Curvature of a Plane Curve y = f(x) + Example on a Parabola

mes (67)in DTube • 2 days ago

In this video, I go over the curvature of the special case of a plane curve with equation y = f(x), by selecting x as the parameter and using our earlier cross product curvature formula. The magnitude of the cross product simplifies due to the standard basis vectors canceling out, giving just |f''(x)| or the absolute value of the second derivative of f. The length of the second derivative of the vector function is just the arc length of a plane curve, thus further simplifying the final curvature formula. I illustrate this with obtaining the curvature of a parabola y = x^2. The curvature is max at the origin, and flattens out to zero at infinity as expected. I also graph it out in the amazing Desmos 2D graphing calculator: https://www.desmos.com/calculator/xfmwdwircp

#math #vectors #calculus #desmos #curvature

Timestamps:

Curvature of a plane curve: choose x as the parameter – 0:00
Cross product of standard basis vectors – 2:00
Cross product of first two derivatives of the vector function – 4:08
Length of the cross product is absolute value of f''(x) – 5:52
Length of the tangent vector – 7:33
Curvature formula for a plane curve – 9:41
Recap on standard basis vectors – 11:37
Example 5: Curvature of a parabola y = x^2 – 12:58
- Curvature at multiple points – 15:28
- Curvature approaches zero as x goes to infinity – 18:20
- Graph of the curvature – 19:33
- Graphing curvature in Desmos: https://www.desmos.com/calculator/xfmwdwircp – 21:57