ðŸ’¬ ðŸ‘‹ Weâ€™re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Report

No Related Subtopics

Missouri State University

Campbell University

University of Nottingham

Idaho State University

0:00

Felicia S.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

Jsdfio K.

Create your own quiz or take a quiz that has been automatically generated based on what you have been learning. Expose yourself to new questions and test your abilities with different levels of difficulty.

Create your own quiz

Okay, So now that we've talked about how to do double and triple integral over volume regions and area regions, there is one more type of integral that's extremely important. And that's just single integral over curves. So if you think you have volume regions like this, So here's volume. Here's some area. So here we're going to do triple integral ALS over the volume. Break it up into little volume elements here. We're going to do double and girls over area regions. Yeah, so functions over area elements. So in other words, we break up the area into small pieces, and then we just add up all the function values on all these little area elements. We do the same thing here, except, of course, we break it into little volume pieces. So the last thing we want to talk about, like I said, is lying in the girls. So if we have a line or a curve rather and now this could be in the plane or in space so called this curve C. And now we want the curve to be oriented. We discussed this a little bit when we talked about Victor valued functions, but specifically And you know another way to think about it oriented, oriented curve, is it? It's a curve given by some premature ization. Or in other words, we have a vector valued function like this our f t that gives this curve and it gives it an orientation. And I like to stick to the physical representation of this. I want to think about this is the motion of the particle moving along this curve C. Okay, so what we can dio is we can break up our curve and a little tiny pieces like this. If you remember, this is really how we found the arc length of the curve or the distance of the particle travels. We broke it up into little small pieces and then added up all those pieces. But now what we can think about is we can think about having a function that's defined on the curves scene. So sort of like when we had a function to find on this volume region V or a function to find on this area region A. So we can actually just think about a function to find on this curve C and then for all of these small little pieces. And now these small little pieces here I'm gonna call DS, so they're gonna be just kind of elements of the curve or small little length elements. So here we have volume elements here. We had area elements now will have little length elements pointing along the curve. We can take the function value at that point, so f of X y z, and we can just multiply it together so we could take the function and multiplied by the little piece of of the curve ds the length element. Then we can add all those up. So the line integral so of this function f over the curves c is this follows. Okay, so now what we want to think is that we have a starting point equals a an unending point. T equals B. So that's sort of like the domain of our vector value function. So what we dio is we said the line integral of F oversee is well, it's the inner girl over the curve c of our function f over these line elements, DS. Okay, so that's not super enlightening. That's just kind of notation. Now, normally, when we have this oriented curve. We have a parameter ization. So another way to think about this vector valued function that's giving us this curve we've said before is that it is a parameter or ization of the curve. So if we have a parameter ization of the curve, then we can actually right this as the Anna Girl from T equals a two t equals B of F of well, f is a function of X, y and Z But I could sort of input in exit T y t and Z f t. So this is kind of like the chain role. You've seen this idea before, and then what I want to do is I wanna multiply by the derivative, specifically the magnitude of the derivative, where you can think about the speed of the particle. DT. Okay, so this should look pretty familiar. We talked about what happens when we integrate just the speed of the particle that actually gives us the distance. And so, just like when we integrated just the volume element over the volume, we got the volume and we integrated just over the area elements of the area. We got the area. In other words, If we take our function to be one, we just get the volume of area. So if this function is just the function one, then we will really will just get the arc length of the curve. So this is the general form of a line integral. And now I should mention that it's really the unexamined of the first type of line integral that we'll talk about. So this is really the first type of line integral and what's special about this type of line. Integral is the function that we're integrating this f right. So the in a grand the immigrant is a scaler valued function, meaning that I'm really just adding up a bunch of numbers. So at each little line element, the function has a number there, and I'm just taking that number and then adding up all of those numbers over all of the pieces of the line, all the line elements and then this. This is just sort of the what happens when I have the premature ization. And it's coming from the fact that D s