# Curvature with vector functions

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##### Intros

###### Lessons

**Arc Length with Vector Functions Overview:**__Formal Definition of Curvature__

- What is a Curvature?
- Formal Formula to calculate Curvature

__2 Alternate Formulas for Curvature__

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

__Examples of using the 2 formulas__- Using the first formula $\kappa = \frac{||T'(t)||}{||r'(t)||}$
- Using the second formula $\kappa = \frac{||r'(t) \times r''(t)||}{||r'(t)||^3}$

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##### Examples

###### Lessons

**Finding Curvature using the alternate formulas**

Determine the curvature of the vector function $r(t)= \lt \sin 2t, \cos 2t, t\gt.$- Determine the curvature of the vector function $r(t)= \lt 2t, 4t^2, \frac{1}{3}t^3\gt$.
- Determine the curvature of the vector function $r(t)= \lt e^{2t}, 2e^{2t}, 3\gt$.
- Determine of the curvature of the vector function $r(t)= \lt a \cos t, a \sin t, 1\gt$, where $a$ is a constant.
**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:$\kappa = \frac{ | f''(x) | } {(1+[f'(x)]^2 )^{\frac{3}{2}} }$

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###### Topic Notes

__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 $\kappa$.

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

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

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