In Lesson 7 "Vector Functions and Space Curves", Tim mentions torque with the "Twisted Cubic" curve. This is a discussion of the twisting forces at a particular place in the curve. It was a casual reference, and he didn't go into details.
In the YouTube video: "Torsion: How curves twist in space, and the TNB or Frenet Frame", Mathematics Professor Dr. Trefor Bazett provides a few more details. The torsional vector of a space curve is the binormal vector: the cross product of the tangent vector and the normal vector. Look at around 3:10 into Trefor's video if you'd like to see it graphically. While I consider myself slow to understand the "big picture" of multivariable stuff, this image and calculation makes perfect sense to me. I'm sure Arben, Luke, and the other Wolfram staff are familiar with the concept of the double-normal vector of a curve and its significance.
Why does this mathematical torque matter? It matters because it corresponds to physical torque for objects following a particular curve! In a lateral jump-rope movement called "RMT Ropes" (AKA "Flow Ropes"), the midline of the rope follows Viviani's Curve (with the rope criss-crossing over the rope-jumper's head). The rope's torque is obviously present and the front-most and rear-most movements of the rope. The frontmost torque is rope-over-the-top; the rearmost is rope-under-the-bottom. I'd felt this for many years of dragon-rolling; it was #!$$ exciting to see that the geometry of space-curves notes the same torsional forces. The rope-forces are noteworthy because they reflect the torsional forces in the musculoskeletal network of the human body. The math gives us insight how the rope moves and forces exist; the rope gives us insight in how a body driving those movements must also work. As a society, we have never understood how torsional forces are manifested and used in our bodies. That should change -- it must change!
I guess we're here for different purposes; I'm trying to understand use computational movement to visualize human movement a bit better. And anyone interested in getting a hands-on visualization to an interesting space-curve can try RMT Ropes.
I hope all are doing well with their progress with quizzes and the final.
