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02:07

Fangjun Z.

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

00:50

Masoumeh A.

0:00

Virendrasingh D.

01:02

Anshu R.

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Okay, so we've discussed Vector Fields and we have discussed line and the girls of Dr Fields over specific types of oriented curves. And so now what we're going to do is we're going to talk about a very important property that vector Fields can have. And now the motivation comes from physics. And so I'm gonna pick up the perspective from physics. So here's the idea. Well, first of all, we're going to live in the plane for the time being. Something similar will be true in space. But as we've seen time and time again, if we can understand what's going on and two dimensions, then that will clear things up quite a bit. So let's say we have a vector field F. And I'm just gonna draw, um, sample arrows of F. So, you know, one thing we can think about is that maybe F is an electric field, something like that, or if you prefer, it's, you know, the surface of the water kind of pushing me around, but I really want to think about it, has a force field. So all of these Arabs, representing forces and as a particle, moves in the vector field the force. Eras are doing work on the particle, so our line into girls will really represent work done on the particle moving through the vector field. Okay, so let's say that we have two points. We have a point here and we have a point down here. And so you know, this is kind of an interesting question. So I want to travel from this point to this point. Now there's kind of a shortest distance geometric distance. I could just go down the street line. Or, you know, I could go out here and then kind of move around and go like that. And so this could be an oriented curve. I'll call this curve C one, but they're really infinitely many paths that I could take. I could also start at this point and go down and around this way like that, and this will also be oriented and I'll call this path C two. So here's the question. So if I started this point and I travel to this point, I have the option to take this path or I have the option to take this path. And in both cases, the force field F the vector field is going to do work as I move along the path. So this is a particle moving on this path. F is doing work on the particle. Same thing over here. And so I can set up the line integral for the work done by the vector field along the path. See, one which recall is just C one f dot with the unit and a vector DS. And then I can look at the work done or the line integral along C two. So physically, this is really significant. Okay, so I take two different paths you would probably expect for the work done to depend on the path. Okay, so maybe if I choose a longer path, then the vector field will do more work. If I take a shorter path, maybe you would expect that the vector field to do less work. So this amount of work will say his work. Number two This amount of work is work number one. And here is three fundamental question that we wanna answer in this lecture. When does the work from Path one equal the work done long path to And now I really want to think I don't want to just pick two special curves, but I really want to think is if I take any curve. So you know I could take a C three or C four. Like I said, there's infinitely many curves I can take from this point to this point. When is the work done? Independent of the path taken and in other words. So it on Lee depends on the starting points here and here. That's the question that we're going to try to address moving forward. Okay, So just to reiterate the point on the last slide, a vector field F is said to be path independent. I really want to reiterate this definition because it's really the essential part of this lecture. So f is path independent if for any two paths connecting any two points. So notice that I'm saying any So I take any two points and I take any two paths connecting those points. So those pasts are gonna be oriented curves you can think about them is past that Ah, particles taking. So if his path independent for any two paths connecting any two points, the work done by S. Or, in other words, the line integral along the two paths is the same. And this is such an important property in physics, because what this is saying, think about F. Let's just reinterpret what this means if F is a force field. So in other words, if F is a vector field that represents the, you know, the forces acting and you get the point, so maybe it's a gravitational force field or its electric force field or even, uh, you know, air resistance. So if I'm up in the sky, you know, the wind pushing me is just giving me a vector field everywhere I am. I have an error pushing you somewhere. So whatever that force is that the vector field represents if f is path independent. So if yeah, is path independent, then f conserves energy. So this is a huge deal, Okay, So conservation of energy is an essential topic in physics. So if f is represents a force field and F has this property of being path independent, then what that means is that F is going to conserve energy, meaning that if I start and end at the same point, if I travel around and I end up at the same point. That means my net energy change is going to be zero. So I haven't lost any energy. And so maybe it's best you know, to kind of think about forces that don't conserve