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OK, so remember,
we've seen Stokes theorem,
00:00:29.000 --> 00:00:37.000
which says if I have a closed
curve bounding some surface,
00:00:37.000 --> 00:00:40.000
S,
and I orient the curve and the
00:00:40.000 --> 00:00:44.000
surface compatible with each
other,
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then I can compute the line
integral along C along my curve
00:00:53.000 --> 00:00:57.000
in terms of,
instead,
00:00:57.000 --> 00:01:04.000
surface integral for flux of a
different vector field,
00:01:04.000 --> 00:01:12.000
namely, curl f dot n dS.
OK, so that's the statement.
00:01:12.000 --> 00:01:18.000
And, just to clarify a little
bit, so, again,
00:01:18.000 --> 00:01:24.000
we've seen various kinds of
integrals.
00:01:24.000 --> 00:01:26.000
So, line integrals we know how
to evaluate.
00:01:26.000 --> 00:01:30.000
They take place in a curve.
You express everything in terms
00:01:30.000 --> 00:01:32.000
of one variable,
and after substituting,
00:01:32.000 --> 00:01:36.000
you end up with a usual one
variable integral that you know
00:01:36.000 --> 00:01:40.000
how to evaluate.
And, surface integrals,
00:01:40.000 --> 00:01:44.000
we know also how to evaluate.
Namely, we've seen various
00:01:44.000 --> 00:01:47.000
formulas for ndS.
Once you have such a formula,
00:01:47.000 --> 00:01:50.000
due to the dot product with
this vector field,
00:01:50.000 --> 00:01:52.000
which is not the same as that
one.
00:01:52.000 --> 00:01:56.000
But it's a new vector field
that you can build out of f.
00:01:56.000 --> 00:02:00.000
You do the dot product.
You express everything in terms
00:02:00.000 --> 00:02:03.000
of your two integration
variables, and then you
00:02:03.000 --> 00:02:06.000
evaluate.
So, now, what does this have to
00:02:06.000 --> 00:02:12.000
do with various other things?
So, one thing I want to say has
00:02:12.000 --> 00:02:18.000
to do with how Stokes helps us
understand path independence,
00:02:18.000 --> 00:02:24.000
so, how it actually motivates
our criterion for gradient
00:02:24.000 --> 00:02:30.000
fields,
independence.
00:02:30.000 --> 00:02:35.000
OK, so,
we've seen that if we have a
00:02:35.000 --> 00:02:40.000
vector field defined in a simply
connected region,
00:02:40.000 --> 00:02:43.000
and its curl is zero,
then it's a gradient field,
00:02:43.000 --> 00:02:47.000
and the line integral is path
independent.
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So, let me first define for you
when a simply connected region
00:02:53.000 --> 00:03:01.000
is.
So, we say that a region in
00:03:01.000 --> 00:03:17.000
space is simply connected -- --
if every closed loop inside this
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region bounds some surface again
inside this region.
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OK, so let me just give you
some examples just to clarify.
00:03:39.000 --> 00:03:46.000
So, for example,
let's say that I have a region
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that's the entire space with the
origin removed.
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OK, so space with the origin
removed, OK, you think it's
00:04:00.000 --> 00:04:06.000
simply connected?
Who thinks it's simply
00:04:06.000 --> 00:04:09.000
connected?
Who thinks it's not simply
00:04:09.000 --> 00:04:14.000
connected?
Let's think a little bit harder.
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Let's say that I take a loop
like this one,
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OK, it doesn't go through the
origin.
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Can I find a surface that's
bounded by this loop and that
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does not pass through the
origin?
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Yeah, I can take the sphere,
you know, for example,
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or anything that's just not
quite the disk?
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So,
and similarly,
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if I take any other loop that
avoids the origin,
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I can find, actually,
a surface bounded by it that
00:04:42.000 --> 00:04:44.000
does not pass through the
origin.
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So, actually,
that's kind of a not so obvious
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theorem to prove,
but maybe intuitively,
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start by finding any surface.
Well, if that surface passes
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through the origin,
just wiggle it a little bit,
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you can make sure it doesn't
pass through the origin anymore.
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Just push it a little bit.
So, in fact,
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this is simply connected.
That was a trick question.
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OK, now on the other hand,
a good example of something
00:05:13.000 --> 00:05:16.000
that is not simply connected is
if I take space,
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and I remove the z axis -- --
that is not simply connected.
00:05:36.000 --> 00:05:39.000
And, see, the reason is,
if I look again, say,
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at the unit circle in the x
axis,
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sorry, unit circle in the xy
plane,
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I mean, in the xy plane,
so, if I try to find a surface
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whose boundary is this disk,
well, it has to actually cross
00:05:58.000 --> 00:06:03.000
the z axis somewhere.
There's no way that I can find
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a surface whose only boundary is
this curve, which doesn't hit
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the z axis anywhere.
Of course, you could try to use
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the same trick as there,
say, maybe we want to go up,
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up, up.
You know, let's start with a
00:06:19.000 --> 00:06:21.000
cylinder.
Well, the problem is you have
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to go infinitely far because the
z axis goes infinitely far.
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And, you'll never be able to
actually close your surface.
00:06:27.000 --> 00:06:30.000
So, the matter what kind of
trick you might want to use,
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it's actually a theorem in
topology that you cannot find a
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surface bounded by this disk
without intersecting the z axis.
00:06:39.000 --> 00:06:44.000
Yes?
Well, a doughnut shape
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certainly would stay away from
the z axis, but it wouldn't be a
00:06:47.000 --> 00:06:50.000
surface with boundary just this
guy.
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Right, it would have to have
either some other boundary.
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So, maybe what you have in mind
is some sort of doughnut shape
00:06:57.000 --> 00:07:01.000
like this that curves on itself,
and maybe comes back.
00:07:01.000 --> 00:07:05.000
Well, if you don't quite close
it all the way around,
00:07:05.000 --> 00:07:07.000
so I can try to,
indeed, draw some sort of
00:07:07.000 --> 00:07:10.000
doughnut here.
Well, if I don't quite close
00:07:10.000 --> 00:07:13.000
it, that it will have another
edge at the other end wherever I
00:07:13.000 --> 00:07:15.000
started.
If I close it completely,
00:07:15.000 --> 00:07:18.000
then this curve is no longer
its boundary because my surface
00:07:18.000 --> 00:07:20.000
lives on both sides of this
curve.
00:07:20.000 --> 00:07:22.000
See, I want a surface that
stops on this curve,
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and doesn't go beyond it.
And, nowhere else does it have
00:07:25.000 --> 00:07:28.000
that kind of behavior.
Everywhere else,
00:07:28.000 --> 00:07:33.000
it keeps going on.
