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- [Voiceover] Hello again, in the last video
we were looking at vector fields that
only have an X component,
basically meaning all of the vectors
point just purely to the left or to the right,
with nothing up and down going on.
Then we landed at the idea that
the divergence of V, you know when
you take the divergence of this vector valued function,
it should definitely have something to do
with the partial derivative of P,
that X component of the output,
with respect to X,
and here I wanna do the opposite,
and say, okay, what if we look at functions
where that P, that first component is zero,
but then we have some kind of positive Q component,
some kind of positive, well maybe not positive,
but some kind of non-zero,
so positive or negative Y component,
and what this would mean,
instead of thinking about vector as just left and right,
now we're looking at vectors that
are purely up or down, kind of up or down.
So kind of doing the same thing we did last time,
if we start thinking about cases where
the divergence of our function
at a given point should be positive,
and an example of that, you might be saying,
nothing is happening at the point itself,
so Q itself would be zero but then below it,
things are kind of going away so
they're pointing down, and above it,
things are going up.
So in this case down here,
Q is a little bit less than zero,
the Y component of that vector is less than zero,
and up here, Q is greater than zero.
So here we have the idea that as you're going
from the bottom up,
the Y value of your input is increasing
as you're moving upward in space,
the value of Q, this Y component of the output,
should also be increasing because
it goes from negative to zero to positive.
So now you're starting to get this idea
of partial Q with respect to Y,
you know, as we change that Y and move up in space,
the value of Q should be positive,
so positive divergence seems to
correspond to a positive value here.
And the thinking is actually gonna be
almost identical to what we did
in the last video with the X component,
because you can think of another circumstance
where maybe you actually have a vector
attached to your point and something's going on,
and there even is some convergence towards it,
where you have some fluid flow
in towards the point,
but it's just heavily outweighed by
even higher divergence, even higher flow
away from your point above it,
and again you have this idea of
Q starts off small, so here's kind of,
Q starts off small, maybe it's kind of like near zero,
and then here Q is something positive
and then here it's even more positive,
and sort of making up notation here
but I want the idea of kind of small
and then medium-sized and then bigger,
and once again, the idea of partial derivative of Q
with respect to Y being greater than zero
seems to correspond to positive divergence,
and if you want, you can sketch out
many more circumstances and think about
what if the vector started off pointing down,
what would positive and negative
and zero divergence all look like,
but the upshot of it all, pretty much
for the same reasons I went through in the last video,
is this partial derivative with respect to Y
corresponds to the divergence.
And when we combine this with
our conclusions about the X component,
that actually is all you need to know for the divergence.
So just to write it all out,
if we have a vector valued function of X and Y,
and it's got both of its components,
we've got P as the X component of the output,
that first component of the output,
and Q, and we're looking at both of these at once,
the way that we compute divergence,
the definition of divergence of this vector valued function,
is to say divergence of V
as a function of X and Y,
is actually equal to the partial derivative of P
with respect to X, plus,
the partial derivative of Q with respect to Y.
And that's it, that is the formula for divergence,
and hopefully by now, this isn't just kind of
a formula that I'm plopping down for you,
but it's something that makes intuitive sense,
when you see this term, this partial P
with respect to X, you're thinking about,
oh yes, yes, because if you have
flow that's kind of increasing
as you move in the X direction,
that's gonna correspond with movement away,
and this partial derivative of Q
with respect to Y term,
hopefully you're thinking, ah yes,
as you're increasing the Y component
of your vector around your point,
that corresponds with less flow in than there is out,
so both of these correspond to that
idea of divergence that we're going for,
and if you just add them up,
this gives you everything you need to know.
And one thing that's pretty neat,
and maybe kind of surprising,
is that the way we just came across this formula,
and started to think about it,
was in the simplified case,
where you have, you know, just pure movement
in the X direction or pure movement in the Y direction,
but in reality, as we know,
vector fields can look much more complicated,
and maybe you have something where
you know, it's not just in the X direction,
and there's lots of things going on
and you need to account for all of those,
and evidently, just looking at the change
in the X component with respect to X,
or the change in the Y component of the output
with respect to the Y component of the input,
gives you all the information you need to know.
And basically what's going on here is
that any fluid flow can just be broken down
into the X and Y components where
you're just looking at each vector,
you know, whatever vector you have,
it could be broken down into
it's own X and Y components and if you
wanna think kind of concretely about
the fluid flow idea, maybe you'd say that
for your point, if you're looking at a point in space,
you picture a very small box around it,
and the reason you only need to
think about X components and Y components
is that you're only really looking at,
you know, what's going on on the left and right side,
and then you can kind of calculate
what the divergence according to fluid flowing
in through those sides is,
and then you just look at kind of,
fluid flowing through the top or the bottom,
and if you kinda shrink this box down,
all you really care about is those two different directions,
and anything else, anything that's kind of
a diagonal into it, is really just broken down
into what's the Y component there,
what's the, you know, how is it contributing to
movement up through that bottom part of the box,
and then what's the X component,
how's it contributing to movement
through that side part of the box.
But anyway, I mean the upshot here
is just that the formula for divergence
only involves these two components.