Let r r 1 0 lc r3 s r 1 s 59 be the induced

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ature of a curve Let C be given by a parametrization: r : [a, b] → R3 , t → r (t). Let R = r ◦ ψ −1 : [0, L(C ] → R3 , s → r ◦ ψ −1 (s) 59 be the induced new parametrization defined at the end of last section, where Ψ(t) = t r ′ (u) du. . Then we define the curvature for C a κ := More precisely, κ(R(s)) = d2 R dt2 d2 R . ds2 (7) (s) for any s ∈ [0, L(C )]. Proposition 13.6 κ(t) = More precisely, κ(r(t)) = Proof: T ′ (t) r ′ (t) T ′ (t) . r ′ (t) for any t ∈ [a, b]. In fact, by the chain rule d (r ◦ ψ − 1 ) dr dψ −1 1 dR = = · =r ′· = T. ds ds dt ds r′ Here we have used dψ−1 ds = (ψ −1 )′ = 1 r′ was at the end of the last section. Then d2 R d d dR d T ◦ ψ −1 = = T= 2 ds ds ds ds ds = dT dψ −1 1 · =T ′· . dt ds r′ The curvature κ is independent of choice of parametrization. [Example](Lines) Consider a line C : x = x0 + d1 t, y = y0 + d2 t, z = z0 + d3 t. Write r (t) = (x0 + d1 t) i + (y0 + d2 t) j + (z0 + d...
View Full Document

This note was uploaded on 10/02/2012 for the course CALCULUS 2433 taught by Professor Shanyuji during the Fall '12 term at University of Houston.

Ask a homework question - tutors are online