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In my textbook, it says: "Consider two curves in the plane, one of which is the x-axis, the other being the graph of a function $f(x)$. The two curves intersect transversally at a point x if $f(x)=0$ (the intersection condition), and $f'(x)\neq0$ (transversality)." I know that: "Two submanifolds of a given finite dimensional smooth manifold are said to intersect transversally if at every point of intersection, their separate tangent spaces at that point together generate the tangent space of the ambient manifold at that point." From: Transversality (wikipedia). In the example of my textbook, what are the spaces tangent to the x-axis and to the graph of the function $f(x)$? What is the tangent space of the ambient manifold?

Thank you very much

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up vote 7 down vote accepted

The tangent space of the x-axis (at a given point) is again the x-axis (because it's horizontal, i.e. derivative is zero). Then the tangent space of the graph of $f(x)$ is whatever it is, but the key is that when it intersects the x-axis, the tangent space at that intersection point is not horizontal (i.e. not the x-axis). So you have a tangent vector on the x-axis (which is necessarily horizontal) and a tangent vector on the graph of f(x) (which is necessarily not-horizontal), and hence these two vectors are linearly independent, hence span the whole plane (agreeing with Wikipedia)!

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I was writing my answer when noticed that your one had appeared. It is very similar, but I decided to publish my version too. I upvoted you answer, of course :-) – Yuri Vyatkin Sep 19 '12 at 8:07

Tangent spaces at points of $\mathbb{R}^n$ are naturally identified with $\mathbb{R}^n$ (see Tangent space in Wikipedia). So at a point $\mathbf{x}=(x,0)$ on x-axis the tangent space of the ambient space (i.e. $\mathbb{R}^2$) is $\mathbb{R}^2$, the tangent space of x-axis is $\mathbb{R}^1$, and the tangent space to the graph is the line passing through point $\mathbf{x}=(x,0)$ at the slope $f'(x)$. If you take a basis in both tangent lines the resulting pair will be a basis in the tangent plane, hence transversality.

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The tangent manifold to $\mathbb{R}^2$ is $\mathbb{R}^2$. The tangent space to the x-axis is spanned by $(1,0)$ and the tangent space to the function is the slope $f'(x)$ line, a space that is spanned by $(1,f'(x))$. These two spaces span $\mathbb{R}^2$ whenever $f'(x) \neq 0$.

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