really is like a change in t along the tangent direction like that. So the length that I moved along the curve from this parameter ization is just the absolute value of the derivative and the times. A little change in team. Okay, so the thing to take away is that this is the first type of line integral, and it's it's really the line integral of a scaler valued function. And so next, after we looked at a couple of examples, we'll talk about the second type, and that will be a line integral over a vector field. So kind of like with doing triple integral over volume regions or double integral over area regions. I want to give you just a very concrete example of when you would do a line integral. So let's say we have a piece of wire and it's kind of bent into this curve shape like this. So the wire is bent along the curve, and let's say that the wire has a linear density. This is just going to be exactly like the volume density or the area density. So the linear density, but we're gonna use lambda, so use sigma for area density, Delta for volume density. We don't really need to use a different letter here, but we'll use the Lambda thrilling your density. Okay, so the linear densities lambda And this is really like mass her unit length. Okay. And of course, the linear density doesn't have to be constant. Now, if you pick up any piece of rope that you confined, most rope has a constant linear density, meaning that if I take any small piece of the rope, the density is going to be the same. But it is possible that this wire that's bent in the shape like this has a density that depends on where I am on the wire. Maybe it's more dense on one end and kind of windows open becomes lessons on the other end. Okay, so let's say I want to find the mass of this piece of wire. Okay, so what I want to do is and this is really standard for doing line intervals. The first step is typically always to give a parameter ization of the curve. So this is something that you know, we we have a little bit of practice doing before, but it's going to be worth looking at some examples in doing some practice how to parameter rise a curve because the first step in doing a line Integral is going to parameter rise the curve. And what do I mean by that? Well, I want to give a vector value function r t 40 between, say, a and B that traces out this curve. So this is like t equals a down here, and this is t equals B And then, you know, for any time in between, this will be our 50. So remember, rt is just tracing out the curve, and here we're using t on, you know, we could think about this Is being the path of a particle in t being time. But here, I mean, it's just t is just a parameter really? T is the parameter that's kind of tracing out the curve. Okay, So once we have a parameter ization of the curve RFP, then what can we do to find the mass of the wire? Well, we just add up the mess elements and the mass elements are d m is well, you take a small piece of the wire ds and you multiply it by the density there. So D m is just lambda times. Yes. Okay, so then the mass will say AM is the integral from well is the integral over the curve. What is kind of right symbolically what we're doing here. And then we add up the mass elements. Well, that's the same thing is just integrating over the curve the function lambda, uh, along thes length elements. So a length element times its density gives us the mass element. But since we have a premature ization, the reason we found a privatization is because we can actually write this in a form that we can evaluate just is a single integral. So the integral from A to B of Lambda of exit t wife T Z f t. And then, of course, we multiply by the derivative are the absolute value of the derivative of the parameter ization. And then we have d t. Okay, so that's how we would find the mass of the wire. Given a linear density function, we prime tries to curve, and then we just add up all of these mass elements very similar to finding the mass of volume element but notice, You know, the thing that comes up is this sort of absolute value of the of the derivative of the privatization or you can think about. It's the speed if this is representing the motion of a particle and we're sort of waiting the integral with this factor and this is a very important factor on what it's really doing is it's allowing us to kind of give it a better description of what this DS is in space. So DS is just kind of a little piece of the curb, but it's what it really is is sort of a little change along the tangent line of the curve. And since we have this premature ization, this is what a change along the tangent line looks like. So without getting into too much detail, that's what's going on. Alright. So I already mentioned that there were two types of Lyon intervals and we already did the first in the first type. The line, integral of a scaler valued function, is just a natural extension of what we did with double and triple under girls. But instead of integrating over a volume region or area region were just integrating over occurred. So it's kind of the one dimensional analog of what we did before. I don't want to say that the second type of line integral is more important, but it might be more important on that's the line integral over a vector field and the this line Integral is going toe have. What's that? We're definitely going to talk about more applications of this type of line integral. But before we can talk about the line integral over over a vector