So,
00:07:33.000 --> 00:07:36.000
actually, I mean,
maybe actually another way to
00:07:36.000 --> 00:07:40.000
convince yourself is to find a
counter example to the statement
00:07:40.000 --> 00:07:44.000
I'm going to make about vector
fields with curl zero and simply
00:07:44.000 --> 00:07:48.000
connected regions always being
conservative.
00:07:48.000 --> 00:07:52.000
So, what you can do is you can
take the example that we had in
00:07:52.000 --> 00:07:55.000
one of our older problem sets.
That was a vector field in the
00:07:55.000 --> 00:07:58.000
plane.
But, you can also use it to
00:07:58.000 --> 00:08:02.000
define a vector field in space
just with no z component.
00:08:02.000 --> 00:08:05.000
That vector field is actually
defined everywhere except on the
00:08:05.000 --> 00:08:08.000
z axis, and it violates the
usual theorem that we would
00:08:08.000 --> 00:08:12.000
expect.
So, that's one way to check
00:08:12.000 --> 00:08:20.000
just for sure that this thing is
not simply connected.
00:08:20.000 --> 00:08:26.000
So, what's the statement I want
to make?
00:08:26.000 --> 00:08:40.000
So, recall we've seen if F is a
gradient field -- -- then its
00:08:40.000 --> 00:08:46.000
curl is zero.
That's just the fact that the
00:08:46.000 --> 00:08:49.000
mixed second partial derivatives
are equal.
00:08:49.000 --> 00:08:53.000
So, now, the converse is the
following theorem.
00:08:53.000 --> 00:09:01.000
It says if the curl of F equals
zero in, sorry,
00:09:01.000 --> 00:09:09.000
and F is defined -- No,
is not the logical in which to
00:09:09.000 --> 00:09:15.000
say it.
So, if F is defined in a simply
00:09:15.000 --> 00:09:30.000
connected region,
and curl F is zero -- -- then F
00:09:30.000 --> 00:09:45.000
is a gradient field,
and the line integral for F is
00:09:45.000 --> 00:09:53.000
path independent -- -- F is
conservative,
00:09:53.000 --> 00:09:55.000
and so on,
all the usual consequences.
00:09:55.000 --> 00:09:58.000
Remember, these are all
equivalent to each other,
00:09:58.000 --> 00:10:01.000
for example,
because you can use path
00:10:01.000 --> 00:10:05.000
independence to define the
potential by doing the line
00:10:05.000 --> 00:10:08.000
integral of F.
OK, so where do we use the
00:10:08.000 --> 00:10:12.000
assumption of being defined in a
simply connected region?
00:10:12.000 --> 00:10:17.000
Well, the way which we will
prove this is to use Stokes
00:10:17.000 --> 00:10:20.000
theorem.
OK, so the proof,
00:10:20.000 --> 00:10:25.000
so just going to prove that the
line integral is path
00:10:25.000 --> 00:10:29.000
independent;
the others work the same way.
00:10:29.000 --> 00:10:34.000
OK, so let's assume that we
have a vector field whose curl
00:10:34.000 --> 00:10:38.000
is zero.
And, let's say that we have two
00:10:38.000 --> 00:10:44.000
curves, C1 and C2,
that go from some point P0 to
00:10:44.000 --> 00:10:49.000
some point P1,
the same point to the same
00:10:49.000 --> 00:10:54.000
point.
Well, we'd like to understand
00:10:54.000 --> 00:11:00.000
the line integral along C1,
say, minus the line integral
00:11:00.000 --> 00:11:04.000
along C2 to show that this is
zero.
00:11:04.000 --> 00:11:06.000
That's what we are trying to
prove.
00:11:06.000 --> 00:11:12.000
So, how will we compute that?
Well, the line integral along
00:11:12.000 --> 00:11:17.000
C1 minus C2, well,
let's just form a closed curve
00:11:17.000 --> 00:11:24.000
that is C1 minus C2.
OK, so let's call C,
00:11:24.000 --> 00:11:36.000
woops -- So that's equal to the
integral along C of f dot dr
00:11:36.000 --> 00:11:44.000
where C is C1 followed by C2
backwards.
00:11:44.000 --> 00:11:50.000
Now, C is a closed curve.
So, I can use Stokes theorem.
00:11:50.000 --> 00:11:52.000
Well, to be able to use Stokes
theorem, I need,
00:11:52.000 --> 00:11:54.000
actually, to find a surface to
apply it to.
00:11:54.000 --> 00:11:57.000
And, that's where the
assumption of simply connected
00:11:57.000 --> 00:12:00.000
is useful.
I know in advance that any
00:12:00.000 --> 00:12:02.000
closed curve,
so, C in particular,
00:12:02.000 --> 00:12:10.000
has to bound some surface.
OK, so we can find S,
00:12:10.000 --> 00:12:21.000
a surface, S,
that bounds C because the
00:12:21.000 --> 00:12:32.000
region is simply connected.
So, now that tells us we can
00:12:32.000 --> 00:12:38.000
actually apply Stokes theorem,
except it won't fit here.
00:12:38.000 --> 00:12:40.000
So, instead,
I will do that on the next
00:12:40.000 --> 00:12:45.000
line.
That's equal by Stokes to the
00:12:45.000 --> 00:12:51.000
double integral over S of curl F
dot vector dS,
00:12:51.000 --> 00:12:54.000
or ndS.
But now, the curl is zero.
00:12:54.000 --> 00:12:58.000
So, if I integrate zero,
I will get zero.
00:12:58.000 --> 00:13:02.000
OK, so I proved that my two
line integrals along C1 and C2
00:13:02.000 --> 00:13:04.000
are equal.
But for that,
00:13:04.000 --> 00:13:08.000
I needed to be able to find a
surface which to apply Stokes
00:13:08.000 --> 00:13:11.000
theorem.
And that required my region to
00:13:11.000 --> 00:13:14.000
be simply connected.
If I had a vector field that
00:13:14.000 --> 00:13:17.000
was defined only outside of the
z axis and I took two paths that
00:13:17.000 --> 00:13:20.000
went on one side and the other
side of the z axis,
00:13:20.000 --> 00:13:21.000
I might have obtained,
actually,
00:13:21.000 --> 00:13:27.000
different values of the line
integral.
00:13:27.000 --> 00:13:35.000
OK, so anyway,
that's the customary warning
00:13:35.000 --> 00:13:43.000
about simply connected things.
OK, let me just mention very
00:13:43.000 --> 00:13:46.000
quickly that there's a lot of
interesting topology you can do,
00:13:46.000 --> 00:13:48.000
actually in space.
So, for example,
00:13:48.000 --> 00:13:50.000
this concept of being simply
connected or not,
00:13:50.000 --> 00:13:55.000
and studying which loops bound
surfaces or not can be used to
00:13:55.000 --> 00:13:58.000
classify shapes of things inside
space.
