V2EX pushforward

Pushforward

释义 / Definition

pushforward(名词,数学/几何/测度论)指在一个映射 \(f: X \to Y\) 下,把 \(X\) 上的对象“推到” \(Y\) 上得到的对应对象;最常见的是:

  • 测度的推前(像测度):把 \(X\) 上的测度 \(\mu\) 通过 \(f\) 变成 \(Y\) 上的测度 \(f_*\mu\)。
  • 在微分几何、代数几何中也常指 (切向量场、微分形式、层等的)推前
    (不同领域定义形式略有差别,但核心都是“沿着映射把结构转移到目标空间”。)

发音 / Pronunciation (IPA)

/pfrwrd/

例句 / Examples

The pushforward of a probability measure under this map gives a new distribution.
这个映射下概率测度的推前会给出一个新的分布。

In measure theory, the pushforward \(f_*\mu\) is defined by \(f_*\mu(A)=\mu(f^{-1}(A))\) for measurable sets \(A\).
在测度论中,推前 \(f_*\mu\) 定义为:对任意可测集合 \(A\),有 \(f_*\mu(A)=\mu(f^{-1}(A))\)。

词源 / Etymology

pushforwardpush(推动)+ forward(向前)构成,是一种“直观命名”:示把某个空间上的结构沿映射“往前推到”目标空间上。数学符号里常写作 \(f_*\)(读作 “f star” 或 “f pushforward”),与“pullback(拉回,\(f^*\))”成对出现。

相关词 / Related Words

文学与经典著作中的用例 / Notable Works

  • John M. Lee, Introduction to Smooth Manifolds(讨论光滑映射下的推前/拉回等概念)
  • Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications(涉及测度的推前/像测度思想)
  • Robin Hartshorne, Algebraic Geometry(大量使用 \(f_*\) 表示层的推前)
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