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Problem 47 Medium Difficulty

Show that the curvature $\kappa$ is related to the tangent and
normal vectors by the equation
$$\frac{d \mathbf{T}}{d s}=\kappa \mathbf{N}$$

Answer

See proof as demonstrated in video.

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Video Transcript

all right. In this problem, we want to show that the curvature of a function K or of occur K is related to the unit Tangent and unit Herbal Vector by the following equation the derivative of T respect s PT over DS is equal to tapa times and this is what we want to prove. I find it easiest to work through this proof backwards. So starting with the right hand side. So you want to look at cava times the unit Normal vector. So one of the things that I'm going Teoh recall is that the way that curvature is defined is, um, as the magnitude of the unit tangent vector over the magnitude of first derivative. We're sorry magnitude of t prime. So first derivative of the unit tangent Vector. Take the magnitude over the magnitude of the first derivative of our which is the curve that represent or the vector function that represents are curved. So I can rewrite this as magnitude of t prime over magnitude of our prime times. The unit normal vector n and then we can think about how our unit normal vector is defined, which is as the, uh, see, the first derivative of T so t prime over the magnitude of tea prime. So I can rewrite this then as magnitude of t prime over magnitude of our prime times t prime over the magnitude of tea per time. And then we see that our magnitude of tea primes cancel out. So we're left with t prime of tea over magnitude of our prime of tea. All right, so what else do we know here? Well, we know that we can write the first derivative of a T as d t over do you lower case T. And then something that we also know from when in our discussions about arc length that show up in the textbook we also knew that the magnitude of our prime is equal to DS over DT. Where s is the Ark link? So what that means is that I can rewrite this as de Capital T Verjee, Lower case t times d t over D s, t t and D two year owing to cancel, we end up with derivative of t over vector t over derivative of s which we could just write this DT over gs. And then we see we have the left hand side of the equation that we wanted to prove, so we can call our proof done.

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