V2EX slope field

Slope Field

释义 Definition

slope field(斜率场/方向场):在微分方程中,用许多小线段在平面上表示每个点处解曲线的斜率(方向)的图形工具,帮助直观理解解的走势。(也常叫 direction field

发音 Pronunciation (IPA)

/slop fild/

例句 Examples

We used a slope field to sketch the solution.
我们用斜率场来勾画解的图像。

By analyzing the slope field of \(y' = y - x\), you can predict how solution curves behave even without solving the equation explicitly.
通过分析 \(y' = y - x\) 的斜率场,即使不显式求解方程,也能预测解曲线将如何变化。

词源 Etymology

slope 来自表示“倾斜”的含义,指曲线在某点的“斜率”;field 在数学中常指“在空间中每一点都赋予一个量”的“场”。合起来 slope field 就是“在平面各点给出斜率(方向)的一种场”。

相关词 Related Words

文学与著作 Literary Works

  • Calculus: Early Transcendentals(James Stewart)微分方程章节常用 slope field/direction field 进行定性分析
  • Thomas’ Calculus(George B. Thomas 等)以斜率场介绍解曲线族与初值问题的直观方法
  • Differential Equations with Boundary-Value Problems(Dennis G. Zill)用方向场/斜率场帮助理解一阶微分方程解的行为
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