 Hi, I'm Zor. Welcome to Nezor Education. I would like to continue talking about fields. Primarily, we were talking about certain characteristics of the fields. Well, let me just very briefly remind you that we were using a concept called gradient of a scalar field. Scalar field is when the value, some value, real value, is defined at any point in three-dimensional world and the gradient was a characteristic which basically explains which direction these values of the field f are growing faster, the fastest actually. So and the formula for this was it's the vector which has coordinates df by dx, df by dy, df by dz. Partial derivatives of function f by x by y and yz. So from a scalar field we get the vector field of its gradient and we also used the symbol nabla, which is kind of a pseudo vector, but it's a set of three operators of partial derivatives by corresponding coordinates and using this symbol pseudo vector, whatever, triplet of operators called nabla. We can use we can basically write this thing as nabla f of x y z. It's like multiplication, basically, because if you multiply scalar by vector, it means this scalar should be multiplied by each component of this vector and that looks like exactly the partial derivatives. It's a symbolic, nothing to it, but it's convenient, basically. Now, the second concept which we were introducing was divergence. Divergence of the, in this case, our subject of interest was a vector field. And in some way it's a reverse operation because here we from scalar field got the vector. This one from the vector gets the scalar field. We were using the concept of air density and the wind. If this is air density, basically it's a higher pressure, which means it's the source of the wind and this is direction of the wind, where the pressure changes the most, from the highest to the lowest. Or from the the question is which one is with F is the lowest part or the highest part. So the wind goes this way or that way. It's increasing or decreasing the pressure. Now, in this case, it's the reverse operation. If you have a wind where are the areas of the highest concentration of air? That's what the divergence gives you. And the formula was dvx by dx plus dvy by dy plus dvz by dz, where vx, vy and vz are components of vector v. And using the symbol nabla, we can actually consider this as a scalar or dot product vector and vector, because if you have three components of one vector and three components of another vector, the scalar product is exactly this. Multiply first by first plus second by second plus third by third. So it looks like this. Again, it's symbolic because it's not like a real vector. It's a pseudo vector or whatever. But if you are using the same rules of multiplication by constant or scalar product of two vectors, it looks exactly what the formula is. So this is basically what we have learned about the field and usage of symbol nabla to express certain characteristics of the field. Our next characteristic which we would like to discuss is called curl. And today I will talk about curl into dimensional world, but the real usage is in a three-dimensional world. And that's what I'm going to start right now. I needed this kind of repetition just to basically resume to restore the memory and introduce basically again the symbol nabla because it will be used. Okay, so we are going to talk about another characteristic of the vector field. It's about vector field and it's usually about three-dimensional vector field. And the characteristic is called curl. In other languages, it's called differently. It's called rotor. And some French word which resembles rotor, which I don't know really how to pronounce, so I will not. But that's also the same thing basically. In English, it's curl. All right, now this lecture is part of the course on unizor.com and the course is called physics for teens. The part of the course is called waves and we are talking about field waves. But before introducing the field waves, I wanted to talk about fields in general and characteristics of the field because the field waves are, well, usually electromagnetic field waves are defined by Maxwell's equation and the most convenient form is related to this symbol nabla which we were using. That's why I wanted to repeat a little bit about it. So, back to curl. Well, just the name of this particular characteristic curl or, again, in other languages, rotor. It actually means that our vector field is somehow turned around. Perfect example of the curl in the three-dimensional world is tornado. When tornado is, you know, spiraling around, it represents each molecule of air which moves in this tornado has a velocity and all these vectors of velocity are surrounding, well, certain axis, if you wish, I don't know. And the whole thing actually is moving all around. So, this particular place where tornado is concentrated is definitely having a very high curl. Now, there are other cases when the air is completely quiet, no wind, nothing, then you would consider this particular vector field with, like, almost zero velocities to have no curl at all. Curl was equal to zero. So, basically, it's a quantitative