field, we're gonna have to say what a vector field is. So what is the Vector field? And we're going to keep this really simple, and I claim that actually, we've already seen a couple different types of examples of vector fields. But, you know, here we're going to give kind of the most general version of what a vector field is. So in layman's terms of Vector Field is a rule or a function that assigns thio each point in space. Or, you know, we can have a similar definition in the plane, a vector. So that's kind of the you know, the general idea. Now, formally, what is a vector field? It's a function from, say, R two R two or R three to r three. So you know the very first examples of vector valued functions that we talked about. We were looking at functions say, from our into our three. So somehow we were taking the rial line and tracing out a curve in our three. So this was like a vector value function just as an example. Now, actually, a vector field is another type of vector valued function because the output is a vector, but a vector field we really want to think about is the input being more than one variable is well, so now we also talked about multi variable functions, something like this where you're you're taking, uh, maybe three variable input and output ing in number. So this was like a multi variant function. But for a vector field, we really want to think about your taking in multiple inputs and output ing vectors. So this is kind of what we want to think about what, the vector field. Just to sort of tell the story of of kind of what we're we've been doing, we spend some time talking about Dr Value functions. Then we spend a lot of time talking about multi variant functions. Now, both of these, we're gonna play a role in what we're going to talk about next. But vector fields are just kind of the next step in this story where we have, you know, we're in space and we're out putting vectors at each point. And if we think about this physically, so I'm going to draw a vector field in the plane. So I said that a vector field is a rule, or I mean, it's really a function that a science to each point in this case in the plane a vector. So let's say let's denote are vector field by f so will typically use capital letters for vector fields and again with arrows over the top just to signify that they are out putting vectors, just like with Victor value functions. And so, what does f look like? Well, you know, a typical vector field f is gonna look like this. So every point I draw a narrow I have a vector. So at every point in space, or in this case, the plane, I have a vector. Okay. And so you can sort of draw this all day, every point. I have a vector. So what in the world does this have to do with anything. Well, vector fields are extremely useful in it, and they model Ah, lot of different things. So let me give you a couple of examples. So this vector field here could represent let's say I'm a leaf sitting on the surface of the water. So I'm here and then what? These arrows are telling me what my vector field is telling me. Are the forces that air pushing me along the water? So maybe I'm right here and there's a vector here pushing me in this direction. And then I get to this point. There's another vector that pushes me in this direction, and then I get to this point. There's another vector pushing me in this direction, and so these arrows are really like forces on the water. And now the same is true if I'm in any force field, if I'm in a gravitational field, so we live in a vector field right now just sitting here, I'm in a vector field, so every point in space there is a sort of imaginary arrow pushing down due to gravity. That's the gravitational field that's a vector field, electric fields or example effective field. So if I'm an electron, then, you know, in the world around me, there are electric fields that are applying force is on, um, on objects that have a charge. And so, at any given point, that electron gets acted on by a vector field. So hopefully that's kind of a snapshot as to what a vector field is and how it could be useful. And what we want to do is we're gonna use vector fields to describe how things move in space and in the plane. And of course, there's a lot of other examples, too. But the general form, I'll just say before we kind of move on and talk about lining the girls over vector fields is that a vector field has this general form. So in the plane of Vector Field, like I said, it is a, um it's a function that has multiple inputs. So let me right the inputs. So in the plane, I'll take into inputs and X Y point. And then what I'll do is I'll output a vector, and so I'm gonna have to component functions. I'm gonna have an X component function that I'm going to denote p of X y and then a white component function that I'm going thio denote que of x y. And now P and Q are scaler value functions. So I take in an X Y point P gives me a number and Q gives me a number and those become the components of my vector field. So these were my out put arrows for every X Y point in the plane, I get a number for the X coordinate. I get a number for the y coordinate and that gives me my outfit Output vector. Now, if I have a vector field in space, I'm gonna input three numbers X, y and Z and I'm gonna have three component functions. So I'm gonna have key on P is going to depend on all three variables X, y and Z in general. And then I'll have Q and then all typically, just use an r for the third component function. So