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The following formulas, called the Frenet-Serret formulas, are of fundamental importance in differential geometry:

$$\begin{array}{l}{\text { 1. } d \mathbf{T} / d s=\kappa \mathbf{N}} \\ {\text { 2. } d \mathbf{N} / d s=-\kappa \mathbf{T}+\tau \mathbf{B}} \\ {\text { 3. } d \mathbf{B} / d s=-\tau \mathbf{N}}\end{array}$$

(Formula 1 comes from Exercise 47 and Formula 3

comes from Exercise $49 .$ ) Use the fact that $\mathbf{N}=\mathbf{B} \times \mathbf{T}$

to deduce Formula 2 from Formulas 1 and 3 .

$$

=\tau \vec{B}-K \vec{T}

$$

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Oregon State University

Harvey Mudd College

Idaho State University

Boston College

{'transcript': "So if we want to prove equation to assuming everything else we have on the board, the first thing I would do is start with this cross product here. Come because we somehow want to get the derivative of n with respect to us. So let's go ahead and just first take the derivative of this because it's the only thing that we really have we're in is just floating around by itself. So doing this, we can go ahead and apply the product rule, which is going to be so first in the N by D s is so it is D B by DS crossed e plus B cross d t I. D s and the order here is important. Um, because otherwise you get something completely different. Now what we're going to do is replace the derivative of this with Equation three, the derivative of this with equation one. And if we do that, we should end up with. So let's just go in and plug these in first also would be negative towel and cross t plus be cross the derivative of Kappa, then Yeah, and one thing I'm going to do is I'm going to use this negative here to switch this around. So instead of it being, uh, negative towel and it's going to be t talent, so we'll write. This is T. Actually, let me go back to rein in blue. Uh, and also, I'll pull that towel out to the towel times t cross and, uh, and then plus, let's pull that cap out. Kampe be cross it. All right. Now for these ends here, um, let's go ahead and replace it with B cross t. So doing that. Will give towel times t cross be cross t. Uh, yeah, here, frosty and then plus Kappa times be cross be cross teeth. Now, there is a nice formula that will help us out here, Um, for breaking these two are and it is if we have a cross the cross product of B N. C. Well, this is going to be equal to So, um, actually let me look again because I always get this reversed. All right? So, yeah, it would be a started with see, And then we have B and then minus a dotted with the times, See, so we can imply this here. Um, So I'll do this first one in green. So this is a This is being This is C So this becomes Tao. Actually, I might need a student so a little bit more So how times so would be t dotted with tea times B and then minus a dotted with b times t And then plus, And I'll do this in black was Kapil. And then this is our A This is our B and this is R C. So be be dotted with tea times the minus mhm, um, be dotted with b times t. Now there's a couple of properties we need to remember about these, um, by normal and tangent vectors. So first and we assume they're both units. So that means be dotted with B is one and t dotted with tea is one since we're assuming their units or a unit vector. So that means TD with tea is one. And the B dollar with B is one. Right now, the second property we need to remember is we assume they are perpendicular to each other. So the perpendicular to t, which would mean the dot product of be with tea, which should be the same thing as t would be is going to be zero. So that means this term is zero. And this term is zero. And then if we go ahead and just write this all out, that would give us a towel, be, uh, and then we'd have K or Kappa times negative t, which would be negative. Kappa T. And so now let's go ahead and see if this is what we were looking for. So, um, it's written the other way around. They have negative cavity plus towel. Be up here, but you can see how I mean, that's the same thing just if we flip. So if we were to just write negative cavity plus Tau B, we get the same thing now that you've proved it can go in and put your little proof box and a smiley face because you're glad you're done with it."}

University of North Texas