energy. So, for instance, friction is not going to conserve energy. So if I think about the vector field that represents friction, it's not gonna be path independent. Because certainly, if I take a longer path, I'm gonna lose mawr energy to heat to that friction. Same thing with drag force drag force isn't going to conserve energy, so it really is a special property to conserve energy. Now, some examples of force fields that are path independent. So gravitational force fields our path. Independent electric fields, our path independent. So there are some very important examples in physics of vector fields. In other words, force fields that our path independent e they conserve energy. So this connection between path independence of the vector field and conservation of energy is just inter linked in this way. So as you can see, it's going to be very important for us to be able to determine whether or not a vector field has this path Independence property. Okay, so unfortunately, before we can answer this question of when a vector field has this path Independence property, we got to go through some definitions. So just kind of some background information about vector fields that's gonna help make everything a little bit easier to state. So the first thing we need to kind of nailed down and this is something we've seen before is what's called the Dell operator. So the Dell de el operator is Well, it's a little bit weird. Okay, so it's this Nabila symbol that we've seen before, like for the Grady int. But what I'm gonna put here is I'm just going to put partial with a partial partial X in the X coordinate Partial partial. Why in the Y coordinate in partial partial Z in the Z Coordinate. And now, of course, I'm assuming here that I'm in space and I'm using Cartesian coordinates X, y and Z, and this is really just a formal definition. Okay, so by itself, this doesn't mean anything. And now it's an operator in the sense that operators in math act on things. So the Dell operator acts on things, so we've already seen the Dell operator act on functions. So if I take the Dell operator and I act on a scaler valued function eth, well, then I just get you know, so the first component, this this partial with respect X operator acts on F then the partial with respect to why acts on F and in the partial with respect to Z acts on F. So the Dell operator acts on scaler valued functions by giving me the gravy int. Okay, Yeah. So the reason I'm entered kind of separating the definition. Just isolating this deal operator is that we're gonna have other operations. So the Dell operator is actually going to be able to act on vector fields, and it's going to be able to act on vector fields in two ways. So the first is going to be what's called the divergence about sorry of a vector field f. Okay, so the divergence of a vector field f is going to be this. Well, I'm going to write it again, just symbolically. So it's the Dell operator, and then I have a dot product with Well, let's actually say that my let's give some kind of content to my victor value My sorry, my vector field. So the vector field is gonna have component functions, so we'll have p, which is a function of scaler, value function of x, y and Z. And then we have Q, which is a scaler valued function of x, y and Z, And then we have our which is also a scaler valued function of X, y and Z. Okay, so that's the typical form of a vector field in three dimensions. And that's just gonna help me write down what? The divergences. So the divergence of this vector field is Okay, well, I'm writing this symbolically, but it's going to help me see exactly what this is. So notice that the Dell operator is sort of a vector now in the components are these derivative operators. But I'm gonna take the dot product just as I would before. But when I multiply this operator partial derivative with respect to X times P, I'm just going to get P X. So the partial derivative with respect to X of the function p and then I'm going to do the y component times the y component. But that's just going to give me the partial derivative with respect to why of the function. Que And then I do the third component, which is e. So I end up with the partial derivative with respect to Z of the component function are so this is the divergence of the vector field and so geometrically. What is this telling me? I mean, I'm giving you this definition, so one way to think about divergence and we'll talk about this a lot more when we talk about green serum is the divergence is basically telling me how much the vector field is spreading out. So a vector field that has a lot of divergence is gonna have arrows pointing away from maybe a source so I can think about it. This is a fluid. If I have a source of the water right here, my vector field is rippling out away from that source. So that's gonna be an example. The vector field that has divergence. So it's diverging away from a point so you can think about a ripple is like a divergence in a vector field. Okay, so there's one more way that the Dell operator can act on a vector field, and that's called the curl okay of the Vector field. And so we'll have another geometric interpretation of what the curl is. But let's first say what it is. And again I'm going thio just right. The F is p Q. Are just to make it a little bit easier to say what the curl is. And first of all, I'm just going to symbolically right that the curl is the Dell operator and then I'm going to write the cross