Hi everyone and welcome to our lecture on vector fields. We have seen Functions before and just as a friendly reminder, of course, what a function is. Usually, write it as f of x equals some equation. I want you to think of this now in a new perspective. I want you to think of it as one number in. Our input is a subset of the real numbers and then one number out so another subset of the real numbers. In this case here, all non-negative reals. When we switch over to multivariable functions, you then started to see functions of the form f of x, y equals and maybe there was some other equation or whatever it did. But the point is both of these functions output, even though it's a multivariable function on the input, they both output a scalar. They had as their output, one single number, even though we changed the inputs and we can often include if we wanted to more than two inputs. It didn't quite matter just how many inputs we had. Now what I want to do is change how the function outputs a number and then this introduces to our notion of what is called a vector field. A vector field is a function that outputs a vector. The keyword here is of course, outputs a vector. I can have a vector field that takes one thing in and outputs a vector or maybe two things in. But the key here is that now I'm outputting a vector. One weird thing about this is we call it a vector field is not called a vector function. Sometimes informally, we can call it a vector-valued function, that just means it outputs the values that are vectors. To distinguish them between their scalar output counterparts, we often use a capital letter F, and depending on the book, they may put a little vector over it even to stress the fact that it's a vector-valued function. It looks like a function, it feels like a Function often multivariable but now the key is I'm going to output a vector. Maybe I'll output X squared comma Y squared and it works just like any other function you ask all the normal questions that you would with any other function. How do you evaluate this thing? What does the graph look like? How do we do that? Just to show you how you would output, if I input zero comma one, this would output the vector zero comma one. Does it have to match? No. Just a coincidence based on easy function, but you can put any function you want inside the component of the vector, exponentials, logs, trig whatever you want. In the same rule, there's an input and output game you give me some numbers and I give you a output back. In addition to evaluating these new function, these vector fields, we'd like to draw a picture to gain some geometric intuition behind them. One of the things that's a little challenging now, especially if I have a multivariable vector function. Let's do one half, let's use i, j notation. I plus J. A friendly reminder that i is one, zero and j is zero, one. If you get the i, j notation, that's perfectly fine if you don't like i, j notation, not my personal favorite. Just remember that this is always easily converted back into vector notation. We can write out all the steps to see it. Wherever you are handed a vector field, I just want you to be comfortable with it. When you add vectors, you add them component-wise. You get one, one, and then remember scalars hit both pieces of the vector. This is a fancy way to write the vector of one half, one half. How do I graph this thing? If you think about the scalar case where I have f of x equals y, the number of dimensions, the number of variables that I need to draw depends on all my input variables plus all my output variables. For f of x equals y of one number in one number out, you would on the XY plane in two dimensions, if I have a multivariable function, z equals f of x, y, that's two numbers in and one number out and I can draw that floating carpet and space. I could draw a three-dimensional thing. Now though I have two numbers in and I have to components out, the graph of this would live in four dimensions. I am unable to draw that actual graph. What I do instead is I draw what's called a vector field. This is the same name as the function, but this corresponds to the picture. This will be clear in context. If I want to draw this vector field F of X equals, and again, I could do 1/2,1/2 where I can present an i, j notation whenever I want. The way you draw these things is you draw the input fields. Here it's two-dimensions because I have x and y. You draw your x and y-axis and then you pick a little point, any point you want. Just to keep it simple, we'll draw maybe the origin or something like that. You ask yourself, what does this function, this vector field due to the origin? Well, it's one half, one half. You draw a little baby graph at one half, one half and you draw a vector in that direction. One half, one half points diagonally up to the right. What happens if I pick the point at one? Well, I'm evaluating this at zero, zero is a constant field. I just get back one half, one half. If I pick it at, let's say one, zero, I'm going to get one-half, one-half again, and I get the same vector back. Because it's a constant vector field at any point, I get the same vector. If I pick one, one, if I pick two, two, it doesn't matter. Every point returns the same vector. You draw at the point the vector that is returned. It doesn't matter if I put any number in, and you start to get these arrows all over the place. The way to interpret this vector field is to think of this as a current, that's how I think of it, like you're staring at the ocean from an airplane, or you're looking in a weather map and this is the wind. You can imagine that at any point or position on this field, there's a push that if I drop some boat in the ocean, it's going to start going that way. Now, these can get as complicated as you want and you can tell for the easy ones, even with a constant vector field, they're a little tedious to draw. We'll do one more and then it'll pass this work off to a computer. But let me just give you one more here. Something a little more complicated. How about if I did a vector field f of x, y equals yi plus one-half j. I'm going to use the i_j notation, for that reminder, i is one, zero and j is zero, one. Let's evaluate some points. Just like we did when we first saw functions for the very first time. I'm going to pick some easy numbers, lots of zeros and ones just to go through them and evaluate what they're going to be, and then we'll draw the vectors here. Let's pick the origin, zero, zero, plug-in zero. Of course, the vector zero, one-half. If I plug in zero, one, I get one, half. What if we pick negative one, negative one. Well, this gives us the vector negative one, one-half, and we could do a bunch of these. If we start to graph these, let's draw another x-y plane here, and let's start drawing these vectors. You'll notice something happen. The values are changing, the vectors are changing in every point. At zero, zero, I had