00:13:58.000 --> 00:14:07.000
So, for example,
one of the founding
00:14:07.000 --> 00:14:15.000
achievements of topology in the
19th century was to classify
00:14:15.000 --> 00:14:24.000
surfaces in space -- -- by
trying to look at loops on them.
00:14:24.000 --> 00:14:33.000
So, what I mean by that is that
if I take the surface of a
00:14:33.000 --> 00:14:39.000
sphere, well,
I claim the surface of a sphere
00:14:39.000 --> 00:14:44.000
-- -- is simply connected.
Why is that?
00:14:44.000 --> 00:14:49.000
Well, let's take my favorite
closed curve on the surface of a
00:14:49.000 --> 00:14:53.000
sphere.
I can always find a portion of
00:14:53.000 --> 00:14:59.000
the sphere that's bounded by it.
OK, so that's the definition of
00:14:59.000 --> 00:15:03.000
the surface of a sphere being
simply connected.
00:15:03.000 --> 00:15:06.000
On the other hand,
if I take what's called a
00:15:06.000 --> 00:15:07.000
torus,
or if you prefer,
00:15:07.000 --> 00:15:10.000
the surface of a doughnut,
that's more,
00:15:10.000 --> 00:15:21.000
it's a less technical term,
but it's -- -- well,
00:15:21.000 --> 00:15:24.000
that's not simply connected.
And, in fact,
00:15:24.000 --> 00:15:26.000
for example,
if you look at this loop here
00:15:26.000 --> 00:15:29.000
that goes around it,
well, of course it bounds a
00:15:29.000 --> 00:15:32.000
surface in space.
But, that surface cannot be
00:15:32.000 --> 00:15:35.000
made to be just a piece of the
donut.
00:15:35.000 --> 00:15:39.000
You have to go through the hole.
You have to leave the surface
00:15:39.000 --> 00:15:41.000
of a torus.
In fact, there's another one.
00:15:41.000 --> 00:15:47.000
See, this one also does not
bound anything that's completely
00:15:47.000 --> 00:15:50.000
contained in the torus.
And, of course,
00:15:50.000 --> 00:15:53.000
it bounds this disc,
but inside of a torus.
00:15:53.000 --> 00:15:56.000
But, that's not a part of the
surface itself.
00:15:56.000 --> 00:16:02.000
So, in fact,
there's, and topologists would
00:16:02.000 --> 00:16:09.000
say, there's two independent --
-- loops that don't bound
00:16:09.000 --> 00:16:15.000
surfaces, that don't bound
anything.
00:16:15.000 --> 00:16:18.000
And, so this number two is
somehow an invariant that you
00:16:18.000 --> 00:16:20.000
can associate to this kind of
shape.
00:16:20.000 --> 00:16:23.000
And then, if you consider more
complicated surfaces with more
00:16:23.000 --> 00:16:24.000
holes in them,
you can try, somehow,
00:16:24.000 --> 00:16:27.000
to count independent loops on
them,
00:16:27.000 --> 00:16:33.000
and that's the beginning of the
classification of surfaces.
00:16:33.000 --> 00:16:39.000
Anyway, that's not really an
18.02 topic, but I thought I
00:16:39.000 --> 00:16:45.000
would mentioned it because it's
kind of a cool idea.
00:16:45.000 --> 00:16:55.000
OK, let me say a bit more in
the way of fun remarks like
00:16:55.000 --> 00:16:59.000
that.
So, food for thought:
00:16:59.000 --> 00:17:05.000
let's say that I want to apply
Stokes theorem to simplify a
00:17:05.000 --> 00:17:08.000
line integral along the curve
here.
00:17:08.000 --> 00:17:11.000
So, this curve is maybe not
easy to see in the picture.
00:17:11.000 --> 00:17:17.000
It kind of goes twice around
the z axis, but spirals up and
00:17:17.000 --> 00:17:20.000
then down.
OK, so one way to find a
00:17:20.000 --> 00:17:25.000
surface that's bounded by this
curve is to take what's called
00:17:25.000 --> 00:17:29.000
the Mobius strip.
OK, so the Mobius strip,
00:17:29.000 --> 00:17:32.000
it's a one sided strip where
when you go around,
00:17:32.000 --> 00:17:35.000
you flip one side becomes the
other.
00:17:35.000 --> 00:17:38.000
So, you just,
if you want to take a band of
00:17:38.000 --> 00:17:41.000
paper and glue the two sides
with a twist,
00:17:41.000 --> 00:17:44.000
so, it's a one sided surface.
And, that gives us,
00:17:44.000 --> 00:17:49.000
actually, serious trouble if we
try to orient it to apply Stokes
00:17:49.000 --> 00:17:53.000
theorem.
So, see, for example,
00:17:53.000 --> 00:17:58.000
if I take this Mobius strip,
and I try to find an
00:17:58.000 --> 00:18:04.000
orientation,
so here it looks like that,
00:18:04.000 --> 00:18:08.000
well, let's say that I've
oriented my curve going in this
00:18:08.000 --> 00:18:11.000
direction.
So, I go around,
00:18:11.000 --> 00:18:13.000
around, around,
still going this direction.
00:18:13.000 --> 00:18:19.000
Well, the orientation I should
have for Stokes theorem is that
00:18:19.000 --> 00:18:22.000
when I, so, curve continues
here.
00:18:22.000 --> 00:18:26.000
Well, if you look at the
convention around here,
00:18:26.000 --> 00:18:31.000
it tells us that the normal
vector should be going this way.
00:18:31.000 --> 00:18:35.000
OK, if we look at it near here,
if we walk along this way,
00:18:35.000 --> 00:18:37.000
the surface is to our right .
So, we should actually be
00:18:37.000 --> 00:18:40.000
flipping things upside down.
The normal vector should be
00:18:40.000 --> 00:18:41.000
going down.
And, in fact,
00:18:41.000 --> 00:18:44.000
if you try to follow your
normal vector that's pointing
00:18:44.000 --> 00:18:45.000
up, it's pointing up,
up, up.
00:18:45.000 --> 00:18:49.000
It will have to go into things,
into, into, down.
00:18:49.000 --> 00:18:53.000
There's no way to choose
consistently a normal vector for
00:18:53.000 --> 00:19:01.000
the Mobius strip.
So, that's what we call a
00:19:01.000 --> 00:19:08.000
non-orientable surface.
And, that just means it has
00:19:08.000 --> 00:19:10.000
only one side.
And, if it has only one side,
00:19:10.000 --> 00:19:14.000
that we cannot speak of flux
for it because we have no way of
00:19:14.000 --> 00:19:17.000
saying that we'll be counting
things positively one way,
00:19:17.000 --> 00:19:19.000
negatively the other way,
because there's only one,
00:19:19.000 --> 00:19:22.000
you know,
there's no notion of sides.
00:19:22.000 --> 00:19:26.000
So, you can't define a side
towards which things will be
00:19:26.000 --> 00:19:34.000
going positively.
