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# Stokes Teoremi

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So I've drawn multiple versions
of the exact same surface S, five copies of that
exact same surface. And what I want to do is think
about the value of the line integral-- let me write this
down-- the value of the line integral of F dot dr, where F
is the vector field that I've drawn in magenta in
each of these diagrams. And obviously, it's different
in each of these diagrams. And the only part of the
vector field that I've drawn is a part that's
along the surface. I could have drawn the
part of the vector field that's off the
surface, but we're only going to be concerned with
what's going on on the surface. So the vector field
could be defined in this entire three-dimensional
space in here as well. And these are obviously
different vector fields, and we can see that visually
based on how we drew it. And the contour that we
care about-- remember, we're going to take a line
integral, so the path matters. The path that we care about is
the counterclockwise boundary of our surface, so it's going
to be this right over here. The counterclockwise
boundary of our surface is what we're going to
be taking F dot dr along. So this right over here, and
let me draw the orientation. It's going to be
counterclockwise. And we're going to do it in
every one of these situations, for every one of these surfaces
and every one of these F's. And what I really want to think
about is how the value of F dot dr over that
contour, how it might change from example to example. And obviously, the
only difference between each of these is what
the vector field F is doing. So first let's think about
this example right over here. At this part of the
contour, this bottom part right over here,
our vector field is going in the exact same
direction as our line, as our contour. So we're going to get positive
values of F dot dr down here, and we're going to
keep summing them up. We're taking an integral. Then as we go up
the curve or as we go kind of uphill
right over here, we see that our vector field is
going essentially orthogonal. It's going perpendicular. It's going perpendicular
to our contour. So our F dot dr, we're not going
to get any value from that. F dot dr and all of these,
this part of the contour, it's all going to be 0. So we're not going to get
anything right over there. Actually, let me
just write nothing. So we get nothing
right over there. Maybe I'll write 0. I'll write we'll get
0 right over there. And then up here, when we're
at this part of the contour, our vector field is going in
the exact opposite direction as our path. Up here our path
is going, I guess, from the right to the left,
while our vector field is going from the
left to the right. And so we're actually going
to get negative values. We are going to get
negative values up here, and they're going to sum up to
a reasonable negative value. And if the vector
field is constant, and I kind of drew
it like it is, and if this length is
equal to this length, then these two values
are going to cancel out. When you add this positive
sum to this negative sum, they're going to be 0. And then once again, when
you go downhill again, the vector field
is perpendicular to our actual path, so
we're going to get 0. So based on the way
I've described it, your line integral of F dot
dr for this version of F in this example right over
here, it might all cancel out. You will get something, if we
make the assumptions that I made, this might be equal to 0. So in this example, F dot
dr could be equal to 0. Now let's think about what's
going on in this situation. In this one, just like the last
one, as we go along the bottom, the vector field is going
in the exact same direction as our contour, so we're
going to get positive values. And once we go up the
hill, the vector field is going perpendicular
to our path, so it's not going to add
anything really to it. So we're just going to
get 0 along this part. But then up here, our vector
field has switched directions. And once again, it's going
in the exact same direction as our path, so we're going to
get more positive values right over there. And then as we go
down here, it's not going to add anything
because our vector field is perpendicular to our path. So we're going to get 0. But notice, now these
two ends don't cancel out with each other. We're going to get
a positive value. We are going to get
a positive value. And what was the difference
between this version of F, this vector field,
and this vector field right over here? Well, this vector field
switched directions so that the top part
didn't cancel out with the bottom part. Or another way to think about
it is, it had some curl. There's a little bit
of spinning going on. If this was describing
the velocity of a fluid, and if you were to put
a stick right over there on the surface, the
stick would spin. It has some spin or
some curl, however you want to describe it. This right over
here has no curl. If you put a stick
right over here, it would just flow
with the fluid, but the stick itself
would not spin. So we got a positive value
for the line integral in this situation. And we also have, it looks
like, a positive curl. Now let's think about this one. In this situation, as we go
along this part of our contour, our vector field F is going
in the exact same direction, so we're going to
get positive values. Now as we go uphill,
our vector field F, it's also kind of turned
in that direction, and so we'll get
more positive values. And now as we go
in that direction, our vector field
F is still going in the direction of
our contour, we're going to get more
positive values. And as we go down, once
again the vector field F is going in the
direction of our contour, so we'll get even
more positive values. So in this situation, the
value of our line integral of F dot dr is even more positive. And we see that the
actual vector field along the surface-- and
remember, the vector field might be doing all sorts of
crazy things off the surface. Actually, let me draw that
in that same magenta color. It might be doing all sorts of