characteristics of this circular type motion. And obviously, the faster we are moving, the curl should be greater, right? I mean, there are certain normal expectations of these characteristics. And what I am talking today about is basically quantifying these characteristics. Now, curl in a three-dimensional world where it is actually most useful, if you have vector field v of x, y, z, v, then to, I hope, to your surprise, but to my surprise, the curl of this vector field is actually expressed as nabla and then the vector cross product with v. So, this is the vector, well, pseudo vector, right? This is also a vector. It has three components. This has a three components. And obviously, the vector product is expressed as their corresponding formulas for all three components of the result. So, if you remember, the vector product is perpendicular to this and this. And, well, there are certain formulas. And we will talk about this when we will talk about three-dimensional world. That will be the next lecture, 3D. Today, I would like to start in a two-dimensional world where I have only two components and nabla will not participate, by the way. But I have to define it properly and have some quantitative expression which characterizes this. So, in two-dimensional, it's kind of easier. First of all, how can we feel that the curl actually exists? I mean, just looking at the vector field, I mean, if this surface of this whiteboard is our two-dimensional world, my vector field would be basically for every point I have to put some kind of a vector here, et cetera. It has direction and magnitude, et cetera. And different fields, obviously, have different curl which I would like to somehow observe and quantitative. Now, this field probably does not have any curl, right? That's kind of expectation, intuitive understanding. And this field seems to be having a curl, right? In this particular direction. It's definitely circular. So, we will express this curl in some way. And instead of kind of deriving the formula for the curl, I will give it to you right now. And then I will explain why this formula is what it is. So, let me just give you the definition. Now, I assume that the vector v has components vx of xy and vy of xy. What it means is if I have x and y, every vector v, it has components. This is vx of xy. And this is vy. So, we have two components. And for this, dv y with dx of xy minus dv xy by dy. And now I will try to explain why. So, what is partial derivative of vy by dx? Well, that's basically a speed of changing in the y direction as we are moving towards x direction. So, let's say we are increment. Now, this is point xy. Now, let's increment x. And we will use another point. This is x. This is x plus delta x. Now, what would be this particular vector in this case? Let's say it goes this way. For example, well, it's y component is changing from this to this. So, it's diminished, right? It was positive, now it's negative. So, as we move towards increasing of argument x, my y component decreasing. Now, what does it mean? It means that the vector field is turning towards x if it's decreasing. If y component is increasing, let's say it goes this way. So, the y component would be this. That means it's increasing. So, it goes away from the x. So, basically, this component gives you whether the y component is moving towards x or out from the x. Now, similarly, this component, when y is changing, we are performing partial derivative by y of the x component. It gives similar. As we are moving along increasing the y argument, does my x increasing or decreasing? If it's decreasing, it means it's moving towards y. If it's increasing, it moves towards x. So, their difference gives you basically towards which axis we are moving left or right, which means we are turning or not turning. If this is equal to zero, it means we have exactly the same direction of the field. It's increasing in as much as it's decreasing. So, that's why it seems to be a convenient characteristic of the curl. I wanted to say the word curling as a property of the field, but there is an Olympic sport curling, which has nothing to do with vector fields. So, I was kind of hesitant to use this word, so people might not really understand what I'm talking about. I'm not talking about sport. You know, when they were sliding on the ice, sliding some kind of a big, whatever, blobs. Anyway, so, this seems to be a good characteristic of whether the field is turning towards x or towards y, as it's slightly infinitesimally moving from the point. So, basically, it's a characteristic of the point. Now, let's go to a physical level of this. The physical level, physical explanation might be like this. If you have a tiny pedal wheel, now, this whiteboard is the two-dimensional surface, and I'm placing this pedal wheel on the axis which goes perpendicularly, so it can spin this or that way. Now, if I will put this into any particular point, this pedal wheel will or will not it turn. That's the question. If it will turn, then the direction and the speed of the turning at angular speed, if you wish, would characterize the field curvature in this particular case, how curly the