# Curvature with vector functions

### Curvature with vector functions

#### Lessons

Notes:

Finding Curvature
The curvature is a way to measure how fast the vector curve $r(t)$ 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 $\kappa$.

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

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

c)
Examples of using the 2 formulas
• Using the first formula $\kappa = \frac{||T'(t)||}{||r'(t)||}$
• Using the second formula $\kappa = \frac{||r'(t) \times r''(t)||}{||r'(t)||^3}$

• 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:

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