全部书籍

1 Primary & Middle School Mathematics1–9 年级在线 2 High School Mathematics10–12 年级在线 3 University Mathematics — Year 1本科一年级在线 4 University Mathematics — Year 2本科二年级在线 5 University Mathematics — Year 3本科三年级在线

1 Primary & Middle School Physics1–9 年级在线 2 High School Physics10–12 年级在线 3 University Physics — Year 1本科一年级在线 4 University Physics — Year 2本科二年级在线 5 University Physics — Year 3本科三年级在线

1 School Chemistry — Grades 1 to 121–12 年级在线 2 University Chemistry — Year 1本科一年级在线 3 University Chemistry — Year 2本科二年级在线 4 University Chemistry — Year 3本科三年级在线

1 Primary & Middle School Biology1–9 年级在线 2 High School Biology10–12 年级在线 3 University Biology — Year 1本科一年级在线 4 University Biology — Year 2本科二年级在线 5 University Biology — Year 3本科三年级在线

1 Primary & Middle School Earth and Space Sciences1–9 年级在线 2 High School Earth and Space Sciences10–12 年级在线 3 University Earth and Space Sciences — Year 1本科一年级在线 4 University Earth and Space Sciences — Year 2本科二年级在线 5 University Earth and Space Sciences — Year 3本科三年级在线

1 Markets I: The Ecosystem and Exchange-Traded Markets市场在线 2 Markets II: Rates, FX and Credit市场在线 3 Markets III: Commodities, Energy and Crypto市场在线 4 Quantitative Methods方法在线 5 Derivatives and Volatility衍生品在线 6 Rates, Credit, XVA and Risk利率、信用与风险在线 7 Research Craft: Predictors, Backtests, Measurement, Portfolios研究在线 8 Strategies I: Equities and Futures策略在线 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desks策略在线 10 Microstructure and Execution交易执行在线 11 Market Making and High-Frequency Trading做市在线 12 Machine Learning for Markets机器学习在线 13 Low-Latency Software技术在线 14 Networks, Hardware and Trading Infrastructure技术在线 15 Research, Data and Risk Platforms技术在线 16 The Desk and the Firm公司运营在线 17 The Industry: Firms, Roles and Careers职业发展在线 18 Interviews I: Maths, Probability and Speed职业发展在线 19 Interviews II: Markets, Trading Games and the Final Round职业发展在线 20 Interviews III: Code职业发展在线

全部游戏 Math Fuse Word Reactor Atlas Dash

应用关于

Coach

English Français Español Nederlands Português हिन्दी العربية Bahasa Indonesia 简体中文 বাংলা

登录开始阅读

书籍暂未推出简体中文版——当前显示英文版。

Mathematics · 术语表

什么是 alternating form？

Definition 2.13 University Mathematics — Year 2 · 第 2 章 — Linear Algebra

A map $f \colon E^n \to K$ is $n$ -linear when it is linear in each variable, and alternating when it vanishes whenever two arguments are equal. Alternating implies antisymmetric: swapping two arguments changes the sign (expand $f(\dots, x + y, \dots, x + y, \dots) = 0$ ); more generally, for $\sigma \in \mathfrak{S}_n$ ,

f(x_{\sigma(1)}, \dots, x_{\sigma(n)}) = \varepsilon(\sigma)\, f(x_1, \dots, x_n),

by decomposing $\sigma$ into transpositions (Theorem 1.21).

示例

Example 2.15 (Sarrus, derived and demolished)

For $n = 3$ the permutation formula has exactly $3! = 6$ terms. Listing $\mathfrak{S}_3$ by signature — $\mathrm{id}$ , $(1\,2\,3)$ , $(1\,3\,2)$ even; $(1\,2)$ , $(1\,3)$ , $(2\,3)$ odd — gives

\det A = a_{11}a_{22}a_{33} + a_{21}a_{32}a_{13} + a_{31}a_{12}a_{23} - a_{21}a_{12}a_{33} - a_{31}a_{22}a_{13} - a_{11}a_{32}a_{23} :

precisely the “diagonals” rule of Sarrus taught in school — now a theorem, with the mysterious signs identified as signatures. The demolition: for $n = 4$ there are $24$ permutations, of which only $8$ are picked up by any diagonal-drawing scheme; Sarrus has no degree- $4$ version, and cofactor expansion (Theorem 2.17 (4)) takes over. Counting terms is also a warning: the permutation formula has $n!$ summands, so it is a definition, not an algorithm — row reduction computes $\det$ in $O(n^3)$ operations instead.

在原文中阅读 →

Definition 21.1 University Mathematics — Year 3 · 第 21 章 — Differential Forms and Stokes’ Theorem

Let $E$ be a real vector space of dimension $n$ . A $k$ -linear alternating form on $E$ is a map $\alpha\colon E^k \to \R$ , linear in each variable, with $\alpha(v_1, \dots, v_k) = 0$ whenever two arguments are equal. Their space is written $\Lambda^k E^*$ ; by convention $\Lambda^0E^* = \R$ . Alternation forces antisymmetry: exchanging two arguments changes the sign (expand $\alpha(\dots, v + w, \dots, v + w, \dots) = 0$ ), and more generally $\alpha(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = \varepsilon(\sigma)\,\alpha(v_1, \dots, v_k)$ for every permutation $\sigma$ .

在原文中阅读 →