product simple. So here comes the cross product of the Vector field. And so let me write down what I mean by this. Well, let me just sort of formally write down the cross product. So I have the I and J and K component I have partial. So from the Dell operator, I have partial partial X have partial partial why I have partial partial Z and then from F. I have my component functions, p que and are and again now when I take the determinant here. So that's what I do, and I take the cross product when I multiply like partial partial X with Q. I'm just gonna take the partial derivative, and so what I end up. Getting here is well. Are why minus cues e And then remember, J I have to do in the opposite order. So I get P Z minus our X and then I have e don't like that are our X There we go. And then I have Q X minus p Y Okay, so my output for my curl is itself a vector field. And here my command component functions. Okay, So again, that's just the definition. What is the curl? So recalled it. The divergence of a vector field basically measured how much the vector field waas or is spreading out away from a point. So this is an example of a vector field that has divergence. So notice that all these arrows air kind of diverging away from a central point, which may be that central point is a source of the water. So the water is flowing out of that point. And so my vector field is spreading out away from that source. So that's the vector field that has divergence. Now the curl has the name suggest basically measures rotation in a vector field. So a vector field that has curl is going to be rotating around a central point like this. So this is an example of a vector field that has curl. And so if you think about like, a whirlpool in the water, that is an example of a vector field around that world pool that has a lot of curl. And so somehow you know, more or less precisely. All vector fields are made up of divergence and curl, and they're sort of, you know, and, uh, the antithesis of each other. So if you have a lot of divergence, then you're not gonna have a lot of curl. And if you have a lot of curl, maybe you don't have a lot of divergence. Uh, and so those are kind of the two extremes. You can either be spreading out or you can be rotating in a vector field. So these air going to be the ideas that we need moving forward now, Right now, we're going to kind of put divergence on holds. The divergence will play a really big role later. But the reason we're introducing Divergence and Curl is because we're actually going to use the curl of a vector field to be able to determine whether or not. It has this path Independence property and which is really cool to think that somehow something about the rotation of the vector field is going to tell me whether or not I have a path Independence property. Okay, so we need to introduce one more concept, and then we'll kind of tie everything together and review kind of everything we said and what we can take away moving forward. And so the last concept that we need to talk about is what's called a potential function, and a potential function is actually going to tell us whether or not a vector field is conservative. Okay, so let me just tell you the definition. So a vector field yes, is called conservative, and now you're probably thinking, Oh, conservative, it conserves energy. Well, there's going to be a connection between our path independence property and conserving energy. So a vector field is called conservative. If there is a scaler valued function, let's just give it a name. Let's just call it fee such that. So here we go. So remember that we had a very important, uh, example of a vector field, and that was the Grady Int Field Okay, so if we have a scaler valued function, then we conform a vector field. But now we're going the opposite direction. We're starting just within arbitrary vector field were saying that Vector Field is conservative if it's the Grady int of a scaler value function. And so the logic here is really important. So it's important to just maybe stop and just think about this. So we said that if we have a scaler valued function, the Grady int of that function is a vector field now. It's not always going to be the case that if I have a vector field, it comes from being the Grady in of a scaler value function. If I just sketched some random arrows, it's going to be very unlikely that there's actually a function that when I take the Grady Int, I get that random combination of Paris or vectors. So it's a really special property, the vector field being conservative, and now it also, if you recall we really thought about the Grady int of a function is being the derivative, and so in the back of your mind, you should also think that what we're finding when we find this function fee. The scaler value function is an anti derivative. And I'm just going to kind of put that in Quote Sophie is and anti derivative of the vector field. Yeah, and I'm putting this in quotes and, you know, I'm putting this in quotes, meaning kind of in the same way. The Grady int is the derivative of fee. You know, fee is really an anti derivative of this vector field F. So the question is, when can I find an anti derivative of a vector field, or when can I find a function that, uh, that might take its radiant? It's equal to the vector field, and this is where the curl comes in. So this is super super cool. So if our vector field is defined on a simply connected could connected domain So what