the vector zero, one-half. This is a vector that points straight up for one-half, a small little vector that points straight up. What happens if I pick zero, one? Well, zero, one we said was one, one-half. That means I go over one and up a little bit at a half. Now, this vector is diagonal to the right, that way. If I picked negative one, one now I get back negative one-half, so I go left one and then up a half, so negative one, one. I start going this way. I'll leave this as an exercise, you can totally fill in the rest, but you'll start seeing that if you drew some other vectors, if you drew as many as you wanted, you don't get a constant vector field anymore, you get a vector field that rotates and in the clockwise direction, and you get these arrows all over the plane. Now, you have to imagine that you're a little boat and this is the ocean current, and if I drop a particle, maybe somewhere down in quadrant four, the current is going to push it, perhaps in a parabolic motion, but up and around. This is a way to get some geometric intuition of what a vector field is. It's not truly the graph, but it's a common picture that you draw when you're trying to understand these functions, these new functions, these vector field functions. You can imagine the application of these functions in Physics. They're extremely important if we have some magnetic field or gravitational field, or really just any other force field that we were thinking of, or we were working with. I told you that these are best done with computers. If you Google vector field grapher or vector field calculator, you will find tons and tons of options. Every online calculator does these. They're great for computers and I just want to show you a couple that I think look cool. X squared minus y squared minus 4. This is the first little complicated function, you have parabolas going on 2xy, and you can see the graph of this thing, it looks like a little spider. But every point, what they're doing should understand how this works. They pick a value x comma y, they plug it into the vector field, and then they just draw the arrow that it makes. At zero, zero, and you can imagine if you drop a particle or if you have a boat on the ocean and you land anywhere, it's going to push you around, and off you go. You can imagine the path that these things will take. Again, a different starting position. Which ways is this vector field pushing the objects through it? This is a nice one. Again, I just made this with a computer graph. Another one here just to show you, so you can do sine and cosine, you can take any value you want. This is the sine of x plus y comma cosine of x minus y. This is just another one that you can do in two-dimensions. Again, let the computer draw the arrows, usually get a better picture than if you do it by hand. The skill that I care about, one introducing this new object, this new function called a vector field. Then just where these arrows come from. I don't want that to be a mystery. It is difficult to draw the stuff in the XY plane and you can certainly do these in three dimensions. Here's a fun one. If you have three inputs. You start doing X, Y, and Z. You draw the x, y, and z plane. Again, let a computer do this. You can draw the vector field in three-dimensions. And now it starts getting cool. Again, when you Google, if you want to do these on your own, you can play around with them. 3-D vector fields. I did a constant vector field, 1-0-0. At every single point, the arrows pointing in the x-direction. You can see all these little arrows are pointing there. If you have a particle that comes in, it's just going to get pushed straight along the x-direction. But you can certainly make more interesting vector fields. For example, here's one XYZ, I did both in vector notation and an IJK notation. This one will point outward. It's like a burst, a radial burst. Wherever you are, you're going to get sent away from the origin. This one, again, looks kind of fun. Play around with these things. Have a good time, make some really cool images. Last but not least, I just want to make a couple more definitions, say something else about vector fields. One way to create or make vector fields is the relationship between the scalar-valued function. If f comma y, so little f comma y. A scalar value function, is a function, multi-variable function, and you can create its gradient. Alright, so friendly reminder, what is the gradient? The gradient is delta f and this is the vector of partial derivatives, f of x, f of y. The partial derivative with respect to x. The partial derivative with respect to y. If I have a z component, I would just add the z component as well. This is a way to create a vector field and you can imagine that delta f, the gradient, can be defined as a vector field where I take in two numbers. You think of f of x comma y as mapping these two numbers to the vector defined by the partial derivatives of f, little f. You can write that using subscript notation or you can certainly write it using Leibnitz notation. Both are perfectly fine. Anyway, this gives you a new way to construct and build vector fields. When you do this, when you're working with a vector field that is built from the gradient of some scalar function, this process is called building a gradient vector field. Gradient vector fields are great. They're going to have special properties. We can use the fact that they're related to the scalar value function. We can exploit that and get more information. We'll say gradient vector fields are vector fields. So scalar outputting functions, that are gradients of some scalar-valued function and we call this little f. We're right on the border of physics here. The scalar-valued function is called the potential function. We're inheriting some names from physics. Little baby f, the scalar-valued function is called the potential function when you view it as building a gradient of the vector field. If that's the case, if you have a gradient vector field, then we say that the vector field is conservative. Another way to describe a gradient vector field is to call it conservative. Again, you want to think about it as f being the gradient of some function. Not every single vector field is conservative. There are plenty of vector fields that are just not built from scalar functions. But the more interesting ones are. These ones with lots of theory that happens. I like to think of it as a Venn diagram. We have all vector fields. Then there's a subset of them that are gradient vector fields. Some are, but not all vector fields are conservative. This is important. We'll see examples of this as you go through the readings and work out some of the problems. But it's one, introduce the vocabulary, the graphs, and introduces a new notion of a function which will be our main object of study as we work through the rest of the course. Alright, great job on this video, and we'll see you next time.
