Intro to VECTOR FIELDS // Sketching by hand & with computers

in this video we're going to talk about vector fields i'm going to introduce the big idea i'm going to show you a bunch of examples of where they come up and we're going to introduce the terminology to talk about vector fields now this video is part of my larger playlist on vector calculus the link to that and all my other courses are down in the description now i want to begin by imagining that i drop a ball now there was a sort of arrow which said where was the vector going to go when i released it from rest it went straight down from my perspective but if you're on the opposite side of the world for me then the direction your ball travels is very different in the direction that my ball travels so what you sort of imagine is that anywhere that you might be either on the surface of the earth or deep anywhere in the solar system if you take a ball and release it from rest there's an arrow it's the arrow that says that is the direction in which the ball is going to fall due to gravity that is the idea of a vector field it is an arrow that tells you what's going to happen at every single spot let me give you a different example this is a standard plot of wind speed i sure feel like a weather person right now where i'm standing in front of a green screen and i've got a weather map behind me but nevertheless what this shows is that at every spot there is an arrow and that arrow indicates the wind speed this happens to be a map at a particular point of time for a tropical cyclone but it has that same feature of dropping the ball where at every spot there is an arrow and the arrow is indicating what is going on at that spot whether it is a wind speed or whether it is the direction in which the ball would fall due to gravity something like this where you want to give a magnitude and a direction at every point in space are called vector fields so with a little more precise mathematical terminology what i might say is that a vector field is just a function from rn to rn it's something with an input domain so at every point in space perhaps r2 or r3 then you get an associated output which is also an r2 or r3 standard examples would be n equal to two or three so for example in two dimensions your function is gonna have a two dimensional output it's going to have an i-hat and a j-hat component and the standard terminology is we'll put m to signify the function the m of x y that's what's going on in the i-hat and n the function that's going to go on in the j-hat so basically your big function f can be decomposed into this component in the i-hat and this component in the j-hat likewise it could be three-dimensional which would also be very standard in which case the standard terminology for the three component functions are m n and p so for example in that plot of wind speed that we had before that was a two dimensional plot and at every location in the x y plane there was a two dimensional vector and that vector tells you the wind speed at a particular point wind speed is both a direction and a magnitude nevertheless you could encode that with these functions m and m and then a lot of the standard terminology from calculus just comes along with these functions so for example we might say a vector field is continuous if the m the n the p are continuous or we might say that a vector field is differentiable if the m the n and the p individually are differentiable functions now what i want to do is actually draw by hand a particular vector field i'm going to draw the vector field y and the i hat and minus x in the j hat and so what i'm going to do is just plug in various values of x and y and see what vectors i get so for instance if i take the vector field and i plot it at the point 0 0 then i'm just going to get the vector 0 0 out and so i can represent that by just putting a little dot right at the origin at the location 0 0. if i come here and do the vector field at the point say 1 0 then by plugging that into the y i hat minus x j hat then this is going to give me the vector 0 in the i hat and minus 1 in the j hat so how do i plot this well i first go to the location of 1 0 which is 1 to the right and 0 down but then i have to draw a vector which is 0 to the right and -1 down so i draw the vector that comes straight down and looks a little bit like this so that is me drawing the vector at the location one zero and the vector itself is the vector zero minus one we could keep on doing these how about instead i figure out what happens now i don't know one one instead in which case if i plug it into my formula then i'm going to get 1 and minus 1. well how do i draw this so for the f of 1 1 i go up to the location of x y equal to 1 1 and then i draw a vector which is 1 to the right and one down and it looks like this vector right there and we could keep doing this as much as we wanted why don't we do just one more here and see what happens i'm going to do the spot that starts right up here i'll start and do zero one it's always nice to combine zeros and plus or minus one is sort of the standard test points either way i put this in and what it starts with is going to be the y value which is one and then minus the x value which is zero and so this is a vector that goes one to the right and zero up so one to the right and zero up goes there now you could keep doing this you could keep on adding new points and plot them around and for example i'll just do a few of them i've i've checked it you can see that this one's going to go over there this one is going to go over here and if you keep doing this you notice that there's sort of a bit of a sort of circular type pattern occurring here so this is useful to do for a few points but it gets a bit tedious so let's let the computer do the rest okay so i'm back now and what i've done is just plotted this with the computer opposed to doing it by hand and that's totally fine and indeed what we get is the sort of circular pattern now when you get a computer to plot a vector field and by the way i'm going to link down below some simple plotters so that you could put in