field is in this case. So, obviously, if the field around this point has this type of vectors, it means it's a circular kind of a field, it's like a tornado, it will move the pedals either this way or maybe that way, depending on the vectors. If vectors would be different, if vectors would be like this, the whole field, let's say it's a wind which goes into the same direction, then obviously this wheel will not spin at all. So, my question is, what kind of, well, let's just exemplify a few examples of the field on the surface, the two-dimensional field, and let's examine how this particular characteristic, the curl would actually work, and whether it corresponds to our intuitive understanding of this. Now, my first intuitive understanding is that if the field is constant, which means v of x, y is equal to a, b at any point regardless of x and y, where a and b are two constants. If they're constant, they're not changing with x and y. I presume this is kind of a field, this is the vector, this is a and this is b for every vector. I assume that this wheel would not spin because we will have the same pressure on left and on the right side, and it should not really turn, but obviously this would be zero because constant partial derivative by x or by y, since it's a constant, is zero, and this is zero. So, obviously the curl is zero as we expect. Now, let's change it slightly. What if my field is slightly different? My field is like this. So, it's still unidirectional field, now assume that I put my wheel somewhere here. Will it spin? Yes, it will, regardless of the fact that all the vectors are unidirectional. Now, on the right side of the wheel, my pressure would be less than on the left, and it will be turning this way. So, you don't really have to this type of circular arrangement of vectors to spin the wheel. This is unidirectional arrangement of the vectors, and it will still spin. So, in this particular case, you would still have a non-zero curl if you calculate it. But what unidirectional field doesn't really have a spinning wheel? Here is my suggestion. What if it's this way? How about this field? You see, in this case, when equal fields are parallel to each other, and basically if this is a direction and this is a perpendicular line to a direction of all vectors. So, all vectors which are on the same distance from this perpendicular line, if all of these have the same magnitude, then my wheel will not turn. And thus, I can actually prove right now. Let's just assume that this is an x-axis and this is a y-axis. I mean, if it's differently, then I can always turn the axis because turning or not turning of the wheel doesn't depend on how I position my axis, right? Because it positions. It depends on the vector. So, I will position my axis in this particular way. So, let's say they are all parallel to y-axis. So, what I can say about vector is that v of xy is vx of xy and vy of xy. Now, what I would like to say is that if these vectors, they all have the same y-coordinates. So, the y of x and y is independent of x. vy, which is vertical component, is independent of x. No matter how far from x I'm moving, so it doesn't have this component to function. It's actually vy of y. And the x of xy is equal to zero, right? Because projection on the x-axis of all these vectors, I chose x to be perpendicular, right? So, I have this and I have this. Now, what if I have these type of arrangements? And what about my formula? Well, my vy does not depend on x. So, this is supposed to be equal to zero. My vx is always equal to zero. So, obviously, since it's a constant, it's also zero. So, again, I have zero minus zero and I have a zero curl. So, in this case, I do have a zero curl. Okay. Now, let's have another example. Let's have a circular, something which you would expect to have a curl. Okay, let's just have this example. So, if this is zero xy and I would like actually to build this type of circular field. Now, what does it mean it's circular? Well, it means that vector towards the point xy should be perpendicular to the vector of the field. So, xy should be perpendicular to vx of xy vy at xy. It's curly brackets. And this is also curly brackets. So, this is the vector which is radial vector into the point. And this is a vector which is vector field at this point xy. They should be perpendicular. Well, what does it mean perpendicular from the vector algebra perspective? It means that x times vx, their scalar product should be equal to zero. That's what it means perpendicularity. So, perpendicular vector have scalar product equal to zero. Okay. Remember, magnitude of one times magnitude of another times cosine of angle. Angle is 90 degree and cosine is equal to zero. Okay. Now, basically, there are many different vector fields which satisfy this equation. And all of them will be, well, kind of circular. I will suggest two vectors. Vector a would be minus y and x. And vector b would be y and minus x. Now, if you will do a times xy, it would be minus y times x plus x times y, which is zero. Same thing here. b times would be y times x plus minus x times y, which is again zero. So, both are good enough. What's their