I mean is that the part of you know whether space or the plane that efforts defined is simply connected and, well, I So what is simply connected mean? Well, I'm not going to tell you precisely what it means. That's more of a top, a logical idea. But you know, for us, I'm just going to say it has no holes. Okay, so it's not like defined on an annual IHS or something that you know we're part of. It's missing. Okay, so for us, it's gonna be every vector field we talk about. But, you know, this is just one of those technical things. So if F is defined on a nice or in other words, if it's just to find everywhere, that's fine. So if it's defined on this simply connected domain and this is super cool, the curl of F is equal to zero. In here. I'm gonna put an arrow over zero to mean Well, the curl of this a vector field is itself a vector field. And so what I mean is that at every single point, the vector field that I get when I take the curl of our vector field F I get exactly is the zero vector. So that's a pretty strong statement, but it's it's also a very simplest okay, So if f is defined on this nice domain and the curl is zero, then F is conservative. And of course what I mean by that Mhm there is a fee. Now I'm just saying a fee. I'm not saying it's unique or anything. There's just a fee such that the Grady int of V equals f. Okay, so this is just about as technicals is going to get, but just kind of step back and think So. I'm saying that a vector field is conservative if it has an anti derivative and the Vector field has an anti derivative, if the curl is zero. So in practice we're going to be given a vector field and I'm going to say Okay, well, I would like for it to be conservative. How do I check and see if it's conservative? Well, I just take the curl. And if I get exactly zero when I take the curl, then F is conservative. And actually there's going to be a way for us to figure out what fee is. So if death is conservative, we can actually figure out what an anti derivative this more or less. We just are literally taking an anti derivative. Now it's a little bit more complicated. We'll do an example, but we just want to find an anti derivative, and we'll be able to do that. Okay, so let's talk a little bit more about physics for a second, and I just want to introduce kind of some terminology. So if the vector Field F is conservative, then the function fee such that well, the Grady int of feet is equal to our vector field. F is called a potential function of our vector field. And so I just wanna make a little note about some of the physics is going going on behind the surface. So if F is a electric field, or if it's the force field kind of induced by gravity, then both of those are conservative. Ivory mentioned, and they come from being the Grady in of potential functions. And those functions air called the electric potential function. So if you take a class on electricity and magnetism, you talk about okay, finding the electric potential, let's just finding the potential function whose Grady Int is the electric field. And then there's also a gravitational potential function, and that's more or less just related to, you know, uh, differences in potential energy. And so the Grady in of the gravitational potential is the gravitational field that's giving the forces the gravity is actually acting on the arrows, that air saying Okay, I'm pushing you towards Mass. So that's just kind of segmentation our terminology to think about in the back of your mind when we're, uh, you know, discussing vector fields and number finding potential functions. We're really finding these functions that are giving us the data once we take the grading it to get the fields. Okay, So let's state a very important theory right now. So I said that a potential function is more or less an anti derivative of F. So if you recall from Cal one we had this little thing called the fundamental dear, um, and one of the fundamental the're, um of calculus say, Well, there were two parts and so I'll say, I guess one of the part of the part that we're going to kind of connect back to here. So it's said that if I took the integral from A to B, the definite integral of the derivative of a function f prime of X dx well, somehow you know, adding up the area underneath the graph of the derivative function, all I really needed was information about the in points A and B. So if I had this anti derivative of F prime I could say the integral was just f f b minus f a b. So really the integral just dependent on the end points. If I had an anti derivative well for line into girls, we have an analog of that. So the fundamental Vera for lying in a girls is going to be almost exactly the same. It's going to say that well, if I take the line integral now, this is gonna be from T equals a two T equals B of the vector field. And then, of course, I have a parameter rised curve. So our f t see and then we will do the dot product with the derivative. Okay, so there's our line integral. So let's suppose that F has a potential function. So there's a function. Fea says that the Grady it fee is equal to F. Well, guess what this is going to equal? Well, it's going to equal basically just the value at the end points. It's going to equal fee of well, wherever I'm starting. So are of a minus fee of where I ent. So this is very much analogous to the fundamental theorem of calculus. If I have an anti derivative of the immigrant. Then I could just evaluate the integral by evaluating