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Problem 51 Hard Difficulty

Use the Frenct-Serret formulas to prove cach of the fol-
lowing. (Primes denote derivatives with respect to $t$ . Start
as in the proof of Theorem $10 . )$
$$\begin{array}{l}{\text { (a) } \mathbf{r}^{\prime \prime}=s^{\prime \prime} \mathbf{T}+\kappa\left(s^{\prime}\right)^{2} \mathbf{N}} \\ {\text { (b) } \mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}=\kappa\left(s^{\prime}\right)^{3} \mathbf{B}} \\ {\text { (c) } \mathbf{r}^{\prime \prime \prime}=\left[s^{\prime \prime \prime}-\kappa^{2}\left(s^{\prime}\right)^{3}\right] \mathbf{T}+\left[3 \kappa s^{\prime} s^{\prime \prime}+\kappa^{\prime}\left(s^{\prime}\right)^{2}\right] \mathbf{N}} \\ {+\kappa \tau\left(s^{\prime}\right)^{3} \mathbf{B}}\end{array}$$
$$(d) \tau=\frac{\left(\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right) \cdot \mathbf{r}^{\prime \prime \prime}}{\left|\mathbf{r}^{\prime} \times \mathbf{r}^{\prime \prime}\right|^{2}}$$

Answer

$$
\tau=\frac{\left[\vec{r}^{\prime} \times \vec{r}^{\prime \prime}\right] \vec{r}^{\prime \prime \prime}}{\left|\vec{r}^{\prime} \times \vec{r}^{\prime \prime}\right|^{2}}
$$

