# Then x 2 y 2 32 x y x y x 2 y 2 t x

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Unformatted text preview: ′′) i 2 1 + y ′′((x′ )2 + (y ′ )2 )−1/2 − y ′ ((x′ )2 + (y ′)2 )−3/2 (2x′ x′′ + 2y ′y ′′ ) j 2 = ((x′ )2 + (y ′ )2 )−3/2 x′′ ((x′ )2 + (y ′ )2 ) − x′ (x′ x′′ + y ′ y ′′) i +((x′ )2 + (y ′ )2 )−3/2 y ′′ ((x′ )2 + (y ′)2 ) − y ′ (x′ x′′ + y ′y ′′ ) j = ((x′ )2 + (y ′ )2 )−3/2 x′′ (y ′ )2 − x′ y ′ y ′′) i +((x′ )2 + (y ′ )2 )−3/2 y ′′ (x′ )2 − y ′x′ x′′ j = ((x′ )2 + (y ′ )2 )−3/2 (x′′ y ′ − x′ y ′′)(y ′ i − x′ j ). Then κ= ((x′ )2 + (y ′ )2 )−3/2 |(x′′ y ′ − x′ y ′′ )| (x′ )2 + (y ′ )2 T′ |x′′ y ′ − x′ y ′′| = = r′ ((x′ )2 + (y ′ )2 )3/2 (x′ )2 + (y ′)2 61 [Example](Circles) Consider a circle C : x = R cos t, y = R sin t, 0 ≤ t ≤ 2π. Proof: Since x(t) = R cos t, y (t) = R sin t, we have x′ = −R sin t,...
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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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