 I want to talk a little bit about curvature and unit tangents and stuff like that. In some sense, this section really belongs a month or two ago, but for some reason it's here in the book. I don't really know why, but we didn't talk about it before. So we have some curves. Let's think of this curve in space. It doesn't really matter. I just think it's going to be some kind of space curve. So I have some function g taking r to rn, and we know that position g prime t is the tangent vector. So at some point here, I guess the way I'm going to measure it, I have a tangent vector in prime t. This is both stuff that we know already. I talked about this in some length last time. So if I take dbs of g, then this should be, at least in length, this should be 1. I've seen a lot of, like, little bit. Are we okay? I'm good. If I take the derivative, I mean, if g is parameterized by arc length, then I take the derivative with respect to that parameter, then of course I should get a vector of length 1 with respect to arc length, sort of captures, let's call it capital T. It depends on my parameter. This is the unit tangent vector, which we could also compute and we could also write g prime of t is the same as saying d of t is dds, tds, dbs, ds, dt, that's this, dt. So in particular, we can calculate this vector t without needing to parameterize s by arc length and just take the derivative of s and divide by its length. It's just g prime is, like, really quiet and, like, crazy. What are you doing? Just drinking. Yeah? I just want to look at what you mean with respect to arc length. Like, can you interact with me with respect to arc length? Can you get the 5th assumption of t and not a 5th one? So, one thing we talked about doing last time, suppose I had some arbitrary function, I could change my parameter so that the thing went with unit speed, right? So if I have some arbitrary curve, like, I don't know, g of t is, I don't want to do it, so let me do one that I can do without thinking, a 3 cosine t, 3 sine t. So there's a circle of radius 3. This is not parameterized with respect to arc length because g prime of t is 3 minus sine t, my cosine of t. And so I would need to change this parameter by dividing by 3. So if I want this, the curve g of t over 3 is parameterized by arc length. It's the same curve, 3 cosine, maybe I should use t over 3, 3 sine t over 3. But the derivative of this guy maybe s is, this is a unit vector, the length of this vector, the length of this vector is 1. So I can change my parameterization and of course I could have done one harder where this change would not just be a linear change of variable, so it may be something nonlinear. So I can, this is what we talked about last time to some extent, I can integrate and do a bunch of pain in the next stuff to make sure that this thing is going at unit speed. If it's going at unit speed, then in a unit of time, it traverses one unit of length. So arc length just varies linearly with arc, the speed is linear with arc length. I mean linear of like times 1. If you go one mile an hour, then in an hour, guess how far you go. So, right? So if I press the function, and it's like an arc length. Right, right, exactly. So that's the point of this discussion. So this is saying, if we change the parameterization of this curve so that it goes one unit of distance per unit of time, then these two things would be the same. The unit tangent vector and the tangent vector, would be the same vector. But even if it's going at a variable speed or some faster speed, then we can still figure out what this unit tangent vector is. And this is a convenient object in a lot of cases to deal with. This unit normal vector is a convenient choice. Well, let me say some more stuff about that. Yeah, this S of t is the new parameter? So S of t is the arc length. It really should be like S of g. It's the arc length of g. It's how the arc length varies with respect to time. Right, so this should be the integral from first time to S of g prime of t. It's how long somehow there should be a t in here. So I guess I'm screwing something up. Maybe this should be t and this should be... It's the distance along the curve. And so really S depends on g but it gets too tedious to write all the subscripts and everything like that. People usually use, say, dS to mean distance with respect to arc length or just S to mean arc length. So we have this other object and really this t also depends on g. Right, because I'll have different tangent vectors for different curves. I don't want to write little sub g's all over the place. But the curve itself, this curve that I just drew, if I just look at the curve, there's a natural parameterization of the curve where one unit of distance equals one unit of time. And it doesn't depend on how I parameterize the curve unless others are going this way or going that way. It's just intrinsic to the curve once I decide what my units are. So that's what I'm trying to get at. And another thing that is intrinsic to the curve, not how I traverse the curve but just the curve itself, would be this tangent vector here which I'm trying to draw one unit long but not so well. This vector t, which varies as I move around the curve. Now, the book writes this. The book writes this as a boldface lowercase t. That's just too hard for me to do. The common notation is actually a capital T. I see capital T's more, but the book likes to use a boldface lower t. A lowercase boldt in the textbook means the unit tangent vector. I think. Pretty sure. But I'm not sure. So that means that also, and so this derivative, s prime, is also the velocity. Distance is changing. This is the velocity along the curve for the parameterization. Maybe it should be speed because this vector is the speed. This together is the velocity. It's the way you're going and how fast you're going. I don't know. Okay, so we have the tangent vector. It's just the derivative divided by its length. And now we can also, I mean, if you think about it, once I've made not have a length that matters, all these vector changes doesn't capture how fast we're going on the curve, but how much the curve is turning. So this vector