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Video Transcript

in this problem, we would like to use the Freenet Saray formulas, which are written on the right hand side of the screen, to prove a Siri's four equations. So let's look at part A So in this we want to show that for some, the vector function are that are double prime is equal to s double prime times t plus kappa s times as prime squared times n we want to show you about the quality and we're gonna use the for necessary formulas to help us out a little bit. So let's get started with our proof. First, we should remember that our unit Tangent Vector T is defined as our prime over the the magnitude of our prime. And we should also remember that the magnitude of our prime can be written as DS over DT, where t some perimeter and s is the arc length of at the curb. Our So using that information, let's go ahead and rearrange this equation here. Do you give us our prime? So we have our prime is equal to the magnitude of our prime times, the vector t. And then if we use this second equation here that will give us. This is equal to D S D over DT Times t All right, so here's their first derivative of our let's go ahead and find our second derivative are double prime. Well, if we use the product rule on this equation here, what we end up getting is d squared s over. D t squared times t class. And just to be clear, we're taking the derivative with respect to t He s ever tt times t prime. So our first term D squared s ever DT squared is the second derivative of s. So I'm gonna raid that is s double prime. I'm Steve and then I'm going to write t prime in the second term as, uh c d t over de lower Case T. And then to manipulate this little bit, I'm gonna multiply this'll by D s every D s, which is equivalent to one. But it's going to make our ah solution look a little bit neater. So here is That s double prime Times t Plus. What we can see is that we have a DS over DT term here, and then I look at this numerator and this denominator we have another DS every DT. So you can write. That is DS over. DT squared times d t a d capital tear unit tangent vector over the surveilling d s. So where do we want to write that out that way? Well, the cause this right here is one of our for an X ray formulas. This is Formula One right here. So this becomes Kappa Times. And so we end up with Kappa Times s prime squared times and which is the right hand side of the equation that we wanted to show. So that concludes our proof of part A Let's go ahead and tackle party in part B. What? We want to dio this show that our prime crossed with our double prime is equal to Kappa times. First derivative of s quantity. Cute times that by normal vector B All right, let's tackle lovers. So what I'm going to do is years. Look, take my full screen back here. We're gonna do is I'm going to use Ah, this expression of our prime here. So d s over. DT times t Let's start this off correctly. So our prime cross with our double prime. Okay, this year is going to give me my expression for our prime. So, yes, every DT times the vector T and I want to take the cross product of that with the expression that we got. Ah, by part A for our double prime, which is s double prime time. Inspector T Plus Kappa Times s prime squared times and our unit Normal vector. And that is by part, a All right, so I can distribute my cross product of it. Um, this is distributive over addition. So what we're gonna have, uh, I'm gonna rate this part as s prime. You can have s prime times. The vector t crossed with s double prime times a vector t plus s primetime Specter tiu crossed with Kappa s prime squared times and right then in parentheses to be consistent here, we know that we can take our scaler multiples out of the cross product. So here we have s prime times s double prime times, the cross product of tea with itself, which we know that the cross product of a vector with itself is going to be equal to zero. And then for this second term over here, we're going to have again pulling our scaler multiples out K or Kappa Times s prime cubed times Vector t crossed with Vector And and we know that buying definition the by normal vector is defined as t crossed within. So we end up with this term become zero. We end up with Kappa Times s prime cured times be and that gives us the right hand side of our equation so we can conclude or proof of part beat. All right, so we're halfway there. Got a part C now and in part C, we want to show the third derivative of our is equal. Teoh s triple prime minus Kappa squared s primes cubed. That quantity is gonna be multiplied by vector T plus three Capa s prime s double prime plus Kappa Peron s prime squared. That is the coefficient four vector n plus Kappa Tao s prime cubed Be all right, so this one looks a little bit bad, But once we get started, the proof should come pretty naturally. So let's start with the expression that we got for the second derivative as, uh, as we found in part A. So are double prime is equal to s double prime times T plus Kappa s prime squared times the unit Normal vector. And that's by part A. So then, let's go ahead and take the third derivative. So take the derivative of second derivatives. All right? And we're taking the derivative with respect to t Remember that the Ark Link s is a function of t. So we do have Teoh treat it like affection. Not like a constant. So for our first term to use the product rule, what we're gonna get is s triple prime times t plus s double prime times t prime. Not right that in brackets just to keep separate. Um, which I think is gonna help us. What? We help us keep things organized. Hurry. Taking the derivative of the second term is a little bit of a bear, so I'm gonna work with it over off to the side. So we have kappa, which is also a function of t times s prime tea times, you know, normal vector. And so what I'm going to dio is I'm going to let you represent the vector Kapa s prime squared. So then if I take the first derivative of you, I have to use the product rule here. So we're gonna end up with K A Kappa Prime Times s prime squared plus Kappa Times two s prime and then derivative of the inside is a double prime. So it's right that a little bit Niedere capper. Prime times s prime squared, plus to Kappa s prime s double prime. So then, if we take the derivative of Kappa s prime squared and which I'm representing as you times end, what I have to do is use the product rules that we have. They use a product rule again. So we have you prime times n plus you times and prime. And that's going to give us Kappa Prime s primes weird. Plus to Kappa s prime s double prime times n plus Kapa s prime squared and prime. All right, now we've taken that derivative. We can substitute it back in for the derivative, this derivative of the second term. Here. So what this is going to give us? It's Kappa Prime s prime squared. That's too Kappa