smooth vector field

光滑向量场

Geometry & Topology · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

A smooth section of the tangent bundle of a smooth manifold: it assigns to each point a tangent vector at that point, with smooth coefficient functions in every coordinate chart.

光滑流形切丛的光滑截面，即在每一点选取该点的切向量，且在每个局部坐标中各系数函数都光滑。

Example

On $\mathbb R^2$, $X=-y\partial_x+x\partial_y$ is smooth and tangent to each centered circle since $X(x^2+y^2)=0$.

Usage and distinctions

A vector field assigns vectors at all points, whereas a tangent vector belongs to one point. Smoothness is stronger than continuity of a section.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§2.1.4, Definition 2.1.27(b) · Checked 2026-10-14

Translation sources

1. Zhang Sirong — Beihang University, 微分流形与黎曼几何, Chapter 3 slidesChapter 3，切向量场，slide 12/54: 特别光滑映射是光滑向量场; PDF p. 3 · Checked 2026-10-14

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