again, for every point X, y and Z in space I If I have a vector field f, then I have an X component function. Why component function and Z component function and I'm out putting a vector at each point X y and z So that's the idea of a vector field. We'll talk more about Vector fields in the next topic, but for now, that's the definition. It's just a function either say, from R two or R two or R three or three. I'm taking in a point in space or the plane, and I'm out putting a vector there. How can we think about that? Well, the best way is to think about forces. Force Fields are examples of vector fields. We live in vector fields. They're everywhere. So hopefully that'll did you primed for what's to come. Okay, so real quick. I want to remind you that we've actually already discussed a type of vector field, and I would say it's probably the most important example of a vector field, which is great. We've already talked about the most important example of Vector Field. Then we should already be in the game. So again, just imagine that I have ah, function of two variables. I could do something similar if I had a function of three variables. But just to help visualize things, let's take a scaler valued function of two variables and so recall that I can plot the level curves of F. So maybe my level curves Look something like this I don't know, something like that. So here my level curves of death. You know, we have our typical a topographical map or mountain, you know? Well, however, you want to think about our level curves Hopefully by now you're getting used to this idea. Okay, What this is and how this represent it's f. So what we have is what's called the Grady Int Field. Here we go. So maybe you already made this connection. But the Grady int Field, Grady in of F is a vector field. And hopefully now this is clicking. So why is it a vector field? Well, because what is the Grady in the Grady int For any point in space, I get a vector pointing uphill like that. So for any point in space, I have a vector showing me where uphill is. So I'll draw a few examples, okay, and and literally any point here I can pick. I have an area. It's pointing me uphill and so recall that, you know, just to kind of think about our previous description of what Vector field is, it is a vector valued function. So the X component function is the partial derivative with respect to X. And then the Y component function is the partial derivative with respect to why. And of course, I could do the same thing if I had a function of three variables. The Grady int field of a function of three variables is a vector field. In fact, the radiant function, radiant field of any any function is a vector field. Okay, so this is a very important example, and hopefully it kind of gives you a little bit of comfort to see that this isn't such a new idea way. Have you know an example of our Grady int? It's just giving us eras and those areas actually mean something, and typically that's going to be the case. We're not just going to consider random vector fields with Victor's going all over the place, they're typically going to represent something that you may have seen before. Andi, you'll see more of that as we go along. All right, so now we're ready to talk about the lying in a girl. The vector field over occurs Alright, so let's just kind of gives them set up. So let's suppose we have a vector field f and we'll sit. Just say it's in two dimensions. Of course, everything I'm going to say will apply to three dimensions as well. But it's a little bit easier to visualize things in two dimensions. So let's maybe just sketch what are vector field might look like. So just have a bunch of heroes. Maybe Maybe I'll kind of give it some feel for what it might represent. So you see that there's kind of emotion. Two, this vector field. These areas should be bent, but there we go. Give me somewhere down here. Okay, So here's an example of a vector field and you see that this might represent some sort of wind blowing, you know? So if I'm here, I'm kind of getting pushed this way. If I'm here, I'm kind of getting pushed this way. So you could just imagine that maybe this is some kind of force field. Or maybe it's an electric field or something where the arrows air sort of showing you where things were getting pushed in the vector field. That's a really good visual way to think about vector fields. Okay, so now within this vector field, suppose that I have an oriented curves like this, okay? And it's gonna be oriented. So let's give it a name. C is an oriented curve. All right, Cool. And now again, let's say that we have a parameter ization of C as an oriented curve. So C is really going to be given by some function rt, which is just accept e y 50. Since we're in the plane, okay? And now we really want to depend on kind of a physical analogy here. When we talk about the line integral, you really want to think about the curve being the path of a particle. So this is really the path of a particle here. So here's maybe t equals a That point in this point will say is t equals B. So it's starting here, and then it's going along and ending here. Then what we want to define is, or I guess what our motivation could be is, well, the work done by this force field on the object. So if you look, let's say right here, So right here on our curve, recall that we have a unit tangent vector that will just call tea with the arrow. So this is a unit tangent, Dr pointing along the direction of motion in the curve. So