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In Hartle'sGravitation, it seems like a given the way he manipulates the "differentials" when he gives a line element. The thing is that it works, and I know it's not mathematically rigorous, but first of all, I'm wondering when treating the differentials in the line elements in this way breaks down, and I'm also wondering if there's some kind of rigorous mathematical framework behind framing line elements in this way.

Example: the standard Euclidean line element ##ds^2 = dx^2 + dy^2##. When I was in calculus II, we covered arc lengths, but there was never a good explanation as to where the formula comes from, but it seems so intuitive to say $$s = \int_a ^b ds = \int_a ^b \sqrt{dx^2 + dy^2} = \int_a ^b \sqrt{dx^2\left(1+\frac{dy^2}{dx^2}\right)} = \int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2} \ dx$$

I guess what I'm asking is: is there anything that justifies these operations with line elements without going into nonstandard analysis in a way where it's not so less-than-rigorous?

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# The rigor of using differentials in line elements

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