# Questions tagged [tangent-spaces]

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55 questions
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### Quotient by a subring and not an ideal

I'm working towards understanding the Zariski tangent space of a $C^k$ manifold, using this pdf. The author defines $\mathcal{O}^{(k)}_{M,p}$ as the set of germs of $C^k$ functions at $p$, which ...
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### Approximating the tangent vector in a phase space (or state space) reconstruction

I am investigating an application of differential geometry in experimental dynamical systems. Given a 1D time series (e.g., one that has been experimentally obtained), $x(t)$, I am considering the ...
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### Tangent Space and Charts

I don't come from a Differential Geometry background but I have been trying to read a bit about Lie algebras. I am using Humphrey's as my main source but just to get a glimpse of the correspondance ...
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### Calculating the derivative of a mapping $\varphi: S^2 \rightarrow S^2$

I have recently learned about tangent spaces and derivatives in the context of manifolds and I am having a hard time solving the following exercise: Let A be a $3\times3$ orthogonal matrix. ...
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### How do I find a tangent plane without a specified point?

I was having a problem finding the points on $z=3x^2 - 4y^2$ where vector $n=<3,2,2>$ is normal to the tangent plane. How do we calculate the tangent plane equation without a specific point to ...
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### How to recover the tangent space from the metric

This seems such an elementary question, but I cannot see how to do this. Say that you are being given a metric (locally of course): $$g =ds^2 = g_{\mu \nu} dx^\mu dx^\nu$$ Since the metric encodes ...
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### Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds. (Guillemin & Pollack p.23)
Why $T_{A}M(n) = M(n)$ and why $T_{f(A)}S(n) = S(n)$ and why $M(n)$ & $S(n)$ are manifolds? $M(n)$ is the space of all $n x n$ matrices and $S(n)$ is the space of all $n x n$ symmetric matrices. ...