t of t, it captures the direction. And so now, I mean, if you want to think about the acceleration of this, think of something with that parameterization, then the acceleration could be the second derivative. This is the velocity vector wherever it went here. This is the velocity. So the acceleration will be the second derivative of t. But since g, so that will be e dt of g prime of t, which is the derivative of s prime of t. So this is a number and this is a vector. And this should be a vector. And so that will be the second derivative of the arc length of that in the t direction. So again, this is a vector. This is a number. Plus the first derivative times the derivative of t. So this is a vector. So let me put that on hold for a minute. So I have, so what is that saying? I don't know what it's saying. Well, if you think about this for a minute, this is a unit vector. Let me just say, I claim it should be obvious but let's just do it that t is perpendicular. This is saying I have some curve. It's going that way. This is always perpendicular to its derivative. This derivative is going to point, the derivative vector is going to point in a perpendicular direction. It's just an easy straightforward calculation. What's the second derivative? The second derivative of my curve is my acceleration vector. What's the derivative? It's the direction of the engine. But why would you say it's the perpendicular? This is the derivative of the tangent. And so this curve, I mean, so I'll do the calculation in just a second, but if you just think about it, this thing doesn't shrink and grow in length. It only turns. As I move around the curve, t just changes direction a little bit. And so its derivative will be the direction in which t has changed direction. That's why it should be perpendicular. If I move t a little bit along this curve, then if I move t a little bit along this curve, then it will move down in the perpendicular direction. But it's also really easy to just calculate. So since the length of t is 1, that means that t dotted with itself is 1. And if I take the derivative of that equation, well, here I get 0. But if I use the product tool for dot products, then I get t prime dotted with t plus t dotted with t prime is 2 dotted with t prime. In other words, t dotted with t prime is 0. t is perpendicular. So it's the point of that. Well, that means that I can decompose the acceleration in terms of the second derivative of the curvature. Think of this as just the number A times this unit tangent vector plus another vector the first derivative of the arc length times some number. Well, let's just write it as the length of t prime, which is kind of stupid, but this is another number B times the vector n that shouldn't be that long. So I have that my curve can be expressed tangential in a normal direction. This n which is just the derivative of t divided by its length, unit normal. When you're dotting s prime of t, call the number over there. Where am I dotting? Oh, this is not supposed to be a dot. This is a number. These numbers, I call them A and B. So I have sort of a built-in so the idea here is that there's a built-in coordinate system to the curves. As this curve moves around there's a coordinate which is intrinsic to the curve itself. There's this tangential direction which is sort of easy to see and then there's this direction of acceleration which is a little more hitting. So I have this sort of so this gives an intrinsic, well at least part of an intrinsic coordinate system in the plane we have something built in and if we do this in space we can pick up the third one. For some reason it's called B and I don't know why. The inertial frame, is that right? Yeah. So I can also so I can get the third coordinate by t cross n. So that is I have my space curve here I have a tangent direction t, I have some normal direction n oops I drew it the wrong way I have my tangent direction t, I have my normal direction n and then I have some third t, which is t cross n. So this gives me these are all perpendicular to one another and this gives me sort of a built in coordinate system to this curve that doesn't depend on the rest of the world it just depends on how you're traversing this curve. I'm seeing a lot of blankness am I just saying blah blah blah we're just all ready to go home for Thanksgiving. Yes. So I have this sort of these built in coordinates the other thing that I can pull out of this is a useful number which is so I guess I forgot to say the words that this number G double prime is what did I write? S double prime S prime T prime length n and some words feel free to forget them right after I write them is that S double prime is the tangential acceleration and that S prime times the length of the derivative of the unit tangent is the centripetal. So again, don't memorize these words, I'm just saying these words so that if you hear them in another context you see what they are I'm not going to ask you this word on a test or anything but so another thing that is sort of built into this when you think about how this tangent is turning that is this coefficient here this gives us the curvature if this thing is not turning then it's not curved and if it is turning, if it turns a lot so this curve this is somewhat curved this guy this is a very sharp, a lot of curvature a little bit of curvature no curvature so we can measure the curvature of G by looking at this quantity but we want to do this in a way that doesn't depend on our parameterizations so what we want to really do is find the curvature it's actually the scalar curvature of our curve G and C with respect to arc length of the unit tangent how much the unit tangent curves but I only want the derivative with respect to arc length because if I do it with respect to T I can traverse this thing so slowly that it doesn't fit that number small even though it's a very tight curve so I want something with respect to arc length I notice here that oh, this is kappa and it needs to be so kappa equals 0 means that there are other notions of curvature I mean the scalar curvature is sort of a standard thing if you think about the surface then there are other