# Questions tagged [tangent-spaces]

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61 questions
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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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### Germs: Why is it sensible to define a function on a collection of equivalence classes by its action on each element?

I am following Loring W. Tu in his second edition of 'An introduction to manifolds'. Here is a pdf-copy of the book. On page 87 he defines $C^\infty_p(M)$ as the set of germs of $C^\infty$-functions ...
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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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### How to show that the tangent space as defined by velocity of curves matces intuitive tangent idea?

I'm reading: https://en.wikipedia.org/wiki/Tangent_space Specifically, "Definition as the velocity of curves" and the definition of tangent space at a point as the set of all tangent vectors of ...
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### Tangent Spaces (Algebraic Geometry)

I'm in my algebraic geometry class and I have the definition of the space tangent to some variety $W=V(F_1,F_2,...)$ as the degree-1 components of $F_1,F_2,...$ . We then introduce the differential at ...
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### Tangent space $T_q(df(M))$ as a subspace of $T_q(T^*M)$

I have been asked to describe the tangents space $T_q(df(M))$ as a subspace of $T_q(T^*M)$ where $f\in C^\infty(M)$ and $df$ is a 1-form (or smooth section of $T^*M$). Here, $df:M\rightarrow T^*M$ ...
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### Tangent space of smooth manifold $M=\{(x,x^3,e^{x-1}) : x \in \Bbb{R}\}$ at $(1,1,1)$

What's the tangent space of $M=\{(x,x^3,e^{x-1}): x \in \Bbb{R}\}$ at the point $(1,1,1)$, where $M$ is a manifold of smoothness $C^\infty$. I know how to find the tangent space of a manifold in the ...
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### Need help understanding differential of function

I have encountered the term differential/pushforward many times in the literature, although I cannot seem to understand just what is meant by it. I still cannot seem to understand the definition of ...
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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. ...
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### Formula for tangent map, proof check

Consider smooth map $f:\mathbb{R}^{n}\to \mathbb{R}$, let $a \in \mathbb{R}^{n}$ be any point, $X \in T_{a}\mathbb{R}^{n}=\mathbb{R}^{n}$ be tangent vector at point $a$. I have probably proven the ...
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### How does this equality come between a point in $\mathbb R^n$ and combination of operators?

My Doubt:- I understood the proof of $T_p(\mathbb R^n)\simeq \mathcal{D}_p(\mathbb R^n)$. $T_p(\mathbb R^n)$ is a space consists of elements from $\mathbb R^n$. $\mathcal{D}_p(\mathbb R^n)$ is a ...
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### Show that if $f$ is constant on a manifold $M$ then $\nabla f$ is orthogonal to the tangent space of each $x \in M$

Let $M \subset R^n$ be a $k$ dimensional manifold. Let $f: R^n \to R$ be a smooth function that satisfies $f(x) = c$, $c \in R$ for every $x \in M$. I need to prove that $\nabla f(x)$ is orthogonal ...
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### KIlling vectors from isometries and orbit spaces

I am currently (trying) to learn more about orbit spaces generated from an isometry group of a manifold. I cannot quite pinpoint what I (don't) understand, so I will try to lay out what I could gather:...
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### Intersection number via tangent spaces

Assume that finite groups $G_1$ and $G_2$ act smoothly on a manifold $M$ in such a way that the fixed point set, $M^{G_1\cap G_2}$, is an oriented closed manifold, $M^{G_1}$ and $M^{G_2}$ are its ...
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### Finding equation of tangent of a vector

hi guys and thanks in advance. I encounter this problem 21 and i couldnt solve it. my answer is so wrong.can anyone guide me on this?
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### Dimension of relative tangent space

Let $k$ be an arbitrary field, $X$ be a $k$-Scheme locally of finite type, $x \in X$ a closed point and $\kappa(x)$ its residue field. Question: Is the dimension of of the tangent space $T_x X$ ...
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### Formula for tangent plane to surface given by parametrization

I am aware of how to find an equation of the tangent place to a surface that is given as the graph of a function $z = g(x,y)$. Here one finds a normal vector by essentially taking the partial ...
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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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### Are the spaces $T_pM$ and $\mathbb R^n$ homeomorphic?

Let $T_pM$ be the tangent space at a point $p$ in a n-dimensional smooth manifold $M$. In addition, if we assume $(M,g)$ as a smooth Riemannian manifold, then $T_pM$ is a n-dimensional real normed-...
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### Distance of Manifold and a vector on its tangent space from it.

Let $M \subset R^n$ be a manifold and $x \in M$ a point on it. I want to prove that if $h \in T_xM$ (the tangent space at x) then for every $\epsilon$: $\text{distance}(x+\epsilon h) = o(\epsilon)$, ...
### Consider the real-valued function $M:=\{(x,y,z)| (2 - (x^2 + y^2)^{1/2})^2 + z^2=1\}$ defined on $\mathbb{R}^3-\{(0, 0, z)\}$.
Show that the manifold $N=\{(x,y,z)\in \mathbb{R}^3|x^2+y^2= 4\}$ is transverse to M. Identify the resulting manifold $N\cap M$. My Attempt: Pardon me for something vacuous, as I am a beginner in ...
### Calculating tangent vector of curve s(P,$\alpha$) at given point $\alpha$ = 0. http://yann.lecun.com/exdb/publis/pdf/simard-00.pdf
I am reading one chapter where tangent vector is calculated for the given curve $s(P,\alpha)$ at $\alpha=0$ by differentiating with respect to $\alpha$; $\frac{\partial s(P,\alpha)}{\partial\alpha}$. ...