basically, at this point, this arrow is just pointing where I'm going as you It's a direction vector. It's a length one vector vector. And since we have a parameter ization of the curve, recall that we actually have tea Big t the tangent vector as a function of T. And so what is it? Well, it's our crime of teeth divided by the length of our prime of tea. Okay, so just a little bit of review there, and so we can think that the work done by this force field is Okay, so we're at this point and I have a force that's pushing me this way, But I'm actually going this way. So the work that the force field does on the particle moving is the dot product of the force vector field vector here and the direction of motion here. So that's how we're going to define our line integral. That's gonna motivate the line. Integral, um, definition. So the line integral and again, it's the line integral of a vector field over an oriented curve or a parameter rised curves see, and it's well, it's an integral over the curve. And I can't add up function values like I did with the line integral of a scaler valued function. What I'm actually going to be adding up is the contribution of the dot product of the vector at the point and the direction I'm moving. So it's sort of just adding up how much of the vector field is moving in the direction of motion of the curve at along the point. So what we have is that it's f dot t d s. And so again we're breaking. We're gonna break the curve into little small pieces, little line elements And then for each one of those lion elements, what are we going toe to sort of multiplied by. We're going to multiply by Theus amount of the vector field that's lying in the direction of motion. Okay. In other words, how much is the force helping us move in? That direction is what we're hiding up. Okay, So since we have a premature ization, okay, this is just kind of notation for the lining or girl wears adding up on all of these dark products. Well, since we have a parameter ization, we can write the integral is the integral from A to B of what? So I'm going to take my Dr Field f okay. And so recall that, um, the vector field we're going Thio evaluate at the point ar 15. And now what? I mean, what do I mean by this? Well, I mean, that recall that f is really going to be a function of X and y so when I write f of are of tea, sometimes this is shorthand notation just to avoid writing all this. But what I mean is f of x of T. Why is t like that? So I have that. And then I'm going to do the dot product with are crime of teeth. Yeah, and so notice. Okay, so it's f and then I have the unit tangent vector. Okay. And I'm going to be, um, dividing out by this our prime of tea the length of our prime of team. So that's really it. And then I have t team. So this is the line integral of the specter field over this parameter rised curve and now there's another way to actually think about this. Okay, So if you remember the chain rule, this is basically what the change will says. So if this vector field were the Grady int of a function, I mean, this would be exactly the changeable. So just write down what the changeable says. So basically, we're just adding up the little changes of the vector field with respect to t. Um, but, I mean, that's only in the case where the vector field is actually a grating appealed, So that's not always going to be the case. But this is always with the line integral. Is it equal Thio when we have a parameter rised curve, So as I are, you alluded to one of the primary applications of the line. Integral of a vector field over a oriented curve is what the work done off of a force field. If the vector field is a force field done on a particle as it traces out a curve, or as it moves either in the plane or in space. So if we think about a little work element done by okay, so we have ah vector. So a force field specifically f. I mean, it's a vector field, but we can think about it is actually represented Force Field. And we haven't oriented curve C than the work done by F on sea or on the particle moving along. See, more specifically is remember, work is a dog product. It really is just okay, it's f And then we just take, uh, the component of F that's along the component emotion like that. Okay. And then we multiply by this D s and the D s there just to signify. Okay, well, we're taking a small were not just taking ah unit step. We're just taking a small differential step in the tangent direction and then adding up the component of that force. And so this gives just a work element in terms of a line element multiplied by, basically just the the component of the force that's in the direction of motion. So that's what work is it z f dot displacement. So that's what this is saying. It's just f dot with, you know, the direction vector times a small change in that direction. Okay, And then if we want the total work done on the motion of the particle moving homes curve by the vector field. Then we just add up all of these little work elements. And, of course, what we get is exactly the line integral. And so again, I mean, the first time you see this, this notation, it's sort of formal. It looks kind of like what is going on? But remember that this doesn't really mean anything until we give some sort of parameter ization of the curve. Once we have a parameter ization of the curve, we sort of actually just fill in the blanks here and actually right in what? This is what this really ends up being is. Well, what we really mean