# Questions tagged [tangent-spaces]

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### For any k-dim. subspace $L$ of $T_p M$, can we find a sub manifold, say $R$, of $M$ containing $p$ s.t $T_p R = L$

Let $M$ be an $n$ dimensional manifold, and $S\subseteq M$ be a k-dim. sub manifold of $M$, where each is in fact a smooth manifold to be precise. We know that $T_p S$ is a k-dim. subspace of $T_p M$....
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### Determining the linear independence of tangent vectors at a point on the manifold

We define the tangent space at a point, say $x_0$, on the manifold $M$ as the set of all derivations, i.e maps which maps smooth maps from a neighbourhood of $x_0$ to real numbers to real numbers. ...
### Finding points on a surface $\{ z= f(x, y)\}$ with horizontal tangent plane
Could someone please explain in detail how this is done? For example there is a surface $$M = \{ (x, y, z) : z = x^4 - 4xy^3 + 6y^2 - 2\}$$ and the question is to find the points on $M$ where this ...