crazy things off the surface, but what we really
care about is what's happening on the surface. And because this
vector field is, I guess you could
say curling, or it's spinning along the
surface, it allows it to go with the boundary
along all the points, and we get a very positive
value for this line integral. So we have a higher curl. So more curl, it looks like, is
leading to a more positive line integral. Now let's think about
what's happening in this situation
right over here. This situation down
here, our vector field is going in the same
direction as our path, so we're going to
get positive values. Just like the first situation,
as we go up the hill like this or up the surface like
that, our vector field is perpendicular to our
surface, so it's really not going to add anything
to our line integral. And then as we go along this
top part, this first part of the top part right over
here, the vector field is going against us. So it's negative
right over here. We're going in the exact
opposite direction of our path. And then, right as
we get to the end, the vector field
flips direction, and we get a little
bit of positive right over here because it's
a little bit of it going in the same direction. And then we go back downhill. When we go back
downhill, it adds nothing because our vector field
is going perpendicular to our path. So the big difference between
this case and the case up here is the case
up here-- well, actually, I could compare
between these two or these two. But the difference between
this one and this one is that at least this
part of the vector field has switched directions. So we get a little
bit of positive value. And one way to think
about it, this one is going to be less positive
than that, if we take the line integral, but more
positive than that. And one other way
to think about it is, we have a little bit of
curl going on right over here. Our vector field switched
directions right around there, or I guess you
could say that it's spinning right around there. So if you put a stick,
if that was in the water, it would start spinning. But everywhere else,
there isn't a lot of curl. So you have some curl, but
it's over a little small region of the surface. While over here, you
had curl going on over a larger portion
of our surface. And so up here, you had
a more positive curl, more positive line integral. Here you have a curl
over less of the surface, and you're going to have a
less positive line integral. Now let's think about
this one over here. This vector field, along the
surface there is some curl. There's some curl going
on right over there. If you put a stick
in the water, if you view that as the velocity of
water, the stick would spin. So you have some curl. But then it switches
direction again, so you have curl there as
well, and it's actually curl in the opposite direction. So to some degree, if you
were to sum all of this up, maybe it would cancel out. And it makes sense. It makes sense that
it would cancel out because when you take
the line integral around the whole thing, just
like this first situation, it looks like it
will add up to 0. Because even though
you have some curl, the curls cancel out each other. And so when you go to this
top part of the surface, the vector field is going
in the exact same direction as this bottom part
of the surface. So if you were to take your line
integral, the same one that we care about, just like
the first situation it would be positive down
here, 0 as we go up the curve. And then as we go down
here, the vector field has switched directions
twice, so it's still going against the
path of our contour, just like this first situation. So it would be
negative up there. And then as we go
down, it would be 0. So this thing right
over here also looks just like the first one
because the curls essentially cancel out. We switched direction twice. So over here, our line
integral might also be 0. Now, the whole reason why I
went through this exercise is to give you an intuition
of why it might make sense that if we have more
curl happening over more of this surface,
why that might make the value of this line
integral be larger. And so, hopefully,
it starts to give you an intuition that maybe, just
maybe, the value of this line integral, the value of F
dot dr over that contour, over this contour that's
going in the counterclockwise direction-- we'll talk
more about orientation in future videos-- maybe
this is equal to the sum of the curls over the surface. And so let's think about this. It could be a surface integral. So we're going to
go over the surface, and what we care about
is the curl of F. We care about the curl
of F. But we don't just care about the curl of F
generally because F might be spinning in a direction, in
a way, that's off the surface. We care about how much it's
curling on the surface. So what we would want to do is
we'd want to take the curl of F and dot it with the
normal vector at any point to the surface and then multiply
that times the surface itself. And this is just to say
that the more surface where we have more curling
going on, the more that the line integral, the
value of the line integral, might be. And we saw that when we
compared these three examples. And another way of
writing all of this is the surface
integral-- let me write the surface in that
same brown color-- the surface integral of the
curl of F, which would just be another vector that tells
us how much we are spinning generally, but we want
to care how much we're spinning along the surface. So we're dotting it
with the normal vector. Or another way to write this
whole thing is to say dot ds. So if you take essentially the
sum across the entire surface of how much we're
curling, how much we're spinning along that surface,
then maybe, just maybe, this will be equal to the
value of the line integral as we go around the
boundary of the surface. And it actually turns out
that this is the case. And obviously, I haven't
proven it to you here, but hopefully you have some
intuition why this makes sense. And this idea that
this is equal to this is called Stokes'
theorem, and we'll explore it more in
the next few videos.