Riemann sum of derivative (something like that)

[quasar987]

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 $t\in [a,b]$, let $\sigma = \{t_0,...,t_N\}$ be a partition of [a,b] and $\delta_{\sigma}=\max_{0\leq k \leq N-1}(t_{k+1}-t_k)$. Also define $\Delta t_k=t_{k+1}-t_k$ Then,

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

Proving this would also amount to proving

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

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

 

[Hurkyl]

If it can be done directly, it looks like a differential approximation should be the obvious thing to do! Remember that for any differentiable f, there is a remainder r such that:

$$f(x + k) = f(x) + k f'(x) + k r(x, k)$$

and, for each x,

$$\lim_{k \rightarrow 0} r(x, k) = 0$$


Of course, what you want to prove is trivial over any interval where c is monotonic.

 

[quasar987]

Hi Hurkyl,

If I make that substitution in the riemann integral, I get

$$\lim_{max(k_i)\rightarrow 0} \sum_i^N \frac{f(x_i+k_i)-f(x_i)}{k_1}k_i = \int_a^b f'(x)dx + \lim_{max(k_i)\rightarrow 0} \sum_i^N r(x_i,k_i)k_i$$

So the problem is essentially the same: In my OP, I knew that the differential ratio was going to the derivative as $\delta_{\sigma} \rightarrow 0$ so the limit of the riemann sum should be $\int_a^b |\frac{dc}{dt}(t)|dt$, but did not know how to prove it. Now I know that as max(k)-->0, r(x,k)-->0, so the limit of the riemann sum should be $\int_a^b 0dx$, but still don't know how to prove it.

 

[Hurkyl]

If you take off the absolute value signs, then it's very easy. Your sum is a telescoping series, and your integral is easily integrated.

If you don't want to use that... do you know about uniform convergence?


Anyways... *bonks self* forget about differential approximation. This is a job for the mean value theorem!

 

[quasar987]

Hurray for Hurkyl and the mean value theorem! :D