tangent space

切空间

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Translation evidence: Literature usage

Meaning in context

The tangent space at an interior point of a smooth manifold, or on a manifold without boundary, is the real vector space of velocities of smooth two-sided curves through that point, identifying curves with equal coordinate derivatives at time zero. At a boundary point it is the full vector space of real-linear derivations on smooth real-valued function germs, obeying the product rule at that point and using smooth extensions of half-space charts, rather than the cone of inward velocities.

在无边光滑流形上一点或带边流形的内点，切空间为经过该点的双侧光滑曲线速度所成的实向量空间，坐标中的零时刻导数相同的曲线等价。在边界点，采用光滑实值函数芽上满足该点乘积法则的实线性导子，借助半空间坐标的光滑延拓得到完整切向量空间，而非仅有朝内速度的锥。

Example

For S2⊂R3S^2\subset\mathbb R^3, TpS2={v:p⋅v=0}T_pS^2=\{v:p\cdot v=0\}; at p=(0,0,1)p=(0,0,1) this is the plane v3=0v_3=0.

Usage and distinctions

The tangent space belongs to one point and is an abstract vector space. A chosen embedding represents it in an ambient space; that representation is additional data.

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Definition sources

  1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§2.1.1, Definitions 2.1.2 and 2.1.4 · Checked 2026-10-14

Translation sources

  1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要二、向量场：切空间；PDF p. 77, printed p. 74 · Checked 2026-10-14
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