is the particle moving from, you know, time equals a two time equals B. And this is where we actually just fill in. Okay, so we compose the vector field with the position of the particle, and then we do a dark product with the derivatives, and that's Katie. So of course, yeah, Once we have a premature ization of the curve, we can actually write it in a way where Okay, this is actually just a scaler value function in here that we're just doing a standard calcula 11 variable integral with respect to the parameter which were typically thinking of as time and again This is the total work done Bye f on So I'm gonna say on C, But really, I mean on a particle moving along. See? Okay, so we have now addressed the two major types of line intervals. So we have the line and a girl of a scaler valued function f over a curve c and then we have the line integral of a vector field oversee. So actually, there's going to be two special cases of the line integral of a vector field oversee. And that's the case where the curve C so is three things. So if C is simple, I'll tell you what all these things mean closed and positively positive. Lee oriented. Okay, so over here, let me tell you what all these things mean. So simple means non intersecting and then closed means we start an end at same point and then positively oriented means the orientation is counter clockwise. Okay, so let's give some examples and non examples. So here is an example. We want to make sure that the orientation is counterclockwise like this, so notice that this curve does not intersect itself. It's closed. So if it starts here, it also ends there and is positively oriented because it moves counterclockwise. So that's our example. So are non examples. So here's a curve. It's not simple because it intersects itself. So not simple. Of course we could have something like this. It's not closed. And then finally we could have a simple closed curve that's actually oriented clockwise. So this is negatively oriented. Okay, so hopefully that's clear. Simple, closed, positively oriented. It doesn't intersect itself. And now we probably I think we talked about this a little bit when we talked about parameter izing curves. I mean, we also wanted to be was called piece wise, smooth. But we've kind of been assuming all that from the beginning. Okay, but I mean, it's just a loop, you know? It's like a circle, maybe with a little bit of bend here, there, but not enough that intersects itself. So if C is a simple, closed, positively oriented curve, then we actually get to sub cases. So we get what's called a circulation and a girl, and then we get what's called a flux. Integral. Okay, so we're going to spend a little bit of time talking about what both of these are. But the idea is okay. So we first talked about line integral of scaler value functions over oriented curves. Then we talked about line Integral a vector fields over oriented curves. Now specifically here, when we have a line in a girl of a vector field over a closed, simple, positively oriented curve, we can also talk about a circulation integral oversee in A and a flux integral oversee. So let's talk about what those mean specifically, all right. So the first type of special integral that we want to talk about is a circulation integral and really, there's not a lot of new content that we're going to talk about here. The only distinction I'm wanting to make is that a circulation integral Onley comes up when you have a closed, simple, positively oriented curb. So that's the set up. So we have a vector field F and we have a curve. See that simple, closed, positively oriented. Okay, so let's just draw what's going on here, and I'll say that it really makes them a sense to talk about circulation integral in the plane and the same thing for flux line into girls. So we're really going to narrow down to just talking about vector fields in the plane. So a vector field in the plane is just a rule that assigns to each point in the plane a vector like this. So all over my plane, I have a vector. But again, you know you can think about this is like the surface of the water to the water is just kind of has ebbs and flows, and it's moving. It's applying force is to whatever is in it, causing things to move and maybe something moving in it. The force field of the water is doing work, etcetera. All the ways we have to sort of conceptualize a vector field. And what I have is this curve and now the curve specifically, I want it to be those three things simple, non intersecting, closed, meaning that it has a start point in an in point. And then it's also positively oriented. So the orientation will be counterclockwise. So actually, this way There we go. Okay. And so the circulation integral. I mean, all I'm really doing is just introducing some notation. Because when I tell you what, what the circulation, integral, actually is you're going to be like, Wait, But what are you telling me? Okay, so I'm introducing this new notation, and I'm putting the circle around the integral sign to signify that the line integral that I'm taking over this curve c is over a curve. See that? It's simple, closed, positively oriented. And so from there, I'm just going to write in the line, Integral. Okay, so this is just notation, but all I mean by this is just the line, integral. But I'm going to reserve this notation and I'm gonna use this notation when I really want to highlight the fact that this is a line integral over a simple, closed, positively oriented