Vector Functions: Arc Length and Curvature

mes (67)in DTube • 11 hours ago

In this video, I go over further into vector functions and this time do a deep dive into the arc length and curvature of a space curve. The arc length for a space curve is calculated in much the same way as for normal plane curves, by approximating the curve with polygons and taking the limit as they approach infinity, but we now have to include the z component. The curvature of a function is defined as the magnitude or length of the derivative of the unit tangent vector in terms of the arc length. The curvature describes how curved or "smooth" a space curve is. I also go over normal, binormal vectors, as well as the TNB frame, normal & osculating planes, and much more, so buckle up!

#math #vectors #calculus #arclength #curvature

Timestamps:

Intro – 0:00
Calculus Book reference – 0:57
Calculus book chapter – 1:17
Topics to cover – 1:50
Arc Length – 3:02
- Example 1: Arc Length of Circular Helix: https://www.geogebra.org/3d/gbvmtdug – 17:19
Parameterizations and Arc Length Function – 24:55
- Example 2: Arc Length Function of a Helix – 38:05
Curvature – 43:50
- Definition of Curvature – 49:36
- Example 3: Curvature of a Circle is 1/radius – 57:46
Theorem: Curvature Cross Product Formula – 1:10:02
- Example 4: Curvature of a Twisted Cubic – 1:29:26
- Curvature of a Plane Curve y = f(x) – 1:39:46
- Example 5: Curvature of a Parabola: https://www.desmos.com/calculator/xfmwdwircp – 1:52:48
The Normal and Binormal Vectors – 2:02:25
- Example 6: Unit Normal and Binormal Vectors of a Circular Helix: https://www.geogebra.org/3d/wyr5mysb – 2:09:42
TNB Frame: Normal and Osculating Planes – 2:30:44
- Example 7: Normal and Osculating Plane of the Circular Helix – 2:33:41
- Osculating Circle – 2:52:25
- Example 8: Osculating Circle of a Parabola: https://www.desmos.com/calculator/tzb5bjsxkl – 2:53:54
Summary of T, N, B, and κ Formulas: 3:06:08
Outro – 3:10:28