 Now, there's another derivation that's quite important, and that is acceleration. We talk about the normal unit tangent vector, the principle normal unit vector. We talk about binormal unit vector, but what about acceleration? What happens to acceleration? And that is a derivation that you should just know something about. So let's just go there. Let's just remember what the unit tangent vector is all about. That is r prime of t divided by its magnitude, divided by its magnitude. So this is this, look, I think the normal tangent, or the unit tangent vector, I should say, and mix these up. So it's just the direction, basically, with a magnitude of one there. But I know something else. I know that, for instance, what this is, what is that? Well, that is just ds dt, nothing major there. And I can now just rewrite this. First for now, let's get the r prime of t, let's just rewrite that r prime of t, that is just going to be t, this times ds dt, that's what we have there for now. But let's think about it, what is this? This is velocity. And I want to know acceleration. Do I get an equation for acceleration? That's going to be double prime of t. So what do I have to do? I've taken the time derivative here, I have to take the time derivative there. But if you think about it, I have to do the product rule. This is the product of two functions. So let's do that, let's do this, let's write it in this form, so that's going to be the second derivative of this, dt squared t of t, plus I'm going to have ds dt, ds dt there, and the second and the derivative of this, t prime of t, t prime of t. I've got to do some side work because I know that I know something about this little guy there. We know that the principle unit normal vector, that is going to be t prime of t divided by that. And if I have to get these two just on its own, if I just had this kind of zone, it's going to be those two there, so let's put them in there. So that's going to be n of t times the norm of t prime of t, norm of t prime of t. Now as it stands, we've got to do something about this guy. Let's do some side work with this guy. Mathematics is beautiful. You can sit there and play and play and play, and eventually you're going to get to stuff that becomes quite useful. So this can just then be rewritten as, remember this is just a dt, like that, dt of t dt. That's what we have there, but that can be expanded, basically remember we did it when we looked at curvature before, that that can be expanded as dt ds times ds dt. And we're dealing here with norms, so that's what we're dealing with. And what is that? That's curvature. That's kappa. So we can replace all of this here with that. So there's another ds dt in there, so that becomes squared, and there's a kappa in there as well. There's a kappa in there as well. So now we have this. That acceleration has got two components. It's the second derivative there, d squared s dt squared, and this tangent, so it's got a tangent component, it's got this tangent component, and it also has this kappa ds dt squared, n of t, it has this component in the normal unit vector, principal normal unit vector. So acceleration is always going to have these two components, and this becomes very helpful when we derive another form of an equation for curvature.