curve. But it really is just the lining your girl. I'm just giving it a new name. But this is going to be a very important application. Is when we have this sort of situation simple, closed, positively wanting to curve. There's gonna be a lot of really good applications. And it's really going to illustrate the idea of, uh, sort of the fundamental theorem or the're ums of vector calculus that we're moving towards. Okay, so the next type of special interest girl we want to talk about is actually a little bit different in the circulation. Integral. So the circulation integral. Basically, it was just a line integral over a simple clothes, positively integral, positively oriented curve. So the next in a girl that we're going to talk about is a flux integral. And this really will be a little bit different than just typical line Integral, and you'll see what I mean. But the set up is exactly the same. We have a vector field, and we have a curve that is simple, closed, positively oriented. So let's draw picture again. And so our vector field, whatever it looks like, all of these arrows pointing all over the place. Maybe there's some method to their madness, but there's a vector field, and then we have our simple, closed, positively oriented Kirk's oriented this way. Okay, and so the idea behind the circulation and a girl sort of the physical interpretation is, what we're really doing is we're adding up. How much of the vector field sort of rotates around the boundary of the curve like this, so you can think about it. Kind of like you know how much rotation is in the vector field. How is how much is it spinning? Sort of around this curve like that? The flux Integral is gonna be completely different. Okay, so the flux inner girl, let me just tell you what it is first, and then we'll talk about what it means again. I'm going to use this notation to signify. Okay, well, this is a sort of line integral over the simple, close, positively oriented curve. But instead of taking the dot product of the vector field over the tangent direction So basically looking, how much you know of the vector field lies along the tangent direction like that, I'm actually going to do the exact opposite. I'm actually going to think about how much of the vector field lies over the normal direction. And so where does the normal direction point? Well, it points perpendicular to the curve like this, and I'm doing the dot product. So I'm really seeing how much of the vector field is leaving this bounded region. So I'm taking instead of taking the dot product of the vector field with the tangential direction Or, in other words, seeing how much of the the vector field is kind of going with the flow going with the direction, my particles traveling or going with the curve. I'm looking at how much is going across the curve, Okay. And so you're probably thinking, Well, what is this? I mean, this is more or less just notation. Now, I'm using this kind of hat notation to signify that this is just a unit vector in the normal direction. Before we had the tangent vector or a unit vector in the tangent direction. And so I can actually write this in terms off the tanta direction. So what this really is is the You know, sometimes this is called a contour integral. Just I might say that from time to time. So this integral over a closed, simple, positively oriented curve. So what I can dio is Aiken say All right, what is the normal direction? Well, the normal direction points out, and the curve is positively oriented. Okay, So if I want the normal direction, what I can do is I can actually do a cross product. Okay? So let's think about this. So if my direction of my curve is pointing this way and I want a vector perpendicular to the curve, what I can do is I can look at t so I can look at my vector t pointing unit vector pointing in the direction of the curve. And I could take the cross product with Okay, a unit vector in the see direction. Okay, so just do kind of your right hand rule if you forgot about that. But if I point in the tangential direction and curl towards the positive Z direction were out of the page towards me, my thumb points normal and out of the curve. So what this really is is f dot the tangent direction? Yeah. Cross. Okay. You know, direction, unit vector in the que direction. And of course, these intervals are overline elements ds like that. Okay, so again, what's the name of the game here? Well, I wanna parameter rise my curve. And once I have a parameter rise curve, I can actually express the tangent unit Tangent Vector is a function of t I can express. This is a function of t. I could just do this cross product and I'll get a normal vector is a function of T. And then I just have my typical line integral. But it's a little bit different because I'm doing the dot product instead of with the tangent direction. Actually, with the normal direction and physically, the interpretation is very important, important. So we think about this is a fluid. Then along this curve, the circulation and a girl gives how much the fluid rotates. Well, the flux integral is giving how much the fluid is sort of moving out across this boundary or how much it is rippling out across the boundary. So that's the picture that I want you to have in mind as we go forward when we talk about circulation and girls and flux integral.

03:31

05:08

08:32

07:44

44:11

04:14

07:28

04:12

10:03

26:52

05:28

06:38

07:48

05:33

28:40

06:44

10:25

08:33

09:29

11:59

07:49

12:44

03:43

04:53