- #1
bananabandana
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Homework Statement
Prove that, given a metric ##g_{ij}## such that ##ds^{2}=g_{ij}dx^{i}dx^{j}##, where ##x^{r} = x^{r}(\lambda)## , we have the following result for the arc length:
$$ L(p,q) = \int_{p}^{q} ds = \sqrt{ g_{ij} \frac{dx^{i}}{d \lambda} \frac{ dx^{j}}{d \lambda} } d \lambda $$
Homework Equations
The Attempt at a Solution
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Honestly not sure about how to attempt this. I've had a look in Riley Hobson and Bence and Schaums Vector analysis for clues, but can't find any. Both simply state the result as self evident. Have I missed something obvious?