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exmachina

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How do I find the length of the line M on this surface? Ideally the expression would be somehow linked to the directional derivative,

IE. I've been thinking of slicing the line M into tiny little components, and taking the directional directive at each point of M, the integrating with respect to.. something (maybe dx1, dx2, dx3...

Any tips? Some recommend arc length, others recommended differential manifolds (Which I have NO IDEA)