Curvature with vector functions

Notes:

Finding Curvature
The curvature is a way to measure how fast the vector curve

r(t)

is changing direction from a point

P

.

The formal definition of a curvature is:

$\kappa = |\frac{dT}{ds}|$

Which is the absolute value of the derivative of unit tangent vector

T(t)

in terms of the arc length

s

.

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

\kappa

$\kappa = \frac{||T'(t)||}{||r'(t)||}$
$\kappa = \frac{||r'(t) \times r''(t)||}{||r'(t)||^3}$

1.
Finding Curvature using the alternate formulas
Determine the curvature of the vector function $r(t)= \lt \sin 2t, \cos 2t, t\gt.$

2.
Determine the curvature of the vector function $r(t)= \lt 2t, 4t^2, \frac{1}{3}t^3\gt$ .

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

4.
Determine of the curvature of the vector function $r(t)= \lt a \cos t, a \sin t, 1\gt$ , where $a$ is a constant.

5.
Deriving the Curvature formula with $y=f(x)$
Suppose we have $y=f(x)$ . Use the formula $\kappa = \frac{|| r'(t) \times r''(t) ||}{|| r'(t)||^3}$ to derive the new curvature equation:
$\kappa = \frac{ | f''(x) | } {(1+[f'(x)]^2 )^{\frac{3}{2}} }$