curvatures like the sectional curvature this is just the curvature of a curve right you know this looks like just like in calculating T it looks like you have to parameterize your curves by arc length first but it's really easy to calculate that the curvature is just the derivative of the unit tangent divided by the derivative of arc length I guess I need to make and this is just this is just the chain rule so if I just calculate actually I want to calculate the derivative of the unit tangent well this is again just the chain rule and so now I just so this is if it's 0 but if the sdT is 0 then this wasn't a smooth curve it would stop somewhere sort of 999 today yeah do you need to go over there no, no kappa is the derivative of the unit tangent with a standard arc length so oh this is not kappa so now solve this for kappa kappa is this divided by this so in that example with the circle which I've already erased so if we do this stupid example take the circle of radius 3 sin T 3 cosine T and cosine first the curvature of this is a third because if I take dsDT it's 3 and the unit tangent this guy has constant acceleration so the curvature does anybody need me to do this calculation it should be apparent what? it will be fine right and in general I mean another definition of curvature oops an older definition of curvature for a plane curve I don't know anybody know that if I have some curves in the plane yeah I know the curvature is here then I take the biggest circle that I can fit in there this is called the osculating circle osculating means kissing so it's the circle that just kisses there and then I take this radius and the curvature is one overall right so an older definition of curvature or a common definition of curvature is you look at you look at the circle the biggest circle that you can put in tangential and then you take one over its radius and that can be the curvature is there a reason you called it older is there a reason that that idea would no longer be the way this existed before calculus so it's difficult to define it in terms of derivatives if you don't have the notion of a derivative it's not an archaic definition it's just one that's been around a lot longer than calculus and with a 3D analog of that it would be like spheres or it does that kind of thing so in 3D you have to figure out which circle you're going to put in there but if you think about it's still the same definition I don't need a sphere necessarily I have some curve here and now I want to fit the tightest circle I can fit in so I have to put that circle in the plane that the curve is infinitesimally lying in and then once I do that it's just that that plane will move around as the curve bends around but it's the same definition if you change the parameterization there to g of t over 3 as we did this course it's still going to be one third you'll still get the same curvature yeah sure so this is 3 cosine t not cot yeah cosine t cosine t so here my s prime of t and here my unit tangent vector is the vector minus sine t cosine it's wrong it's derivative so now I take its derivative and I get minus cosine t minus sine t and its length is the square root of cosine square plus sine square which is 1 so I get 1 so this formula just says it's this length 1 which is that length 3 if I change this to 3 cosine notice that if I get the red pen the red pen is gone so if I do this again with 3 cosine t over 3 3 sine t over 3 so let's call it h of t then when I take the derivative h prime of t I get now the 3 is cancelled and I get cosine minus sine t over 3 cosine t over 3 so here s prime 1 because it was parameterized with respect to arc length t my unit tangent of the length of this guy is a third length 1 I have to take his derivative so t this is t minus sine t over 3 cosine t over 3 its derivative is 1 third minus cosine t over 3 minus sine t over 3 and so the 3 sort of moves from the bottom to the top so my curvature is 1 third over 1 so the parameter and obviously there's nothing magic about this if I put any function phi here the derivative is going to move around in the right way this doesn't depend on how I parameterize which was built into the definition the whole point was it shouldn't depend on how fast you go on the curve right that has a curve like this it turns the same amount whether you drive really fast or really slow its intrinsic to the curve not how you drive on blah blah blah blah blah blah blah blah blah I guess I should let me point out one thing maybe you've seen this before so suppose I have a graph in the plane y equals f of x then you can write a formula and this is one of the homework problems that it's the second derivative of y 1 plus the first derivative squared if I have the graph of y equals f of x in the plane you can write a formula for the curvature in this way I'm going to change gears here for ok with this idea and like I said we could have done all of this quite sure where it's y here in the book I guess because we need we'll need the unit normally so I'm going to change gears a little now if there's no questions I'm going to go back to talking about vector fields so we have some vector fields my pictures and will probably be two but sometimes a vector field is often used for a vector field that is supposed to be suggestive of the fact that I should think of this like fluid some fluid motion and if I take and in fact a significant some part of what you do in 308 is looking at flows you have some initial condition here so starting point curve suppose I have some curve here which is everywhere tangent so if I have a curve let's call it phi of t so that let's just write it as x vector of phi of t so I have some curve phi taking r into rn and if it happens that f of phi of t have too many parentheses there tangent to phi of t if at any t then we say that b kind of fading as much for you as it is for me okay so we'll try a different one sorry this board is so horrible they told me I was going to fix it but I don't believe them anymore so such a phi is called a flow line from my vector field f or sometimes it's just called a solution such a flow actually a differential equation and such a phi is actually