So, that's actually a situation
00:19:34.000 --> 00:19:44.000
where flux cannot be defined.
OK, so as much as Mobius strips
00:19:44.000 --> 00:19:48.000
and climb-bottles are exciting
and really cool,
00:19:48.000 --> 00:19:51.000
well, we can't use them in this
class because we can't define
00:19:51.000 --> 00:19:54.000
flux through them.
So, if we really wanted to
00:19:54.000 --> 00:19:57.000
apply Stokes theorem,
because I've been telling you
00:19:57.000 --> 00:19:59.000
that space is simply connected,
and I will always be able to
00:19:59.000 --> 00:20:01.000
apply Stokes theorem to any
curve,
00:20:01.000 --> 00:20:05.000
what would I do?
Well, I claim this curve
00:20:05.000 --> 00:20:10.000
actually bounds another surface
that is orientable.
00:20:10.000 --> 00:20:11.000
Yeah, that looks
counterintuitive.
00:20:11.000 --> 00:20:16.000
Well, let's see it.
I claim you can take a
00:20:16.000 --> 00:20:22.000
hemisphere, and you can take a
small thing and twist it around.
00:20:22.000 --> 00:20:26.000
So, in case you don't believe
me, let me do it again with the
00:20:26.000 --> 00:20:28.000
transparency.
Here's my loop,
00:20:28.000 --> 00:20:31.000
and see, well,
the scale is not exactly the
00:20:31.000 --> 00:20:33.000
same.
So, it doesn't quite match.
00:20:33.000 --> 00:20:35.000
But, and it's getting a bit
dark.
00:20:35.000 --> 00:20:41.000
But, that spherical thing with
a little slit going twisting
00:20:41.000 --> 00:20:46.000
into it will actually have
boundary my loop.
00:20:46.000 --> 00:20:50.000
And, that one is orientable.
I mean, I leave it up to you to
00:20:50.000 --> 00:20:56.000
stare at the picture long enough
to convince yourselves that
00:20:56.000 --> 00:21:00.000
there's a well-defined up and
down.
00:21:00.000 --> 00:21:11.000
OK.
So now, I mean,
00:21:11.000 --> 00:21:14.000
in case you are getting really,
really worried,
00:21:14.000 --> 00:21:18.000
I mean, there won't be any
Mobius strips on the exam on
00:21:18.000 --> 00:21:24.000
Tuesday, OK?
It's just to show you some cool
00:21:24.000 --> 00:21:29.000
stuff.
OK, questions?
00:21:29.000 --> 00:21:34.000
No?
OK, one last thing I want to
00:21:34.000 --> 00:21:38.000
show you before we start
reviewing,
00:21:38.000 --> 00:21:41.000
so one question you might have
about Stokes theorem is,
00:21:41.000 --> 00:21:44.000
how come we can choose whatever
surface we want?
00:21:44.000 --> 00:21:47.000
I mean, sure,
it seems to work,
00:21:47.000 --> 00:21:52.000
but why?
So, I'm going to say a couple
00:21:52.000 --> 00:22:02.000
of words about surface
independence in Stokes theorem.
00:22:02.000 --> 00:22:08.000
So, let's say that I have a
curve, C, in space.
00:22:08.000 --> 00:22:11.000
And, let's say that I want to
apply Stokes theorem.
00:22:11.000 --> 00:22:16.000
So, then I can choose my
favorite surface bounded by C.
00:22:16.000 --> 00:22:18.000
So, in a situation like this,
for example,
00:22:18.000 --> 00:22:21.000
I might want to make my first
choice be this guy,
00:22:21.000 --> 00:22:25.000
S1, like maybe some sort of
upper half sphere.
00:22:25.000 --> 00:22:28.000
And, if you pay attention to
the orientation conventions,
00:22:28.000 --> 00:22:31.000
you'll see that you need to
take it with normal vector
00:22:31.000 --> 00:22:34.000
pointing up.
Maybe actually I would rather
00:22:34.000 --> 00:22:36.000
make a different choice.
And actually,
00:22:36.000 --> 00:22:41.000
I will choose another surface,
S2, that maybe looks like that.
00:22:41.000 --> 00:22:44.000
And, if I look carefully at the
orientation convention,
00:22:44.000 --> 00:22:47.000
Stokes theorem tells me that I
have to take the normal vector
00:22:47.000 --> 00:22:52.000
pointing up again.
So, that's actually into things.
00:22:52.000 --> 00:22:57.000
So,
Stokes says that the line
00:22:57.000 --> 00:23:04.000
integral along C of my favorite
vector field can be computed
00:23:04.000 --> 00:23:09.000
either as a flux integral for
the curl through S1,
00:23:09.000 --> 00:23:16.000
or as the same integral,
but through S2 instead of S1.
00:23:16.000 --> 00:23:21.000
So, that seems to suggest that
curl F has some sort of surface
00:23:21.000 --> 00:23:24.000
independence property.
It doesn't really matter which
00:23:24.000 --> 00:23:27.000
surface I take,
as long as the boundary is this
00:23:27.000 --> 00:23:29.000
given curve, C.
Why is that?
00:23:29.000 --> 00:23:31.000
That's a strange property to
have.
00:23:31.000 --> 00:23:36.000
Where does it come from?
Well, let's think about it for
00:23:36.000 --> 00:23:40.000
a second.
So, why are these the same?
00:23:40.000 --> 00:23:42.000
I mean, of course,
they have to be the same
00:23:42.000 --> 00:23:44.000
because that's what Stokes tell
us.
00:23:44.000 --> 00:23:48.000
But, why is that OK?
Well, let's think about
00:23:48.000 --> 00:23:53.000
comparing the flux integral for
S1 and the flux integral for S2.
00:23:53.000 --> 00:23:57.000
So, if we want to compare them,
we should probably subtract
00:23:57.000 --> 00:24:02.000
them from each other.
OK, so let's do the flux
00:24:02.000 --> 00:24:09.000
integral for S1 minus the flux
integral for S2 of the same
00:24:09.000 --> 00:24:12.000
thing.
Well, let's give a name.
00:24:12.000 --> 00:24:18.000
Let's call S the surface S1
minus S2.
00:24:18.000 --> 00:24:21.000
So, what is S?
S is S1 with its given
00:24:21.000 --> 00:24:26.000
orientation together with S2
with the reversed orientation.
00:24:26.000 --> 00:24:32.000
So, S is actually this whole
closed surface here.
00:24:32.000 --> 00:24:37.000
And, the normal vector to S
seems to be pointing outwards
00:24:37.000 --> 00:24:39.000
everywhere.
OK, so now, if we have a closed
00:24:39.000 --> 00:24:41.000
surface with a normal vector
pointing outwards,
00:24:41.000 --> 00:24:44.000
and we want to find a flux
integral for it,
00:24:44.000 --> 00:24:47.000
well,
we can replace that with a
00:24:47.000 --> 00:24:57.000
triple integral.
