V2EX implicit curve

Implicit Curve

释义 Definition

隐式曲线:在平面上由一个方程 F(x, y) = 0(而不是显式形式 *y = f(x)*)所定义的曲线。它常用于描述无法(或不方便)直接写成“y 关于 x 的函数”的曲线,例如圆、椭圆、双曲线等。

发音 Pronunciation (IPA)

/mplst kv/

例句 Examples

An implicit curve can be defined by an equation like \(x^2 + y^2 - 1 = 0\).
隐式曲线可以由像 \(x^2 + y^2 - 1 = 0\) 这样的方程来定义。

In multivariable calculus, we often study an implicit curve by differentiating \(F(x,y)=0\) to find the slope without solving for \(y\).
在多元微积分中,我们常通过对 \(F(x,y)=0\) 做隐式求导来研究隐式曲线,从而不必先解出 \(y\) 也能得到斜率。

词源 Etymology

implicit 源自拉丁语 implicitus,有“卷在里面的、含在其中的”之意,引申为“未直接说出但被包含在内的”;在数学里表示“变量关系被方程整体隐含”。curve 源自拉丁语 curvus(弯曲的),指“曲线”。合起来 implicit curve 就是“由方程隐含给出的曲线”。

相关词 Related Words

文学与典籍 Literary Works

  • James Stewart, Calculus(微积分教材中常用“implicit curve/implicit function”讨论隐式求导与曲线几何)
  • Walter Rudin, Principles of Mathematical Analysis(在多元分析与函数理论相关语境中出现)
  • Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces(曲线与曲面理论中涉及由方程定义的几何对象)
  • William Fulton, Algebraic Curves(代数曲线以方程 \(F(x,y)=0\) 为核心表述,常与“implicit”语境相连)
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