exterior derivative

外微分运算

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

Translation evidence: Literature usage

Meaning in context

The degree-one real linear operator on smooth differential forms that agrees with the differential on functions, squares to zero, and satisfies the graded product rule; in local coordinates it differentiates each coefficient and wedges its differential with the coordinate differentials.

光滑微分形式上的升一次实线性算子，在函数上等于通常的微分，平方为零并满足分次乘积法则；在局部坐标中对系数微分，再与坐标微分取外积。

Example

For $\alpha=x\,dy$ on $\mathbb R^2$, $d\alpha=dx\wedge dy$ and $d(d\alpha)=0$. Also $d(xy)=y\,dx+x\,dy$.

Usage and distinctions

This operation does not require a metric or a connection. It raises form degree, while a covariant derivative produces an additional covector slot without imposing alternation.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Liviu I. Nicolaescu — Lectures on the Geometry of Manifolds§3.2.1, Proposition 3.2.3 and coordinate construction of d · Checked 2026-10-14

Translation sources

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

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