If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 6: Curvature

# Curvature of a helix, part 2

This finishes up the helix-curvature example started in the last video. Created by Grant Sanderson.

## Want to join the conversation?

• At , I'm wondering why dS(vector valued function)/dt is equal to ds(arc length)/dt ? • Grant answers this in the first Curvature formula video. We need a formula for arc length in terms of t (time). So any ideas of how to approximate the (arc) length of a curve/function?

Well, Sal does a whole series of videos on this in Integral calculus. A tangent line would give you length of a curve at that instant. We have a method to calculate tangent lines, it's called derivatives. The sum of the magnitude of all the tangent lines would give you the arc length of the curve. We use the magnitude because we want the length of the tangent line. In Sal's video on the subject, he shows that:

* arc length (s) = ∫ || dS/dt || *
Where S is the equation of the curve.

Since we want the change in arc length, we can get rid of the integral sign. Leaving us with:

* ds/dt = || dS/dt || *

I hope this makes sense. :)
• I tried to solve the problem, using the last formula you derived in the previous video (Curvature Formula, part 5), where you divided the cross product of the first derivative and second derivative of S(t) by the magnitude of the first derivative of S(t) to the third power, and I got a different result. I did not get a constant number, but rather a vector valued function in the denominator, divided by the third power of the same number you derived for the magnitude of the first derivative of S(t). Why? I checked the arithmetic, and everything seems to be in place. • Do you ever expand on curvature and discuss the TNB frame with torsion
(1 vote) • Not intuitive why the kappa is less than 1.
(1 vote) • ah yes, if my instruments in space lock up imma just be wondering the circle ill be traveling for the rest of my life XD

fr tho i like the way u explain things, wish there was more 3b1b x khan
(1 vote) 