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

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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...
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## 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.

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