Hi, maybe someone can help. When I think about it, I'm pretty sure that the following is true: Let c be a curve parametrized by [itex]t\in [a,b][/itex], let [itex]\sigma = \{t_0,...,t_N\}[/itex] be a partition of [a,b] and [itex]\delta_{\sigma}=\max_{0\leq k \leq N-1}(t_{k+1}-t_k)[/itex]. Also define [itex]\Delta t_k=t_{k+1}-t_k[/itex] Then,(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\lim_{\delta_{\sigma}\rightarrow 0}\sum_{k=0}^{N-1}\frac{|c(t_k+\Delta t_k)-c(t_k)|}{\Delta t_k}\Delta t_k=\int_a^b |\frac{dc}{dt}(t)|dt[/tex]

Proving this would also amount to proving

[tex]\lim_{\delta_{\sigma}\rightarrow 0}\sum_{k=0}^{N-1}\frac{|c(t_k+\Delta t_k)-c(t_k)|}{\Delta t_k}\Delta t_k=\lim_{\delta_{\sigma}\rightarrow 0}\sum_{k=0}^{N-1} |\frac{dc}{dt}(t_k)|\Delta t_k[/tex]

Is there a way to do this using a finite succession of arguments?

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# Riemann sum of derivative (something like that)

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