曲率知识总结

Introduction

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弧微分

  设函数f(x)在区间(a,b)内具有连续导数,图形如图2.1中$\widehat{AB}$所示.

图2.1
  现在存在一段弧s, $s(x)=\widehat{AM}$.则曲线s从M到M’点关于x的变化率为 $\frac{ds}{dx}$:

  $\frac{ds}{dx}=\frac{\widehat{M{M}^{\prime }}}{\Delta x}=\frac{\widehat{M{M}^{\prime }}}{|M{M}^{\prime }|}\ast \frac{|M{M}^{\prime }|}{\Delta x}=\frac{\widehat{M{M}^{\prime }}}{|M{M}^{\prime }|}\ast \frac{\sqrt{(\Delta x{)}^2+(\Delta y{)}^2}}{\Delta x}$

  因为${\lim }_{\Delta x\to 0}\frac{\widehat{M{M}^{\prime }}}{|M{M}^{\prime }|}=1$

  所以 lim ⁡ Δ x → 0 M M ′ ^ ∣ M M ′ ∣ ∗ ( Δ x ) 2 + ( Δ y ) 2 Δ x = ( Δ x ) 2 + ( Δ y ) 2 Δ x = 1 + ( Δ y Δ x ) 2 \lim_{\Delta x \to 0}\frac{\widehat{MM'}}{|MM'|} * \frac{\sqrt{(\Delta x)^2 + (\Delta y)^2}}{\Delta x} = \frac{\sqrt{(\Delta x)^2 + (\Delta y)^2}}{\Delta x}=\sqrt{1+(\frac{\Delta y}{\Delta x})^2} limΔx→0​∣MM′∣MM