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I've a doubt about the tangent space to a manifold. Let $M$ be a $n$-manifold and let $p\in M$, I've heard that the tangent space $T_pM$ at $p$ is the first order approximation of $M$ near $p$ in the same way that the tangent hyperplane to the graph of a function $f : \mathbb{R}^n \to \mathbb{R}$ is the first order approximation to the graph of $f$.

This is really intuitive, but how do I show that ? I mean, I'm using the definition of tangent space with derivations, how do I show that that abstract set associated with each point of the manifold gives the first order approximation to the manifold ? Is this fact already built in into the definition somehow or we should prove it ? If we should prove it, can someone give a hint ? I don't want the full proof, just a hint to begin the proof.

Thanks in advance for your aid, and sorry if this question is too trivial.

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What is your definition of first order approximation to a manifold? – Andrew Oct 9 '12 at 20:54
@Andrew that is really the point. It's something intuitive that I want to make precise. For instance, If I have a surface, it's tangent plane gives a first order approximation to the surface near the tangency point. In particular, I mean that it gives a linear approximation to the surface. It's like the example I gave: if $f : \mathbb{R}^n \to \mathbb{R}$ is differentiable, then Taylor's first order formula allows us to approximate the graph of $f$ by it's tangent plane. Intuition tells that the same relationship exists between manifolds and tangent space, but I want to make it precise. – user1620696 Oct 9 '12 at 20:59
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In the case of a function $f:\Bbb R^n\to\Bbb R,$ the first order approximation to $f$ at a point $x$ is a linear function, say $Df,$ which agrees with $f$ at $x,$ and closely approximates nearby values of $f.$ Geometrically, it turns out that the graph of $Df$ is a hyperplane locally tangent to the graph of $f.$

In the case of manifolds, a priori we may not have a local embedding, as for graphs, and more importantly, the manifold itself need not be defined as the graph of a particular function. So strictly speaking, there is no first order approximation of some function lurking around. But, it still makes sense to consider a vector space which locally best approximates the manifold. The way this is done is via derivatives and derivations, which ultimately carry much more structure than a simple vector space would, letting us do calculus locally on manifolds, but intuitively one can just think of the tangent space as a local approximation of the manifold.

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Andrew, so it's really built in into the definition the fact that the tangent space approximates the manifold ? I think I understood the point. Thanks for your answer. – user1620696 Oct 10 '12 at 22:22
That's exactly it. You're welcome! – Andrew Oct 10 '12 at 22:37

In principle the tangent space T_pM is an abstract thing. But any manifold can be embedded in some R^n (Whitney theorem), and there T_pM is a true linear subspace (up to a translation) and the tangent vector to a curve in M is a true vector and so on... so you can do the same reasoning that you do in analysis

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