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I am reading Tu's Introduction to Manifolds. Given a smooth function $F:N \to M$ between manifolds and $p \in N$, he defines the differential $F_*:T_pN \to T_{F(p)}M$ as follows.

If $X_p \in T_pN$ is a tangent vector (i. e. a derivation on the germ of $C^{\infty}$ functions at $p$) and $f: M \to \Bbb{R}$ is a smooth function, then $F_*(X_p)$ is the tangent vector at $F(p)$ which acts on $f$ as $F_*(X_p)f = X_p(f \circ F)$.

He shows that this definition generalizes the derivative of a smooth function between Euclidean spaces, and that the operator taking $F$ to $F_*$ satisfies a functorial property (the chain rule).

Clearly this is the "correct" generalization of the derivative. However, just looking at the formula for the differential $ (F_*(X_p) = X_p(\cdot \circ F))$, it is difficult to immediately see that it has anything to do with the derivative of $F$-- particularly, the identification of tangent vectors with point derivations does a lot to obscure it for me.

For lack of a better phrasing, is there some way to arrive at this formula from first principles, or to show that it satisfies some kind of desirable universal property? I just need some motivation-- Tu merely presents it and proves its properties afterwards. Also, when did this definition originate?

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    In Calc 1, the derivative of $f(p)$ regarded as a number can be interpreted as the slope of the best linear approximation to $f$ at $p$. Well, this new derivative, i.e. the differential $f_*$, *is* the best linear approximation to $f$ at $p$. – 2017-01-21
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    I know that, and the local formula for the differential that Tu gives (the Jacobian) is proof enough of that for me. I am more wondering why the definition of the differential takes this form in its full, abstract generality, and how it was found. Unlike the Calc 1 derivative, which you can interpret as the limiting form of the average rate of change, this formula appears to come from nowhere. Maybe I just need to get used to it, but it would be nice to see this formula derived from more basic principles (esp. A universal property) somehow. – 2017-01-21
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    The thing is, the Jacobian is chart-dependent, and I would prefer to understand this formula in a coordinate-free way. – 2017-01-21
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    How else could one define it? We have a germ of a smooth function at $F(p)$, and we have a derivation on $C^{\infty}(M)_p$. What can we do? Well, we can pull back the germ via $F$, and then we can apply our derivation to the pulled-back germ. I don't see any other way to bring things together that could stand a chance of being meaningful. – 2017-01-21
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    Well thanks, because that is exactly the sort of algebraic reasoning for why it looks the way it does that I'm looking for. Do you know if there's any universal property the differential satisfies? I ask because the Calc I derivative is the "best possible" linear approximation of a function at a point, and given that this is a more algebraic definition, I would expect it to be the "best possible" construction in an algebraic sense, i. e. It would satisfy a universal property. – 2017-01-21
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    I don't see a universal property there (that doesn't mean there isn't one, I'm just not seeing one). Maybe the functoriality already determines it? – 2017-01-21
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    Regarding your first response to my comment, my suggestion is to forget the Calc 1 definition as the limiting form of the average rate of change. Focus instead on the next level of abstraction, namely the best linear approximation. Generalize that as much as possible to maps between open sets in Euclidean space, and their best linear approximations. Then generalize further to maps between submanifolds of Euclidean space and their best linear approximations, which forces you to consider tangent spaces.... – 2017-01-21
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    ... Study the mathematical properties of those generalizations. Abstract the statements and formulations of those properties. Repeat this of abstraction and generalization. That's what leads to this definition. – 2017-01-21
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    Thanks, i guess that is the best way to proceed. This discussion has been very helpful. – 2017-01-21

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