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Given a path (just say a continuous function) [tex]p(t):[a,b] \rightarrow \mathbb{R}^n [/tex], the "length" of the path is defined as

[tex]\Lambda(p) = \ sup \ \sum_{i = 1}^k |p(t_i)-p(t_{i-1})|[/tex]

where the sup is taken over all partitions [tex]P = \{ a = t_0 <... < t_k = b \}[/tex] of [a,b]. The theorem mentioned in Cal 3 is that if p is piecewise differentiable, then [tex]\Lambda(p) = \int_a^b |p'(t)| dt[/tex].

Once upon a time I would have only tried to remember the integral, and not necessarily the definition. But if you give any thought to it at all, it's clear that the definition of length is much more interesting than the integral which you use to calculate length. The definition shows you the straightforward process to find length: you choose points on the path, and add up the lengths of the lines connecting the points to come up with an approximation. As you choose a finer partition, the approximation gets closer, and by the triangle inequality, the approximation keeps getting larger.. And a monotonic increasing limiting process either converges or goes to infinity.. And that is the length.

Now to the punchline... In calculus of variations it is proved that "the shortest path between two points is a straight line". But that is almost obvious from the definition of length. You first have to check that for a line p connecting p(a) to p(b), you get [tex]\Lambda (p) = |p(b)-p(a)|[/tex]. By the triangle inequality, [tex]|p(b)-p(a)| \leq \sum_{i=1}^k |p(t_k)-p(t_{k-1})| \leq \Lambda(p)[/tex].

etc., etc., etc...