Curvature with vector functions

Curvature with vector functions

Lessons

Notes:

Finding Curvature
The curvature is a way to measure how fast the vector curve r(t)r(t) is changing direction from a point PP.

The formal definition of a curvature is:

κ=dTds\kappa = |\frac{dT}{ds}|

Which is the absolute value of the derivative of unit tangent vector T(t)T(t) in terms of the arc length ss.

We will not be using this formula since it's very complicating to use. Instead, we will be using these two formulas to calculate κ\kappa.

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}

  • Introduction
    Arc Length with Vector Functions Overview:
    a)
    Formal Definition of Curvature
    • What is a Curvature?
    • Formal Formula to calculate Curvature

    b)
    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 }

    c)
    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}


    • 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.

    • 2.
      Determine the curvature of the vector function r(t)=<2t,4t2,13t3>r(t)= \lt 2t, 4t^2, \frac{1}{3}t^3\gt .

    • 3.
      Determine the curvature of the vector function r(t)=<e2t,2e2t,3>r(t)= \lt e^{2t}, 2e^{2t}, 3\gt.

    • 4.
      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.

    • 5.
      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}} }