times\underline{T}=\underline{0}$, we get \begin{align}\underline{T}(t)s'(t) \times (\underline{T}'(t)s'(t) + \underline{T}(t)s''(t)) &= \underline{T}(t)s'(t) \times \underline{T}'(t)s'(t) + \underline{T}(t)s'(t) \times \underline{T}(t)s''(t)\\ &=s'(t)s'(t)(\underline{T}\times\underline{T'})+s's''(\underline{T}\times\underline{T})\\ &=[s'(t)]^2(\underline{T}\times\underline{T'}), \end{align} as desired.