different types of vector fields and see what it looks like there's often choices that are made so one thing for example is that this looks pretty messy because the arrows are sort of overlapping and so one thing that you might want to do is instead of having a vector which we think of as a length and a direction is keep the direction but maybe play with the lengths a little bit so this is for example a re-scaled version of the same thing these lengths are not accurate that is the length of the vector at some random point like 2 3 is not the length of the vector that would be 3i minus 2 j what it's done is it just proportionately scaled everything so that it just looks a little bit prettier and you can see the circular pattern a little bit better and a lot of plots of vector fields end up having these kind of features similarly from the perspectives of computer graphics there's a question of density what i plotted here is that i divided the domain the x's and the y's by integer values so at every point like 1 0 2 0 1 1 2 1 all with integer values i've gone and plotted an arrow but maybe i wanted to see with a bit more density so now i'm going to double the density that means i'm going to split up say the x axis not into integer values but into a half one one and a half two and so on and so i just plotted way more arrows i mean i could plot even more than this it would just depend on my density that's a choice about my computer graphics not a choice about the underlying vector field okay so this was one let's just take one more this is just one that i think is kind of pretty it's sine of x y and the i hat and cos of x minus y in the j hat and as you can see they just get kind of interested in with a little bit of length scaling and densities adjustments you can make these really pretty plots and so often you play around with these features just to get it to something that you can visualize nicely the next vector field that we're going to do is the vector field x i hat plus y j hat and this one i'd actually encourage you to pause the video and see whether you can quickly sketch it yourself using the methods we've done before however if you don't wish to pause and just want to enjoy the show well then what it looks like is just this sort of looks like a starburst here by the way i haven't applied any length scaling and i'm using this integer density that i started with but it kind of looks like the starburst basically at any point say one one then the vector which is also one one is just sort of pointing straight outwards and the reason i'm showing this one is if you can sort of see that starburst behavior that's happening with this vector field i can do the same thing in three dimensions this is what it looks like it's a three dimensional plot now and now this is the vector field x i hat y j hat and z k hat so it's got three components and then what i get is a three-dimensional vector at every point in three dimensions and again it kind of looks like this burst uh sticking out from the origin so indeed you can plot two-dimensional or three-dimensional vector fields and again i'll put a link down below to a plotter for a three-dimensional vector field if you want to play around with changing the component functions away from say the x y and z that i have here now while we're on that theme of how you can play around with the computer graphics a little bit to plot these i want to go back to that original wind speed vector field that we saw at the beginning of the video because notice it's not just the fact that at every point in the x y plane there's some two-dimensional vector that tells you the magnitude and direction of the wind speed there's also color in this plot so what's going on with the color here well the color is a way to signify easily quickly visually what the magnitude is so in this plot the sort of whiter pinker tones are ones that have faster wind speeds well going back from red to orange to yellows to greens to blues are progressively slower and slower wind speeds so you can sort of see that in the middle this is where that cyclone is going where it's got the fastest speeds going on there in the middle so you can either color code the background which is what's done in this plot or sometimes you might just even go and color clothe the vectors themselves this is a really good way to be able to visually and quickly see what's going on as well now previously in single or multi-variable calculus you'd often have the domain either be a line or perhaps a plane and then the co domain the sort of the target of the function would be some height above that line or above that plane and those were the things that we could graph nicely however for a vector field even with n equal to two so that's a two dimensional input and a two dimensional output a graph would be four dimensional and therefore is hard to graph in that normal way however as we've seen this video we've had this wonderful new way to visualize things where at every point in either the plane or in space you just draw a vector you draw an arrow and that signifies what's going to be going on in the output at that particular input spot all right i hope you've enjoyed that introduction to vector fields this is just the first video in a large series of videos talking about vector calculus and playing around with vector fields i encourage you to check out the link in the description for that entire playlist i hope you enjoyed this video and if you did please give it a like for the youtube algorithm because we're all mathematicians here and we appreciate algorithm just as much as youtube does if you have a question leave it down in the comments and we'll do some more math as always in the next video

Channel: Dr. Trefor Bazett

Views: 99,996

Keywords: Solution, Example, math, vector fields, 2D, 3D, plotting vector fields, graphing, definition, rotation, draw vector fields, sketch, continuous, differentiable, scale, density, velocity field, gravitational field, electric field, magnetic field, vector calculus, calc IV

Id: 0QTDisIl9bo

Length: 12min 9sec (729 seconds)

Published: Sun Oct 11 2020