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## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 6: Curvature

# Curvature formula, part 1

Curvature is computed by first finding a unit tangent vector function, then finding its derivative with respect to arc length. Here we start thinking about what that means. Created by Grant Sanderson.

## Want to join the conversation?

• What's wrong with defining curvature(kappa k) as the norm(absolute value) of difference quotient of unit tangent vector function 'T' and the parameter 't', instead of the arc length 's'. • The way I understand it if you consider a particle moving along a curve, parametric equation in terms of time t, will describe position vector. Tangent vector will be then describing velocity vector. As you can seen, it is already then dependent on time t. Now if you decide to define curvature as change in Tangent vector with respect to time, then it would be more like acceleration or change in velocity vector rather than define a characteristic of curve like curvature.

If a tangent vector changes with time more, then it just means particle is moving faster along the curve and does not tell actually anything about the curvature itself.

So I think, if you define dT with respect to arc length, you will get more accurate representation of curvature.
• Is this curve a cycloid? The curve described by a point of a purely rotating body? • 1-cos(t) varies between -1, and 1, but your curve has only positive y, and yet you say that y(t) = 1-cos(t). • I understand the analogy of driving along and getting "locked in" to a position and then drawing out a circle, but why does considering dT/ds correspond to "holding your steering wheel fixed in one place"? Does it have to do with how dT/ds tracks how a vector is changing? I was getting confused because dT/ds gives the vector tangent to the curve, but if you travel along just the vector, it is a straight line?
(1 vote) • "Holding your steering wheel fixed in one place" and looking at the circle this makes actually corresponds to the concept of curvature. We want to know the radius of the circle created, or rather 1/R, which is curvature.
The unit tangent vector is not given by dT/ds, but rather by T. dT/ds is asking how fast the tangent vectors are changing direction relative to the arc length, or to the distance travelled. In other words, how much curve do you get for your distance? How fast are you turning? What is your curvature?
If this doesn't make sense yet, feel free to ask for clarification. Have a great day!
• Hi, I am a Hungarian math enthusiast, invsntor of Spidrons and SpHidrons, I am working on thetheory behind of these geometric fenomena.
My question is about the curvature of surfaces. Is it necessary to be perpendicular for the two (maximal and minimal) main normalvector section?
Why? 