>>16538677
>Once you have a scalar function of 2 (or more) variables then you have an infinite number of tangent lines
That's now what he is claiming. He is saying that there are infinitely many points 0 distance away from a given point. That's now what distance is. You can have this occur on pseudo-Riemannian manifolds, in which case the set of all such points forms an algebraic variety in that manifold. The most well-known example is the lightcone in special relativity. But I am 100% sure this faggot doesn't know any of this. He's just saying retarded shit because he never got beyond "infinitesimals" in his engineering degree.

As for your note, yes, obviously this generalizes to tangent spaces on points of a manifold. These tangent spaces, however, are unions of tangent lines of the set of all geodesics that pass through that point, so the construction is still formally valid.

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>Derivative is the slope of the tangent lineWhat the fuck does that even mean and how does it relate to solving real world problems?

>Derivative is the slope of the tangent line
What the fuck does that even mean and how does it relate to solving real world problems?