So, that's the divergence
00:24:57.000 --> 00:25:03.000
theorem.
So, that's by the divergence
00:25:03.000 --> 00:25:09.000
theorem using the fact that S is
a closed surface.
00:25:09.000 --> 00:25:13.000
That's equal to the triple
integral over the region inside.
00:25:13.000 --> 00:25:26.000
Let me call that region D of
divergence, of curl F dV.
00:25:26.000 --> 00:25:34.000
OK, and what I'm going to claim
now is that we can actually
00:25:34.000 --> 00:25:41.000
check that if you take the
divergence of the curl of a
00:25:41.000 --> 00:25:47.000
vector field,
you always get zero.
00:25:47.000 --> 00:25:50.000
OK, and so that will tell you
that this integral will always
00:25:50.000 --> 00:25:53.000
be zero.
And that's why the flux for S1,
00:25:53.000 --> 00:25:58.000
and the flux for S2 were the
same a priori and we didn't have
00:25:58.000 --> 00:26:01.000
to worry about which one we
chose when we did Stokes
00:26:01.000 --> 00:26:06.000
theorem.
OK, so let's just check quickly
00:26:06.000 --> 00:26:10.000
that divergence of a curve is
zero.
00:26:10.000 --> 00:26:12.000
OK, in case you're wondering
why I'm doing all this,
00:26:12.000 --> 00:26:13.000
well, first I think it's kind
of interesting,
00:26:13.000 --> 00:26:17.000
and second, it reminds you of a
statement of all these theorems,
00:26:17.000 --> 00:26:19.000
and all these definitions.
So, in a way,
00:26:19.000 --> 00:26:25.000
we are already reviewing.
OK, so let's see.
00:26:25.000 --> 00:26:30.000
If my vector field has
components P,
00:26:30.000 --> 00:26:39.000
Q, and R, remember that the
curl was defined by this cross
00:26:39.000 --> 00:26:47.000
product between del and our
given vector field.
00:26:47.000 --> 00:27:04.000
So, that's Ry - Qz followed by
Pz - Rx, and Qx - Py.
00:27:04.000 --> 00:27:14.000
So, now, we want to take the
divergence of this.
00:27:14.000 --> 00:27:19.000
Well, so we have to take the
first component,
00:27:19.000 --> 00:27:23.000
Ry minus Qz,
and take its partial with
00:27:23.000 --> 00:27:28.000
respect to x.
Then, take the y component,
00:27:28.000 --> 00:27:35.000
Pz minus Rx partial with
respect to y plus Qx minus Py
00:27:35.000 --> 00:27:41.000
partial with respect to z.
And, well, now we should expand
00:27:41.000 --> 00:27:43.000
this.
But I claim it will always
00:27:43.000 --> 00:27:44.000
simplify to zero.
00:28:11.000 --> 00:28:24.000
OK, so I think we have over
there, becomes R sub yx minus Q
00:28:24.000 --> 00:28:39.000
sub zx plus P sub zy minus R sub
xy plus Q sub xz minus P sub yz.
00:28:39.000 --> 00:28:47.000
Well, let's see.
We have P sub zy minus P sub yz.
00:28:47.000 --> 00:28:54.000
These two cancel out.
We have R sub yx minus R sub xy.
00:28:54.000 --> 00:28:58.000
These cancel out.
Q sub zx and Q sub xz,
00:28:58.000 --> 00:29:04.000
these two also cancel out.
So, indeed, the divergence of a
00:29:04.000 --> 00:29:10.000
curl is always zero.
OK, so the claim is divergence
00:29:10.000 --> 00:29:17.000
of curl is always zero.
Del cross F is always zero,
00:29:17.000 --> 00:29:26.000
and just a small remark,
if we had actually real vectors
00:29:26.000 --> 00:29:30.000
rather than this strange del
guy,
00:29:30.000 --> 00:29:33.000
indeed we know that if we have
two vectors,
00:29:33.000 --> 00:29:37.000
U and V,
and we do u dot u cross v,
00:29:37.000 --> 00:29:40.000
what is that?
Well, one way to say it is it's
00:29:40.000 --> 00:29:43.000
the determinant of u,
u, and v, which is the volume
00:29:43.000 --> 00:29:45.000
of the box.
But, it's completely flat
00:29:45.000 --> 00:29:47.000
because u, u,
and v are all in the plane
00:29:47.000 --> 00:29:50.000
defined by u and v.
The other way to say it is that
00:29:50.000 --> 00:29:53.000
u cross v is perpendicular to u
and v.
00:29:53.000 --> 00:29:56.000
Well, if it's perpendicular u,
then its dot product with u
00:29:56.000 --> 00:29:59.000
will be zero.
So, no matter how you say it,
00:29:59.000 --> 00:30:02.000
this is always zero.
So, in a way,
00:30:02.000 --> 00:30:09.000
this reinforces our intuition
that del, even though it's not
00:30:09.000 --> 00:30:15.000
at all an actual vector
sometimes can be manipulated in
00:30:15.000 --> 00:30:20.000
the same way.
OK, I think that's it for new
00:30:20.000 --> 00:30:26.000
topics for today.
And,
00:30:26.000 --> 00:30:30.000
so, now I should maybe try to
recap quickly what we've learned
00:30:30.000 --> 00:30:34.000
in these past three weeks so
that you know,
00:30:34.000 --> 00:30:39.000
so, the exam is probably going
to be similar in difficulty to
00:30:39.000 --> 00:30:42.000
the practice exams.
That's my goal.
00:30:42.000 --> 00:30:45.000
I don't know if I will have
reached that goal or not.
00:30:45.000 --> 00:30:48.000
We'll only know that after
you've taken the test.
00:30:48.000 --> 00:30:53.000
But, the idea is it's meant to
be more or less the same level
00:30:53.000 --> 00:30:58.000
of difficulty.
So, at this point,
00:30:58.000 --> 00:31:06.000
we've learned about three kinds
of beasts in space.
00:31:06.000 --> 00:31:12.000
OK, so I'm going to divide my
blackboard into three pieces,
00:31:12.000 --> 00:31:16.000
and here I will write triple
integrals.
00:31:16.000 --> 00:31:20.000
We've learned about double
integrals, and we've learned
00:31:20.000 --> 00:31:26.000
about line integrals.
OK, so triple integrals over a
00:31:26.000 --> 00:31:33.000
region in space,
we integrate a scalar quantity,
00:31:33.000 --> 00:31:35.000
dV.
How do we do that?
00:31:35.000 --> 00:31:41.000
Well, we can do that in
rectangular coordinates where dV
00:31:41.000 --> 00:31:46.000
becomes something like,
maybe, dz dx dy,
00:31:46.000 --> 00:31:52.000
or any permutation of these.
