# Questions tagged [tangent-spaces]

This tag is for questions regarding to the tangent space, the linear space that best approximates an object at a given point. Intuitively, the tangent space $T_p(M)$ at a point $p$ on an $n$-dimensional manifold $M$ is an $n$-dimensional hyperplane in ${\mathbb{R}}^m$ that best approximates $M$ around $p$, when the hyperplane origin is translated to $p$.

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### Show that $T_{(p,q)}(M\times N)\cong T_p M\oplus T_q N$. Tangent space to the product manifold.

I want to check my arguments for the following proof. Let $M$ and $N$ be smooth manifolds where $\dim M=n$, $\dim N = m$, and $\pi_1:M\times N\to M$, $\pi_2:M\times N\to N$ be corresponding projective ...
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### Is it possible to define a vector field of non-coordinate basis vectors?

I'm studying non-coordinate basis of (pseudo-)riemannian manifolds and I found the following definition from Nakahara - Geometry, topology and physics: a non-coordinate basis $\{\hat{e}_\alpha\}$ is ...
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### How to show that $T_{(1,0)}\mathbb S^1 \cong \operatorname{span}(\{e_2\})$?

I want to show that $T_{(1,0)}\mathbb S^1 \cong \operatorname{span}(\{e_2\})$ using the stereographic chart and using the definition that $T_xM$ is the set of velocity vectors $v$ where each vector $v$...
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### Material derivative on moving boundary

Let $T_t$ be a $C^1$-diffeomorphism on $\mathbb R^d$, $$v_0(x):=\left.\frac{\rm d}{{\rm d}t}T_t(x)\right|_{t=0}\;\;\;\text{for }x\in\mathbb R^d,$$ $\Omega$ be a $k$-dimensional embedded $C^1$-...
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### Why is it important the manifold has codimension $1$ in order to prove this identity for $\operatorname{div}fV$ on $\partial M$?

I've seen the following claim in some lectures notes which let me think that I might have a major misunderstanding: The claim is that if $M$ is an embedded submanifold of $\mathbb R^d$ with boundary ...
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### Show an identity for the Laplace-Beltrami operator

Let $\partial M$ denote the boundary of a $k$-dimensional embedded $C^1$-submanifold $M$ of $\mathbb R^d$, $T_x(\partial M)$ and $N_x(\partial M)$ denote the tangent and normal field of $\partial M$ ...