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When reading this post, I become confused about what it means for a map to intersect transversally. To help conceptualize things better here's my question.

Let $f:\mathbb{R}^d \rightarrow \mathbb{R}^n$ be a once continuously differentiable function, $d\geq n>0$ be (positive) integers, and let $x \in \mathbb{R}^n$. When does $f$ intersect transversally with $x$?

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    Let $g:\mathbb{R}^0\to\mathbb{R}^n$ be defined by $g(0)=p$. The maps $f$ and $g$ are transversal when for all $t\in\mathbb{R}^n$ such that $f(t)=x=g(0)$ one has that the image of $df$ and the image of $dg$ (which in our case is only the $0$ space), generate all of the tangent space of $\mathbb{R}^n$ at $x$. Since $dg$ only contributes $0$ then the image of $df$ would have to do all the job. One would need $df$ to be onto for all $t$ such that $f(t)=x$. – MoonLightSyzygy Jan 21 '20 at 14:54
  • I think that some other examples are going to be more interesting. For example (1) $f,g:\mathbb{R}\to\mathbb{R}^2$ given by $f(x)=(x,x^2)$, $g(x)=(x,0)$. (2) With the same spaces are before, but with $f(x)=(x,x^2)$ and $g(x)=(x,x)$. (3) $f,g:\mathbb{R}\to\mathbb{R}^3$ defined by $f(x)=(x,x^2,0)$, $g(x)=(x,x,0)$. – MoonLightSyzygy Jan 21 '20 at 15:16
  • @MoonLightSyzygy Thanks; I would happily accept these comments as a formal answer. –  Jan 21 '20 at 15:35
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    You can write it out properly, maybe even work out the other examples and add your own answer. – MoonLightSyzygy Jan 21 '20 at 15:36
  • two maps $f,g:M\rightarrow N$ intersect transversally at $m\in M$ if $f(m)=g(m)=n$ and $f_{,m}(T_mM)+g_{,m}(T_mM)=T_{*,n}N$.. can you now guess what should be definition for transversal intersection as you mentioned? –  Jan 21 '20 at 15:59
  • So g in this case is taken to be the identity map? –  Jan 21 '20 at 17:12

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