We've seen how to do it also in
00:31:52.000 --> 00:31:59.000
cylindrical coordinates where dV
is maybe dz times r dr d theta
00:31:59.000 --> 00:32:02.000
or more commonly r dr d theta
dz.
00:32:02.000 --> 00:32:06.000
But, what I want to emphasize
in this way is that both of
00:32:06.000 --> 00:32:09.000
these you set up pretty much in
the same way.
00:32:09.000 --> 00:32:12.000
So, remember,
the main trick here is to find
00:32:12.000 --> 00:32:15.000
the bounds of integration.
So, when you do it,
00:32:15.000 --> 00:32:18.000
say, with dz first,
that means for fixed xy,
00:32:18.000 --> 00:32:23.000
so, for a fixed point in the xy
plane, you have to look at the
00:32:23.000 --> 00:32:25.000
bounds for z.
So, that means you have to
00:32:25.000 --> 00:32:28.000
figure out what's the bottom
surface of your solid,
00:32:28.000 --> 00:32:31.000
and what's the top surface of
your solid?
00:32:31.000 --> 00:32:34.000
And, you have to find the value
of z at the bottom,
00:32:34.000 --> 00:32:37.000
the value of z at the top as
functions of x and y.
00:32:37.000 --> 00:32:40.000
And then, you will put that as
bounds for z.
00:32:40.000 --> 00:32:43.000
Once you've done that,
you are left with the question
00:32:43.000 --> 00:32:45.000
of finding bounds for x and y.
Well, for that,
00:32:45.000 --> 00:32:49.000
you just rotate the picture,
look at your solid from above,
00:32:49.000 --> 00:32:52.000
so, look at its projection to
the xy plane,
00:32:52.000 --> 00:32:56.000
and you set up a double
integral either in rectangular
00:32:56.000 --> 00:33:02.000
xy coordinates,
or in polar coordinates for x
00:33:02.000 --> 00:33:04.000
and y.
Of course, you can always do it
00:33:04.000 --> 00:33:08.000
a different orders.
And, I'll let you figure out
00:33:08.000 --> 00:33:11.000
again how that goes.
But, if you do dz first,
00:33:11.000 --> 00:33:15.000
then the inner bounds are given
by bottom and top,
00:33:15.000 --> 00:33:20.000
and the outer ones are given by
looking at the shadow of the
00:33:20.000 --> 00:33:23.000
region.
Now, there's also spherical
00:33:23.000 --> 00:33:28.000
coordinates.
And there, we've seen that dV
00:33:28.000 --> 00:33:32.000
is rho squared sine phi d rho d
phi d theta.
00:33:32.000 --> 00:33:35.000
So now, of course,
if this orgy of Greek letters
00:33:35.000 --> 00:33:39.000
is confusing you at this point,
then you probably need to first
00:33:39.000 --> 00:33:41.000
review spherical coordinates for
themselves.
00:33:41.000 --> 00:33:44.000
Remember that rho is the
distance from the origin.
00:33:44.000 --> 00:33:47.000
Phi is the angle down from the
z axis.
00:33:47.000 --> 00:33:49.000
So, it's zero,
and the positive z axis,
00:33:49.000 --> 00:33:53.000
pi over two in the xy plane,
and increases all the way to pi
00:33:53.000 --> 00:33:59.000
on the negative z axis.
And, theta is the angle around
00:33:59.000 --> 00:34:02.000
the z axis.
So, now, when we set up bounds
00:34:02.000 --> 00:34:04.000
here,
it will look a lot like what
00:34:04.000 --> 00:34:07.000
you've done in polar coordinates
in the plane because when you
00:34:07.000 --> 00:34:09.000
look at the inner bound down on
rho,
00:34:09.000 --> 00:34:12.000
for a fixed phi and theta,
that means you're shooting a
00:34:12.000 --> 00:34:15.000
straight ray from the origin in
some direction in space.
00:34:15.000 --> 00:34:17.000
So, you know,
you're sending a laser beam,
00:34:17.000 --> 00:34:20.000
and you want to know what part
of your beam is going to be in
00:34:20.000 --> 00:34:23.000
your given solid.
You want to solve for the value
00:34:23.000 --> 00:34:26.000
of rho when you enter the solid
and when you leave it.
00:34:26.000 --> 00:34:29.000
I mean, very often,
if the origin is in your solid,
00:34:29.000 --> 00:34:33.000
then rho will start at zero.
Then you want to know when you
00:34:33.000 --> 00:34:34.000
exit.
And, I mean,
00:34:34.000 --> 00:34:38.000
there's a fairly small list of
kinds of surfaces that we've
00:34:38.000 --> 00:34:41.000
seen how to set up in spherical
coordinates.
00:34:41.000 --> 00:34:44.000
So, if you're really upset by
this, go over the problems in
00:34:44.000 --> 00:34:47.000
the notes.
That will give you a good idea
00:34:47.000 --> 00:34:53.000
of what kinds of things we've
seen in spherical coordinates.
00:34:53.000 --> 00:34:56.000
OK, and then evaluation is the
usual way.
00:34:56.000 --> 00:35:01.000
Questions about this?
No?
00:35:01.000 --> 00:35:08.000
OK, so, I should say we can do
something bad,
00:35:08.000 --> 00:35:15.000
but so we've seen,
of course, applications of
00:35:15.000 --> 00:35:19.000
this.
So, we should know how to use a
00:35:19.000 --> 00:35:24.000
triple integral to evaluate
things like a mass of a solid,
00:35:24.000 --> 00:35:29.000
the average value of a
function,
00:35:29.000 --> 00:35:37.000
the moment of inertia about one
of the coordinate axes,
00:35:37.000 --> 00:35:54.000
or the gravitational attraction
on a mass at the origin.
00:35:54.000 --> 00:35:58.000
OK, so these are just formulas
to remember for examples of
00:35:58.000 --> 00:36:01.000
triple integrals.
It doesn't change conceptually.
00:36:01.000 --> 00:36:04.000
You always set them up and
evaluate them the same way.
00:36:04.000 --> 00:36:11.000
It just tells you what to put
there for the integrand.
00:36:11.000 --> 00:36:15.000
Now,
double integrals: so,
00:36:15.000 --> 00:36:18.000
when we have a surface in
space,
00:36:18.000 --> 00:36:21.000
well, what we will integrate on
it,
00:36:21.000 --> 00:36:26.000
at least what we've seen how to
integrate is a vector field
00:36:26.000 --> 00:36:31.000
dotted with the unit normal
vector times the area element.
00:36:31.000 --> 00:36:38.000
OK, and this is sometimes
called vector dS.
00:36:38.000 --> 00:36:48.000
Now, how do we evaluate that?
Well, we've seen formulas for
00:36:48.000 --> 00:36:55.000
ndS in various settings.
And, once you have a formula
00:36:55.000 --> 00:37:01.000
for ndS, that will relate ndS to
maybe dx dy, or something else.
