# Curvature with vector functions

0/3
##### 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
• $\kappa = \frac{||T'(t)||}{||r'(t)||}$
• $\kappa = \frac{||r'(t) \times r''(t) || } {|| r'(t)||^3 }$
4. Examples of using the 2 formulas
• Using the first formula $\kappa = \frac{||T'(t)||}{||r'(t)||}$
• Using the second formula $\kappa = \frac{||r'(t) \times r''(t)||}{||r'(t)||^3}$
0/5
##### Examples
###### Lessons
1. Finding Curvature using the alternate formulas
Determine the curvature of the vector function $r(t)= \lt \sin 2t, \cos 2t, t\gt.$
1. Determine the curvature of the vector function $r(t)= \lt 2t, 4t^2, \frac{1}{3}t^3\gt$.
1. Determine the curvature of the vector function $r(t)= \lt e^{2t}, 2e^{2t}, 3\gt$.
1. Determine of the curvature of the vector function $r(t)= \lt a \cos t, a \sin t, 1\gt$, where $a$ is a constant.
1. 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:

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