s prime s double prime times. And plus Kabah s prime squared and prime. All right, let's go ahead and simplify this out a little bit. So here, we're gonna end up with s triple prime T. You were gonna leave that How it is, Plus s double prime. And I'm gonna rate this a right t prime as d t over de Lower Case T. We'll come back to that a little bit later. We're going to add I don't believe this. Ah, the spit right here as is. So we have capper Prime s prime squared plus two cava s prime as double prime times. And plus en prime is where we want to rewrite something. So Capa s prime squared times de and over d t for end prime. And then we're gonna do something similar to what we did in one of the earlier proofs, and I'm gonna multiply DT over D lower case T and multiply that by DS over DS, which is equivalent to one. I'm gonna do the same thing for d n over d t multiply by ds evds. All right, here is why So our first term will stay the same our second term. If I group my DS every d t. That's the same as s prime. So what, we're gonna get is s prime times as double prime, right? Those in parentheses so they don't get all jumbled. And then we're left with DT over D s, which by the first Freenet Saray equation. This one right here. You know that's equal to Kappa. And so I can rewrite the second term. Here. It's Kappa Times n and that's by friend a equation one that There. All right, now, uh, second, it's going to stay the same. We're not gonna mess with it yet. So Kappa Brian s prime squared plus two Kappa s prime has doubled prime. And then here for the third term, we're gonna do something kind of similar. So again, if we group d s over d T, that's another s prime. So we end up with Kappa Times s prime cubed and then the n over D s. We're gonna deal with that in a minute. That's gonna fall out from another one of our friend. A equations there, Not Right. So what I want to do is I want to combine these two terms, so we're gonna have s triple prime tea and then if I combine these in tow or with one coefficient time is the unit normal vector. We see that this and this least do things their common terms. So what, we're gonna end up with this K Prime Times s prime squared plus three. That should be capital, not K three times cap s prime. That's double prime all that this times. And and then let's deal with this last bit right here. So Kapa s prime here is going to stay the same, but the end over ds if we scroll back up to our friend a equations here from Eysseric equations, that's this quantity right there. So I'm gonna plug that in. That's for day equation number two. So this is going to be negative. Kappa T plus Thao bee. All right, we're almost there. If I combined my terms that involve being a tangent vector T what I'm going to get is s triple prime minus kappa squared times s prime cubed. That's gonna be multiplied by the unit tangent Vector t my unit normal vector coefficient will stay the same. So Kappa Prime s Prime squared plus three Kappa s prime at stubble Prime times E and it normal. Vector end. And then, for our last term, involving the by a normal vector. Get Kappa Times s. Well, let's see. Kappa Times, Tao Times s prime Cubed times A by normal vector B And if you look closely and you'll wanna verify this But we look closely, we can see that this is equal to the right hand side of the equation that we wanted to approve. So we can see that are trouble. Prime can indeed be expressed by this equation so we can call that proof completed. All right, we've got one more problem for this. So what we're gonna do is we're going to prove parte de that Tao is equal. Teoh, our prime crossed with our double prime dot product of that are triple prime over. The magnitude of our prime crossed with our double prime all squared. It looks like it's gonna be bad, but it's not. We have already computed most of these quantities, and all we have to do is, um combine them in the way that this equation is showing. So to prove this, I'm going to start with the right hand side. Now we're gonna share that simplifies to tell. So our prime crossed with our double prime dotted with our triple prime. Not quite durable prime. That's too many. Never Magnitude of our prime crossed with our double grime. All squared. OK, so what I'm gonna do is I'm going to use part B to substitute in the cross product. So, see, let's make some notes here. So this is going to come from part B, as is this. And then our triple prime is going to come from part C. All right, sir, What we yet is for the cross product. We have Kappen Times s prime cubed times be and we're taking dot product that with this crazy thing that we got for part C for the third derivative. So if s triple Prime minus Kappa squared s prime cube Times T plus Kappa Prime s prime squared plus three Kappa s prime s double prime times Vector n plus Kappa Tau s prime. Huge times be. And then that is all going to be over the magnitude. Kappa s prime cubed be, I swear, quantity squared. Okay, so what do we need to do with this? Well, we need Teoh distribute this term into our some here. So what we end up with is I'm gonna leave my Capa s prime cubed on the outside. We don't need to factor it with the, um with the by normal vector, it's what will end up with it's s triple prime my escap a squared s prime. Huge times be dotted with tea. Plus Kappa Prime s Prime Square plus three Kapa s prime s trip or double prime. We're gonna have to buy a normal vector, be dotted with n and last but not least, will have Kappa Tao s prime cute and be dotted with itself. All of that. It's going to be over magnitude of cap. It s prime cubed, be quantity squared. So one thing that will remember is that the by normal vector the normal unit normal vector and the unit tangent vector all orthogonal to one another or they're all perpendicular, which means that they're dot products for zero. So that means that this term is zero and this term zero that saves us a whole bunch of simplification. So what we're left with is Kappa s prime cubed times. Kappa Tau s prime here, But times the by normal, uh, factor times itself dotted with itself. Eso we know that by definition that is the norm of B squared or the magnitude of B squared. Then that's over. I'm gonna split this into two terms. So laugh caba s prime cubed quantity squared times the magnitude would be squared. We can do that because, um, the norm that we would be taking, um I believe treaties as scale. Er's. So then what's gonna happen? Is these two terms of going to cancel out we're going to be left with Kapa s prime cubed squared times Tao over the same thing in the denominator minus the towel. So Capa s prime cubed quantity squared. Those are gonna cancel out. We're going to be left with Tao. That simplifies quays of it. And that's the relationship that we wanted to show. So we can say that our proof is complete.

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