a function which solves the differential equation so this is why it's relevant for 308 which is a differential equation of course but you can also think of this as is it clear what this means flow line business the direction of the maximum flow I don't know about maximum flow but it's the way you're being pushed by this so if you think of this as current I guess I could be electric current too if you're thinking of this as water current then flow will be pushed in the direction that the thing is going right so this is why it's a solution curve because if I think of that vector field as a thing that I want to solve in such a curve and so I guess at this point let's call this phi of 0 and this is with initial condition that if you dropped a ball on a spring you would follow a flow line so let's assume the function didn't have a continuous derivative the vector field is not continuous then we may not have well defined flow lines so so I mean one can't show the condition you actually need to be smooth continuous isn't good enough it needs to have bounded variation so it can even if it's continuous but it turns sharply then that won't be good enough to guarantee if this needs to be a little better than that so if the vector is very smoothly then for sure you have a solution but if the vectors vary a little it's nastier than smoothly have a unique solution yes so an example of this imagine water leaking out of a bucket that will depend continuously but when the bucket's empty it's empty and so there's lots of if we solve the differential equation corresponding to the water coming out it varies nice and smoothly until it hits 0 but if it's 0 it's still continuous but you can't tell from the bucket being empty when it was full at this time then I know how to go backwards so this varies continuously but not quite smoothly enough to guarantee a unique solution because these solutions all come together at the line empty and the same thing happens in the plane or in oriented order you need slightly stronger than continuous so here we're going to mostly think about things that are twice differentiable okay so I mean we already saw vector fields corresponding to gradients for gradient vector fields so my big f is a gradient for some potential little f that my x, y is perpendicular to level curves so here f takes r r so for example squared plus 2y squared so the level curves of this guy are going to be an ellipse like that the vector field gradient the vector field normal for these ellipses and the solution curves are going to be these are my flow lines so I mean it happens in a lot of situations that you have some kind of a vector field because really a vector field is a differential equation and so differential equations occur in lots of physical situations and you want to know how stuff would behave and so there are some things that you might want to know about these flows so in particular in particular one kind of question is how much does this flow stretch or contract stuff so yeah vector field which one would be the acceleration or velocity field you should think of it as a velocity field so wait what is the derivative of p of t have to be equal and I'll switch to velocity then if we're going to have the curve be a solution because yeah does that just change the way it's also equal it does no it needs to be b prime of t equals yeah you can I mean so just like with the curves it's often common to rescale things so that the flow the vectors are all length one direction so they give me something called a direction field rather than a vector field so I might care only about the direction of flow and I might make a direction field rather than a vector field and I can still say stuff about direction but then I don't care about the parameterization p of t I just care about the curve that corresponds to it so one thing that you might want to know I might have a little I don't know box of stuff here and they're not tied together and I'm going to let them flow will they expand under this flow this box should become something much bigger but under some other flows maybe it will get contracted if I run backwards it should get contracted maybe I don't lose any area things like that but if you think about it it's really just so this is the sum of the derivative of the vectors so how much do the vectors infinitesimally move apart or move together and of course it doesn't have to be the same everywhere I could have a flow where stuff let me just draw the flow lines goes like that so here here the divergence and here the divergence is negative and here it's changing so at one point here it's zero right so it's sort of infinitesimally measures how much the vector field stretches stuff out or squishes it together so this is not a form well this is a formal definition but let me write that as a definition so if field f let's say it takes rn rn is I at least need it to be differentiable let me say it's twice differentiable well do I need it twice it's continuously differentiable and let's say that f looks like let me let me just put little sectors f1 x let's just think of that as a vector x plus f2 this notation is horrible these are the coordinate functions f is that then divergence of f is just the derivative with respect to the first guy the derivative with respect to the second guy the third guy so this looks like just take the gradient just add all the coordinates of the gradient there's some sort of useful notation that I can think of this guy so this is the definition now I'm doing notation let's think of as now being a function not gradient which takes functions to derivatives so this is an operator it's a function on functions taking some function this often goes by the name of del so in other words in R3 think of del as the vector with respect to x partial with respect to y partial with respect to z this notation seems a little weird until you get used to it and then it's very handy and then that means that we can write this as the dot product and this just means but this notation is pretty common actually we write this as del dot f so I guess I mean I have that guy laying around still so if I have my vector field xy