Curvature with vector functions

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Intros
Lessons
  1. Arc Length with Vector Functions Overview:
  2. Formal Definition of Curvature
    • What is a Curvature?
    • Formal Formula to calculate Curvature
  3. 2 Alternate Formulas for Curvature
    • κ=T(t)r(t) \kappa = \frac{||T'(t)||}{||r'(t)||}
    • κ=r(t)×r(t)r(t)3 \kappa = \frac{||r'(t) \times r''(t) || } {|| r'(t)||^3 }
  4. Examples of using the 2 formulas
    • Using the first formula κ=T(t)r(t)\kappa = \frac{||T'(t)||}{||r'(t)||}
    • Using the second formula κ=r(t)×r(t)r(t)3 \kappa = \frac{||r'(t) \times r''(t)||}{||r'(t)||^3}
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    Examples
    Lessons
    1. Finding Curvature using the alternate formulas
      Determine the curvature of the vector function r(t)=<sin2t,cos2t,t>. r(t)= \lt \sin 2t, \cos 2t, t\gt.
      1. Determine the curvature of the vector function r(t)=<2t,4t2,13t3>r(t)= \lt 2t, 4t^2, \frac{1}{3}t^3\gt .
        1. Determine the curvature of the vector function r(t)=<e2t,2e2t,3>r(t)= \lt e^{2t}, 2e^{2t}, 3\gt.
          1. Determine of the curvature of the vector function r(t)=<acost,asint,1>r(t)= \lt a \cos t, a \sin t, 1\gt, where aa is a constant.
            1. Deriving the Curvature formula with y=f(x)y=f(x)
              Suppose we have y=f(x)y=f(x). Use the formula κ=r(t)×r(t)r(t)3 \kappa = \frac{|| r'(t) \times r''(t) ||}{|| r'(t)||^3} to derive the new curvature equation:

              κ=f(x)(1+[f(x)]2)32 \kappa = \frac{ | f''(x) | } {(1+[f'(x)]^2 )^{\frac{3}{2}} }