00:37:01.000 --> 00:37:07.000
And then, you will express,
so, for example,
00:37:07.000 --> 00:37:15.000
ndS equals something dx dy.
And then, it becomes a double
00:37:15.000 --> 00:37:21.000
integral of something dx dy.
Now, in the integrand,
00:37:21.000 --> 00:37:23.000
you want to express everything
in terms of x and y.
00:37:23.000 --> 00:37:26.000
So, if you had a z,
maybe you have a formula for z
00:37:26.000 --> 00:37:28.000
in terms of x and y.
And, when you set up the
00:37:28.000 --> 00:37:30.000
bounds, well,
you try to figure out what are
00:37:30.000 --> 00:37:33.000
the bounds for x and y?
That would be just looking at
00:37:33.000 --> 00:37:35.000
it from above.
Of course, if you are using
00:37:35.000 --> 00:37:37.000
other variables,
figure out the bounds for those
00:37:37.000 --> 00:37:40.000
variables.
And, when you've done that,
00:37:40.000 --> 00:37:44.000
it becomes just a double
integral in the usual sense.
00:37:44.000 --> 00:37:46.000
OK, so maybe I should be a bit
more explicit about formulas
00:37:46.000 --> 00:37:52.000
because there have been a lot.
So, let me tell you about a few
00:37:52.000 --> 00:37:56.000
of them.
Let me actually do that over
00:37:56.000 --> 00:38:02.000
here because I don't want to
make this too crowded.
00:38:24.000 --> 00:38:28.000
OK, so what kinds of formulas
for ndS have we seen?
00:38:28.000 --> 00:38:32.000
Well, we've seen a formula,
for example,
00:38:32.000 --> 00:38:37.000
for a horizontal plane,
or for something that's
00:38:37.000 --> 00:38:42.000
parallel to the yz plane or the
xz plane.
00:38:42.000 --> 00:38:47.000
Well, let's do just the yz
plane for a quick reminder.
00:38:47.000 --> 00:38:52.000
So, if I have a surface that's
contained inside the yz plane,
00:38:52.000 --> 00:38:56.000
then obviously I will express
ds in terms of,
00:38:56.000 --> 00:39:01.000
well, I will use y and z as my
variables.
00:39:01.000 --> 00:39:05.000
So, I will say that ds is dy
dz, or dz dy,
00:39:05.000 --> 00:39:11.000
whatever's most convenient.
Maybe we will even switch to
00:39:11.000 --> 00:39:15.000
polar coordinates after that if
a problem wants us to.
00:39:15.000 --> 00:39:16.000
And, what about the normal
vector?
00:39:16.000 --> 00:39:21.000
Well, the normal vector is
either coming straight at us,
00:39:21.000 --> 00:39:26.000
or it's maybe going back away
from us depending on which
00:39:26.000 --> 00:39:30.000
orientation we've chosen.
So, this gives us ndS.
00:39:30.000 --> 00:39:32.000
We dot our favorite vector
field with it.
00:39:32.000 --> 00:39:38.000
We integrate,
and we get the answer.
00:39:38.000 --> 00:39:48.000
OK, we've seen about spheres
and cylinders centered at the
00:39:48.000 --> 00:39:54.000
origin or centered on the z
axis.
00:39:54.000 --> 00:40:00.000
So, the normal vector sticks
straight out or straight in,
00:40:00.000 --> 00:40:05.000
depending on which direction
you do it in.
00:40:05.000 --> 00:40:09.000
So, for a sphere,
the normal vector is 00:40:14.000
y, z> divided by the radius
of the sphere.
00:40:14.000 --> 00:40:17.000
For a cylinder,
it's 00:40:21.000
0>, divided by the radius of
a cylinder.
00:40:21.000 --> 00:40:25.000
And, the surface element on a
sphere,
00:40:25.000 --> 00:40:28.000
so, see, it's very closely
related to the volume element of
00:40:28.000 --> 00:40:31.000
spherical coordinates except you
don't have a rho anymore.
00:40:31.000 --> 00:40:37.000
You just plug in a rho equals a.
So, you get a squared sine phi
00:40:37.000 --> 00:40:41.000
d phi d theta.
And, for a cylinder,
00:40:41.000 --> 00:40:47.000
it would be a dz d theta.
So,
00:40:47.000 --> 00:40:51.000
by the way, just a quick check,
when you're doing an integral,
00:40:51.000 --> 00:40:55.000
if it's the surface integral,
there should be two integral
00:40:55.000 --> 00:40:56.000
signs,
and there should be two
00:40:56.000 --> 00:40:59.000
integration variables.
And, there should be two d
00:40:59.000 --> 00:41:03.000
somethings.
If you end up with a dx,
00:41:03.000 --> 00:41:10.000
dy, dz in the surface integral,
something is seriously wrong.
00:41:10.000 --> 00:41:18.000
OK, now, besides these specific
formulas, we've seen two general
00:41:18.000 --> 00:41:24.000
formulas that are also useful.
So, one is,
00:41:24.000 --> 00:41:29.000
if we know how to express z in
terms of x and y,
00:41:29.000 --> 00:41:33.000
and just to change notation to
show you that it's not set in
00:41:33.000 --> 00:41:36.000
stone,
let's say that z is known as a
00:41:36.000 --> 00:41:42.000
function z of x and y.
So, how do I get ndS in that
00:41:42.000 --> 00:41:45.000
case?
Well, we've seen a formula that
00:41:45.000 --> 00:41:51.000
says negative partial z partial
x, negative partial z partial y,
00:41:51.000 --> 00:41:54.000
one dx dy.
So, this formula relates the
00:41:54.000 --> 00:41:57.000
volume, sorry,
the surface element on our
00:41:57.000 --> 00:42:01.000
surface to the area element in
the xy plane.
00:42:01.000 --> 00:42:08.000
It lets us convert between dS
and dx dy.
00:42:08.000 --> 00:42:11.000
OK, so we just plug in this,
and we dot with F,
00:42:11.000 --> 00:42:14.000
and then we substitute
everything in terms of x and y,
00:42:14.000 --> 00:42:17.000
and we evaluate the integral
over x and y.
00:42:17.000 --> 00:42:22.000
If we don't really want to find
a way to find z as a function of
00:42:22.000 --> 00:42:29.000
x and y,
but we have a normal vector
00:42:29.000 --> 00:42:35.000
given to us,
then we have another formula
00:42:35.000 --> 00:42:39.000
which says that ndS is,
sorry, I should have said it's
00:42:39.000 --> 00:42:42.000
always up to sign because we
have a two orientation
00:42:42.000 --> 00:42:45.000
convention.
We have to decide based on what
00:42:45.000 --> 00:42:48.000
we are trying to do,
whether we are doing the
00:42:48.000 --> 00:42:51.000
correct convention or the wrong
one.