that same one where did it go I wrote it down I didn't write it down this is the vector x4y this is the same one who is the gradient of x2 plus 2y2 it's the one who pictures over there then the divergence of f at any point is just the gradient here 2 I mean it's just plus d dy of 4y which in this case is 2 plus 6 2 plus 4 which is 6 so this vector field which is a linear vector field everywhere kind of infinitesimally expands by a factor of 6 of course I could have a more complicated I could have a more complicated thing I could have something like g of xy x2y so I could have some vector field like that and then the divergence of g it's usually easier to use this I mean it's usually better to write this rather than writing this because sometimes you might forget that that dot is there and now you think it's a gradient and it's a vector better just to write that but think that so this will be dx of x2y plus d dy of yz plus d dz of x2y plus y2 so this is 2xy so at the origin the divergence is 0 the divergence is 3 and at negative 1 0 the divergence is negative 1 so this vector field is not linear because the divergence varies as we move around this divergence thing yes maybe I should do an example where the divergence is 0 so let's do another easy example suppose I take I guess h let's call it r for rotation xy so again this is a linear vector field flow lines are just circles so my solution curves are just circles and if you take some little block of area and let it flow forward for a while you get the same area this guy is area preserving and if we compute the divergence of r then we get minus 1 plus 1 which is 0 so r is divergence free yes this is like a multi-variable calculus way of showing that rotation preserves area well I used rotation since it obviously preserves area but I mean I can come up with other ones that also preserve area I mean if you take something that you know where the flow lines look like this comes weakly vector field like that and if I take a thing here I flow it up to there and it changes so this is called divergence free or area preserving because it preserves area so this divergence thing sort of measures the loss or gain of area at a given place I have time to introduce curl so I guess maybe so I'll have to introduce it again next time I'm going to try so really today is like a lot of words so the next time we can do stuff so there are important theorems that relate divergence for line integral stuff but let me see if I can say something about curl so another thing that you could want to measure and this really only makes sense in the most amount of sense in only two and three dimensions divergence makes sense in any number of dimensions to make sense of curl in more than three dimensions is hard we look at that vector field which turns well it turns so we might want to measure the turniness of the vector field divergence measures how much stuff spreads out or collapses imagine one which rotates around and imagine I stick a little how much that wheel gets spun around this is a very intuitive definition so that would be well if you think about this a little bit this is going to be the difference in the mixed portions let's say in the plane suppose I have a vector field then so this is F1 and if I look at the partial of the Y guy with respect to X minus the partial of the X guy with respect to Y this should give me some measurement of how turny this vector field is how much it wants to twist stuff around this is actually the planar curl three dimensions F2 X minus F1 Y so this will measure how much this thing wants to cause things to spin around I'm sure it is yeah but I don't think it's torn it might be torn matter how big your sphere is your virginity of a really small sphere yeah so there's no part like this you can't work okay so I don't want to work at the same time the radius the length of the radius so anyway this is a measurement of how much the vector field wants to cause things to turn let me just write it down in three dimensions because I only have two minutes now I'm going to crank this up to a three dimensional thing instead of imagining you're sticking a little paddle in here imagine you have some little sphere that you're going to let float a tiny infinitesimal sphere it's going to want to turn somehow and if you measure the amount of turns with respect to the axis that it turns on that will give you the definition of curl in our three so I have f r3 so it's this guy the same guy but now it's a cross product so I can't remember the formula but it's the determinant if I put an i j k here then I put bell here one f2 f3 so I now want to take the determinant of that then I will get f 3 minus d d z f2 I'm going to write this as a vector then here I get d dz f1 minus d dx f3 and then the last one is d dx 2 minus d1 notice this is the same if I have a planar vector field I can think of this as 3D vector field sliced and it can't spin in these other directions because it's all the same here so this component this is the scalar so the normal if I think of that as sitting in space and it wants to spin or counterclockwise but it can't spin yeah, I have no time for that at all so it's this object so what the last week of the class is about is so we have two more classes left after this one is about versions of the fundamental theorem related to vector fields that relate that say let me say something about so the Stokes theorem in full generality says that if I know what goes on inside this curve I know how the derivative depends inside the curve I can relate that to aligning the goal around the outside or the derivative of what goes on in the boundary tell me what goes on in the inside so it's a very it's really the fundamental theorem calculus which says that if I know what goes on on the ends then I know how the function behaves inside but cranked up to higher dimensions the point of the next two classes which are the last two classes of the course is to sort of turn this up the standard fundamental theorem to higher dimensional things where we have the boundary being a curve or the boundary being a surface we want to understand integrating what integration is important