Vector fields, introduction | Multivariable calculus | Khan Academy

- [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.
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.