the in points that the anti derivative, same thing here, if I have an anti derivative of F or a potential function of F. Or, in other words, if F is conservative, then I can just evaluate the line integral by evaluating the potential function at the two endpoints. And this is where, if you've studied physics or if you study gravitational potential, you know this is true that the change in gravitational potential energy Onley depends on where you start and where you end. And that's really what this is saying. That yes, there's all these forces acting and there's all these choices of past that I could take between the two points. But somehow the change in gravitational potential energy Onley depends on the end points. And so that's the fundamental the're, um, their line intervals. All right, the grand finale. So now we're finally ready to go back and discuss the problem of path independence. Okay, so let me draw my picture again because pictures are awesome and I wish I could draw more of them. But sometimes we just have to say things. Okay, so we have some sort of vector field. I'm drawing it randomly. But, you know, normally there's some sort of order, of course, but okay, here's our vector field. Anyways, in recall, we have two in points here and here. And so I take one path traveling the sweat and we call that C one and then we take another path here again, traveling this way. And we call that C two and we want to know when the work done or in other words, the line integral are equal. So our vector field is F So we want to know when the line in her girl oversee one of f dot t yes, equals the line Integral oversee too for any two arbitrary paths. So we want have to be path independent. Well, here's the punchline. If f is conservative, then guess what? Well, the hint is sort of in the name conservative f will conserve energy etcetera than f is path independent. So, in other words, the any line integral of F over any two curves connecting the same points will be equal. So we actually will have equality for these two line inter girls, regardless of the two endpoints, regardless of the path taken and so we can, you know, take advantage of that. Just choose the easiest path. All right, so let's see for a second. Why, that's true. Well, if ethics conservative, then we know that f is equal to the Grady int of some potential function that we call feet. Okay. And so here's what I want to consider. I want to consider the closed loop, See which is given by taking see one. So going See one going this path and then traveling seed to backwards. So I'm just going to kind of write that in a strange way. So c is just see one, then C two backwards. Okay. And so notice that. Let me trace out what I'm saying. CSO see is I take C one and then I go back backwards along C two. So I'm traveling this way. And then this way. Okay. So, notice I'm traveling C two backwards. Okay, so what does that mean? Well, let's look at this circulation. Integral. Oversee. And now notice that C is a, um, closed loop. Okay, now it's the way I've drawn it. It's actually negatively oriented, but That's not really a problem. That's not gonna affect on the line, integral. So I do this line integral over this closed loop. And what is this going to equal? Well, I just do see one. So it's just the line. Integral. Oversee one, But I do. C two backwards. So what happens when you do see two backwards? Well, think about the direction. So I'm just projecting. When I do a line integral, I'm projecting the vector field along the direction. As I go along the curves. I'm just finding the component of the vector field that's pointing in the direction of motion. But now the direction of motion here is going to be backwards. So in other words, I'm just replacing the direction vector T unit Tangent Vector with minus T. So I just get minus the inner girl of C two. Okay, but this is a closed loop. I'm starting and ending at the same point. I'm starting here, and I'm ending here. So by the fundamental, the're, um if I look at this in a girl this and a girl over the closed loop. So, in other words, this circulation integral, it's on Lee going to depend on his in points. It's actually going to equal fee of the starting point minus fee of the ending point. But starting point, an ending point are the same. So this is zero. So again, because F has a potential function, it has an anti derivative. I can apply this fundamental theorem for line into girls and say that actually, this line integral just equals the difference at the end points evaluated by this potential function. But that's just the same number. It's just fee of this point minus fee of this point, which is zero. But that means this minus this equals zero. So the integral oversee one must equal the integral oversee, too, and there's the finale. So if F is a conservative vector field, then F will be path independent. It will conserve energy, so that's why grab the gravitational field has a gravitational potential function. That's why gravity conserves energy. The change in energy inter gravitational field Onley depends on where you start and where you end, not the elaborate path that you take in between. Now, unfortunately, there are non conservative force fields acting on you. There's drag force and friction so we don't live in an ideal world. But at least the part of the vector field, coming from gravity we know actually works. Ideally, it conserves energy.

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