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I'm reading a book on computer science/math and I found this formula for arc lengths that I've not been able to decipher: $$\left|\int_p^q\left\| {df(x)\over dx} \right\| dx\right|$$ where $\lVert \cdot\rVert$ is the Euclidean norm.

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I believe this is the arc length of a parametric curve $f:[p,q]\to \mathbb R^n$. You may be more familiar with the arc length of the graph of a function from $\mathbb R$ to $\mathbb R$, which can be realized as a special case. – Alex Becker Sep 1 '12 at 7:47
@AlexBecker Thanks! Any links to sites that explain this in details? – saadtaame Sep 1 '12 at 7:49
@Alex is correct, though the more usual form I see is (omitting the limits) $$\int\|\mathbf f^\prime(u)\|\mathrm du$$ where the vector-valued nature of the function $\mathbf f$ is indicated by the use of boldface. – J. M. Sep 1 '12 at 7:51
Also, the absolute value outside isn't needed; things are set up such that the integral's value is nonnegative... – J. M. Sep 1 '12 at 8:10
up vote 3 down vote accepted

Yes that's the usual definition. It depends on your setting, for simplicity let's assume you are in $\mathbb R^n$. You need a scalar product (dot product) $< . >$ on $\mathbb R^n$ (seen as the tangent space at each point of the first $\mathbb R^n$). Then if $f : [a,b] \longrightarrow \mathbb R^n$ is a piecewise $\mathcal C^1$ curve, its length is defined by:

$$ \int_a^b \sqrt{ \Big< \frac{df}{dx}(x), \frac{df}{dx}(x) \Big>} dx $$

Notice that we ca take the square-root since $<x,x>$ is positive for all $x \in \mathbb R^n$.

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What is the tangent space? In the book it's $\mathbb R^2$. – saadtaame Sep 1 '12 at 7:59
Ok, so forget it. In the case of $\mathbb R^n$ it's not really needed. This notion is useful in more general settings: Manifolds. Given a point $p$ on the image of your curve $\mathcal C$ (the set $f([a,b]) \subset \mathbb R^2$, the tangent space $T_p \mathcal C$ of $\mathcal C$ at $p$ is the set of all possible speeds a point moving along the curve can have at the point $p$ (all tangent vectors at $p$). So this space is juste a line tangent to $\mathcal C$ at $p$, a copy of $\mathbb R$. – Ilies Zidane Sep 1 '12 at 8:09

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