differential form

外微分形式

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Translation evidence: Literature usage

Meaning in context

For a smooth manifold and a nonnegative integer k, a smooth section of the kth exterior power of its cotangent bundle; pointwise it is an alternating real multilinear function of k tangent vectors.

在光滑流形上，余切丛的第 kk 次外幂的光滑截面，kk 为非负整数；在每一点，它是关于 kk 个切向量的交替实多线性函数。

Example

On R2\mathbb R^2, α=x dy\alpha=x\,dy satisfies α(2,3)(∂y)=2\alpha_{(2,3)}(\partial_y)=2; dx∧dy(∂y,∂x)=−1dx\wedge dy(\partial_y,\partial_x)=-1.

Usage and distinctions

A differential form is alternating; a general covariant tensor need not be. The phrase exterior form does not assert that the form is an exterior derivative.

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Definition sources

  1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§2.3.2, Definition 2.3.4(b) and Example 2.3.5 · Checked 2026-10-14

Translation sources

  1. Peking University — 2022 mathematics teaching plans课程号00130190，内容提要三、外微分形式；PDF p. 77, printed p. 74 · Checked 2026-10-14
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