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This is a question on notation

Can I express the total distance $s=s(t_1,t_2)$ that a moving body has traveled using the following (conditional) sum?

$$s(t_1,t_2) = \sum_{\forall [t_a,t_b] \subseteq [t_1, t_2]: x'(t) \neq 0} \big |x(t_a) - x(t_b) \big |$$

Note that the total space travelled is given by the summation of the absolute values of the displacements for every subspace of $[t_1,t_2]$ that the velocity has a constant sign.

bolzano
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1 Answers1

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The presented notation (what is written under the $\sum$) is not standard, not self-explaining, and does not reflect your intention. What you mean is the following (it cannot be done without text): One has $$s(t_1,t_2)=\sum_{k=1}^N|x(\tau_k)-x(\tau_{k-1})|$$ for any partition $${\cal P}:\quad t_1=\tau_0<\tau_1<\ldots<\tau_N=t_2$$ with $x'$ semidefinite on each subinterval $[\tau_{k-1},\tau_k]$.

Here it was tacitly assumed (as in your question) that $t\mapsto x(t)$ is real- (not vector-) valued.

Christian Blatter
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