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I'm reading:

https://en.wikipedia.org/wiki/Tangent_space

Specifically, "Definition as the velocity of curves"

and the definition of tangent space at a point as the set of all tangent vectors of embedded curves at that point.

How do we show that the tangent space defined as such matches our intuitive idea of tangent... ie: barely touching the manifold. I'm referring to this idea:

https://en.wikipedia.org/wiki/Tangent

"just touches the curve at that point".

Is there a rigorous way to show that these two ideas match?

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Note that the tangent plane does not necessarily touch the manifold at just one point (consider the tangent plane to a cylinder, or to a plane itself). The definition is actually very close to what I would consider intuitive for "tangent": the tangent plane at a point $p$ on a manifold passes through $p$, and for every curve $\gamma$ on the manifold through $p$, the tangent line to $\gamma$ at $p$ lies within the tangent plane...

Here's another way to look at it: suppose you choose coordinates for your ambient space so that the origin is at $p$, and the $x$ and $y$ directions span the tangent plane, and now locally parameterize the surface by a height map $z(x,y)$ on the tangent plane. Then $z$ has a critical point at $(0,0)$. To see this, notice that for any vector $(a,b)$ in the tangent plane, the curve $\gamma(t) = z(at, bt)$ must have velocity tangent to the plane at $(0,0)$, so that $$\nabla z \cdot (a,b) = 0.$$ The case where the critical point is a strict local minimum or maximum is what you usually think of as a "just touching" tangent plane, but notice that $z$ could have a saddle point instead, or there could be a whole line or patch containing $(0,0)$ consisting of critical points (as in the case of the tangent plane to a cylinder.)

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