New answers tagged tangent-spaces
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How can I find a family of hypersurfaces in $\Bbb A^n$ with normal singular points for all $n\geq 3$?
Your proposed solution at the end of the post will work, but I think the point of the exercise here is to reduce the problem of finding a hypersurface in $\Bbb A^n$ for each $n\geq 3$ to finding a ...
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The tangent plane of Roman surface at any point intersects the surface in a pair of conics
Since a fourth degree curve with four singular points is reducible it suffices to prove that $\cal C$ has four singular points, one of which is $(x_0,y_0,z_0)$, the other three singular points are the ...
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Tangent space vs tangent plane
I'm gonna go ahead and try to talk about this all in $\mathbb R^2$. Let's look at a point $P = (3, 5)$, just to be concrete. A typical tangent vector at $P$ is something like $$
v = \pmatrix{2\\1},
$$
...
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Cotangent space using various definition of tangent space $T^{\text{glue}}_pM$, $T^{\text{paths}}_pM$ and $T^{\text{der}}_pM$
Here is a terse take on this:
The definition of $T^{\operatorname{glue}}_p$ is based on the fact that local coordinates $(x^1, \dots, x^n)$ induce a basis $(\partial_1, \dots, \partial_n)$ of $T^{\...
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Why is $v\left( O(\|x\|^2) \right) = 0$ (where $v$ is a derivation on $T_0 \Bbb R^n$)?
You need a stronger version of Taylor. Show that if $f$ is smooth, then there are smooth functions $g_i$ so that $$f(x)-f(0)=\sum x_ig_i(x),$$ and $g_i(0) = \partial f/\partial x_i(0)$.
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