- [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.