difference? Well, the difference is in the direction. For instance, if I will take this point, let's say one zero, then this would be, so, x is equal to one, y is equal to zero. Now, this would be zero, one. So, the projection onto the x would be zero and projection on y would be one, which is this projection on x zero, projection on y is one. So, this is vectors which are directed upwards. How about this one? Well, this one at point one zero would be zero minus one, which means it would be this direction. So, it would be a circular motion into an opposite direction this way. Now, this way is counterclockwise and this way is clockwise. All right. So, let's just consider one of them, which is this one. What is its curl? Okay. Well, this is vx of xy and vy of xy. So, dvy by dx. So, this by dx, which is x by dx, it's obviously equal to one. Now, this vx by dy with minus. So, it would be, again, minus and then minus one. Right? d of minus y by dy is minus one and this minus would be minus. So, it would be, right? Now, what would be, what's the problem with that? Okay. Now, what would be with b vector? With b vector, which is y minus x. So, this is vx and this is vy. Derivative of vi by dx, derivative of minus dx by dx, it's minus one. Now, derivative of dy by dy, this is vx by dy is one. So, we have minus one, minus one, which is minus two. So, what's interesting is that circular field with a counterclockwise direction of, well, direction, counterclockwise direction has a positive curl, which is two in this case, and the clockwise is negative minus two, which we kind of expect basically, right? Because this is positive direction, usually of angles and this is negative direction of angles. Now, what's interesting is that our curl is constant. So, this is a field with these vectors, this vector field is the vector field of constant curvature, so to speak. Well, curl, constant curl, let's talk it this way. Okay, why? Look, if I'm moving further from point zero, my, now let's talk about curvature. The curvature of the circle is diminishing because the radius is greater. The greater the radius, the less circular, so to speak, each particular part of the circle. Now, this is definitely a circle. This is also a circle of greater radius, but the curvature here is much less than the curvature here. So, it seems to be that we should not really feel the same curling, I will use the same, the same curling, the same curl of the field the further we are, because it's not curved. However, don't forget another thing, with this type of vector the magnitude of this vector is increasing. So, here we have small vectors, here we have greater vectors. So, our vector field not only is circular, but it's also increasing the magnitude proportional to a distance from the origin. So, it looks like, consider you are moving on a carousel. If you are moving, if you are standing somewhere close to the axis, but it spins very fast, you feel about the same, well, it spins not very fast, I'm sorry, you feel the same type of centripetal force as if you are further from the axis, but it spins faster. So, basically, again, if the speed is proportional to the radius, then you feel the same no matter where you are. Your centripetal force is relatively the same. So, that's why you have the constant curl in this particular case. So, the further you are, well, if our field vector is velocity, the faster you move. And that's why your curl would be the same, because you would feel the same kind of a curvature of, you will feel the same pressure if you are standing at the wall, let's say, which is spinning cylinder. Okay, now, there will be probably some exercises about calculating the curl. But, again, this is a two-dimensional curl. It's just a preparation for a three-dimensional case. Now, it's simpler, because right now, since you have a surface, a flat surface, your spinning wheel, this panel wheel, is kind of natural to place in place and see whether it spins or not. Whenever you are in a three-dimensional field, it's kind of difficult, because if you position at any point in space, and there is a wind, let's say, it will probably spin. But, if you turn it a little bit, it will still spin maybe with a different speed. So, it's not as easy to explain what the curl actually is, because it all depends on many different factors. Three-dimensional world is significantly more complex in this particular case than two-dimensional world. And in this particular case, your axis is fixed. If you fix the point and put the spinning wheel, the axis is fixed. In three-dimensional world, well, there are many ways you can put this panel wheel. And that's why it's kind of difficult to define exactly, quantitatively, what the curl actually is in that particular place. But we will do it in the next lecture. I do recommend you to read the text notes for this particular lecture. You go to unisor.com, the course is physics for teens. You go to direction, it's called waves. That's the part of the course. And within the waves, you have field waves. One of the lectures in that category is curl in two-dimensional world. That's it for today. Thank you very much and good luck.