– [Voiceover] Hello

everyone, so in this video I’m gonna introduce vector fields. Now these are a concept

that come up all the time in multi variable calculus,

and that’s probably because they come up all the time in physics. It comes up with fluid

flow, with electrodynamics, you see them all over the place. And what a vector field is,

is its pretty much a way of visualizing functions

that have the same number of dimensions in their

input as in their output. So here I’m gonna write a function that’s got a two dimensional input X and Y, and then its output is going to be a two dimensional vector and each of the components

will somehow depend on X and Y. I’ll make the first one

Y cubed minus nine Y and then the second component,

the Y component of the output will be X cubed minus nine X. I made them symmetric here,

looking kind of similar they don’t have to be, I’m

just a sucker for symmetry. So if you imagine trying to

visualize a function like this with a graph it would be really hard because you have two

dimensions in the input two dimensions in the output so you’d have to somehow visualize this thing in four dimensions. So instead what we do, we

look only in the input space. So that means we look

only in the X,Y plane. So I’ll draw these coordinate axes and just mark it up,

this here’s our X axis this here’s our Y axis and for each individual input point like lets say one,two so lets say we go to one,two I’m gonna consider the

vector that it outputs and attach that vector to the point. So lets walk through an

example of what I mean by that so if we actually evaluate F at one,two X is equal to one Y is equal to two so we plug in two cubed whoops, two cubed minus nine times two up here in the X component and then one cubed minus nine times Y nine times one, excuse me down in the Y component. Two cubed is eight nine times two is 18 so eight minus 18 is negative 10 negative 10 and then one cubed is one,

nine times one is nine so one minus nine is negative eight. Now first imagine that this was if we just drew this vector where we count starting from the origin,

negative one, two, three, four, five, six,

seven, eight, nine, 10, so its going to have

this as its X component and then negative eight,

one, two, three, four, five, six, seven, we’re gonna

actually go off the screen its a very very large vector so its gonna be something here and it ends up having

to go off the screen. But the nice thing about vectors it doesn’t matter where they start so instead we can start it

here and we still want it to have that negative ten X component and the negative eight, negative one, two, three, four, five, six, seven, eight, negative eight as its Y component there and a plan with the vector field is to do this at not just one,two but at a whole bunch of different points and see what vectors attach to them and if we drew them all

according to their size this would be a real mess. There’d be markings all over the place and this one might have some

huge vector attached to it and this one would have some

huge vector attached to it and it would get really really messy. But instead what we do, just

gonna clear up the board here we scale them down, this is common you’ll scale them down and

so that you’re kind of lying about what the vectors themselves are but you get a much better feel for what each thing corresponds to. And another thing about this drawing that’s not entirely faithful to the original function that we have is that all of these

vectors are the same length. I made this one just kind of the same unit this one the same unit, and over here they all just have the same length even though in reality

the length of the vectors’ output by this function

can be wildly different. This is kind of common practice

when vector fields are drawn or when some kind of software

is drawing them for you so there are ways of getting around this one way is to just use

colors with your vectors so I’ll switch over to a

different vector field here and here color is used to

kind of give a hint of length so it still looks organized

because all of them have the same length but the difference is that red and warmer

colors are supposed to indicate this is a very

long vector somehow and then blue would indicate

that its very short. Another thing you can

do is scale them to be roughly proportional

to what they should be so notice all the blue vectors scaled way down to basically be zero red vectors kind of stay the same size even though in reality this

might be representing a function where the true vector

here should be really long or the true vector should

be kind of medium length its still common for people

to just shrink them down so its a reasonable thing to view. So in the next video I’m

gonna talk about fluid flow a context in which vector

fields come up all the time and its also a pretty

good way to get a feel for a random vector field that you look at to understand what its all about.

Woah, woah, woah — hold on a minute. When did 3Blue1Brown start working at Khan Academy? It's great to see such great creators being brought together.

The audio volume is way too low.

very comprehensive video, many thanks again. I'm getting closer to my goal everyday!

why are you so quiet

thanks man!

What program are you using to visualize vector fields?

very clear and concise, thanks!

What program is this?

Thanks. I still don't understand what the point of these are, but I can identify the correct one for my class!

i thought khan academy was in india but here american accent is being used.

what?!

Thanks!

This reminds me of the grid I used to levitate objects

please do not change the calculus volume of audio 🙂

Fix the audio vol

Why on earth would you equate vectors of different sizes? What if theyre exponentially larger, would you still consider em equal? Its counterintuitive

Hi guys, use a microphone like Blue Yeti or Rode Broadcaster and turn up gains if you want proper volume.

2:41 why does he say it doesn't matter where vectors start? I thought all vectors have to start at (0,0)?

Personally, I feel it would give a better sense of the length relationship of each vector to each other if they had the same colour but different degrees of darkness.

Sounds like 3brown1blue

Every time I watch this video, yeah, I feel like I am pretty clear about the stuff. But after a while, when I start thinking and try to visualise the vector field as the way he does, It feels pretty absurd despite the fact that he says that visualising the graph precisely needs 4 dimensions.

What I mean is, In the video, he is clearly taking a function with inputs x and y. Yeah, it is easy and clear about the plotting of the input points in the x and y axes. But, what about the outputs?? Those are completely different quantities and clearly can't be plotted along x and y axes as they have their own distinct values. So, Is the way he visualises the vector field in the video precise?? I don't understand.

he has a throat ache?? 😛

Chiming in to mention that the audio is far too quiet, again. Great to see you on KA, 3blue!

its unfortunate that for many of these videos, youtube doesn't recommend the next ones in the series… so i got to go to your channel manually and search for the next video you mention in the end of one video..

I don't understand how could I thank khan academy for clearing my concept in a crystal clear way…

May Allah bless you for sharing your knowledge.

I guess I am confused as to why your graph has x value -10 and y value -8. Why wouldn't your y value be -10 considering arithmetically 2^3-18 is -10.

Why it doesn't matter from where vector starts?

All those going vector fields pointing inward and outward remind of the g2g shear in a tornado. You calculate it by adding the inflow bound and the outflow bound. Tornadoes seem to be.very hard to understand because some of them are very wide and extremely intense while some may be rope like and extremely intense.

This is Maxwell language. It is beautiful!

Audio quality problem