Curvature Cross Product Formula: Proof and Example on Twisted Cubic

In this video, I derive a more convenient formula for the curvature of a space curve, which involves the cross product of the first two derivatives of the vector function. I derive this using the fact that the unit tangent vector has a constant length of 1 (i.e. hence the term "unit"), thus it is perpendicular to its derivative, allowing us to simplify the length of the cross product. The new formula states the curvature is equal to the length or magnitude of the cross product of the first two derivatives of the vector function divided by the length of the tangent vector cubed. I illustrate this with an example on finding the curvature of the twisted cubic space curve that I had covered prior.

#math #vectors #calculus #crossproduct #curvature

Timestamps:

Theorem: Curvature Cross Product Formula – 0:00
Proof: Obtain our unit tangent vector and derivatives of tangent vector – 1:07
- Recap on product rule for vector functions – 4:43
- Recall that cross product of a vector with itself is the 0 vector – 5:30
- Cross product of derivatives of tangent vector – 6:52
- Recall that tangent vector is perpendicular to a vector function of constant length – 10:28
- Recall the length of a cross product formula – 12:05
- Length of the cross product of derivatives of the tangent vector – 12:42
- Rearrange to get length of the derivative of the unit tangent vector – 17:10
- Plug results into the earlier formula for the curvature – 17:54
Example 4: Curvature of a Twisted Cubic – 19:24
- Cross product of the tangent vector of the derivatives – 22:20
- Plugging out results yields the curvature of a twisted cubic – 27:35
- At the origin, the curvature is 2 – 28:21