00:42:51.000 --> 00:43:02.000
So, the other formula is n over
n dot k dx dy.
00:43:02.000 --> 00:43:09.000
Sorry, are they all the same?
Well, if you want,
00:43:09.000 --> 00:43:12.000
you can put an absolute value
here.
00:43:12.000 --> 00:43:16.000
But, it doesn't matter because
it's up to sign anyway.
00:43:16.000 --> 00:43:22.000
So, I mean, this formula is
valid as it is.
00:43:22.000 --> 00:43:24.000
OK, and, I mean,
if you're in a situation where
00:43:24.000 --> 00:43:26.000
you can apply more than one
formula,
00:43:26.000 --> 00:43:32.000
they will all give you the same
answer in the end because it's
00:43:32.000 --> 00:43:37.000
the same flux integral.
OK, so anyway,
00:43:37.000 --> 00:43:40.000
so we have various ways of
computing surface integrals,
00:43:40.000 --> 00:43:44.000
and probably one of the best
possible things you can do to
00:43:44.000 --> 00:43:48.000
prepare for the test is actually
to look again at some practice
00:43:48.000 --> 00:43:51.000
problems from the notes that do
flux integrals over various
00:43:51.000 --> 00:43:55.000
kinds of surfaces because that's
probably one of the hardest
00:43:55.000 --> 00:43:58.000
topics in this unit of the
class.
00:43:58.000 --> 00:44:05.000
OK, anyway, let's move on to
line integrals.
00:44:05.000 --> 00:44:11.000
So, those are actually a piece
of cake in comparison,
00:44:11.000 --> 00:44:17.000
OK, because all that this is,
is just integral of P dx Q dy R
00:44:17.000 --> 00:44:24.000
dz.
And, then all you have to do is
00:44:24.000 --> 00:44:35.000
parameterize the curve,
C, to express everything in
00:44:35.000 --> 00:44:42.000
terms of a single variable.
And then, you end up with a
00:44:42.000 --> 00:44:46.000
usual single integral,
and you can just compute it.
00:44:46.000 --> 00:44:48.000
So, that one works pretty much
as it did in the plane.
00:44:48.000 --> 00:44:52.000
So, if you forgotten what we
did in the plane,
00:44:52.000 --> 00:44:56.000
it's really the same thing.
OK, so now we have three
00:44:56.000 --> 00:44:58.000
different kinds of integrals,
and really, well,
00:44:58.000 --> 00:45:01.000
they certainly have in common
that they integrate things
00:45:01.000 --> 00:45:03.000
somehow.
But, apart from that,
00:45:03.000 --> 00:45:05.000
they are extremely different in
what they do.
00:45:05.000 --> 00:45:08.000
I mean, this one involves a
function, a scalar quantity.
00:45:08.000 --> 00:45:11.000
These involve vector quantities.
They don't involve the same
00:45:11.000 --> 00:45:13.000
kinds of shapes over which to
integrate.
00:45:13.000 --> 00:45:16.000
Here, you integrate over a
three-dimensional region.
00:45:16.000 --> 00:45:19.000
Here, you integrate only over a
two-dimensional surface,
00:45:19.000 --> 00:45:21.000
and here, only a
one-dimensional curve.
00:45:21.000 --> 00:45:24.000
So, try not to confuse them.
That's basically the most
00:45:24.000 --> 00:45:27.000
important advice.
Don't get mistaken.
00:45:27.000 --> 00:45:30.000
Each of them has a different
way of getting evaluated.
00:45:30.000 --> 00:45:34.000
Eventually, they will all give
you numbers, but through
00:45:34.000 --> 00:45:36.000
different processes.
So now, well,
00:45:36.000 --> 00:45:38.000
I said these guys are
completely different.
00:45:38.000 --> 00:45:40.000
Well, they are,
but we still have some bridges
00:45:40.000 --> 00:45:42.000
between them.
OK, so we have two,
00:45:42.000 --> 00:45:46.000
maybe I should say three,
well, two bridges between these
00:45:46.000 --> 00:45:49.000
guys.
OK, so we have somehow a
00:45:49.000 --> 00:45:54.000
connection between these which
is the divergence theorem.
00:45:54.000 --> 00:46:02.000
We have a connection between
that, which is Stokes theorem.
00:46:02.000 --> 00:46:20.000
So -- Just to write them again,
so the divergence theorem says
00:46:20.000 --> 00:46:25.000
if I have a region in space,
and I call its boundary S,
00:46:25.000 --> 00:46:27.000
so, it's going to be a closed
surface,
00:46:27.000 --> 00:46:31.000
and I orient S with a normal
vector pointing outwards,
00:46:31.000 --> 00:46:35.000
then whenever I have a surface
integral over S,
00:46:35.000 --> 00:46:40.000
sorry,
I can replace it by a triple
00:46:40.000 --> 00:46:47.000
integral over the region inside.
OK, so this guy is a vector
00:46:47.000 --> 00:46:49.000
field.
And, this guy is a function
00:46:49.000 --> 00:46:52.000
that somehow relates to the
vector field.
00:46:52.000 --> 00:46:54.000
I mean, you should know how.
You should know the definition
00:46:54.000 --> 00:46:55.000
of divergence,
of course.
00:46:55.000 --> 00:46:59.000
But, what I want to point out
is if you have to compute the
00:46:59.000 --> 00:47:01.000
two sides separately,
well, this is just,
00:47:01.000 --> 00:47:04.000
you know, your standard flux
integral.
00:47:04.000 --> 00:47:07.000
This is just your standard
triple integral over a region in
00:47:07.000 --> 00:47:09.000
space.
Once you have computed what
00:47:09.000 --> 00:47:13.000
this guy is, it's really just a
triple integral of the function.
00:47:13.000 --> 00:47:16.000
So, the way in which you
compute it doesn't see that it
00:47:16.000 --> 00:47:19.000
came from a divergence.
It's just the same way that you
00:47:19.000 --> 00:47:23.000
would compute any other triple
integral.
00:47:23.000 --> 00:47:28.000
The way we compute it doesn't
depend on what actually we are
00:47:28.000 --> 00:47:32.000
integrating.
Stokes theorem says if I have a
00:47:32.000 --> 00:47:35.000
curve that's the boundary of a
surface, S,
00:47:35.000 --> 00:47:40.000
and I orient the two in
compatible manners,
00:47:40.000 --> 00:47:52.000
then I can replace a line
integral on C by a surface
00:47:52.000 --> 00:47:57.000
integral on S.
OK, and that surface integral,
00:47:57.000 --> 00:48:00.000
well, it's not for the same
vector field.
00:48:00.000 --> 00:48:03.000
This relates a line integral
for one field to a surface
00:48:03.000 --> 00:48:06.000
integral from another field.
That other field is given from
00:48:06.000 --> 00:48:10.000
the first one just by taking its
curl So, after you take the
00:48:10.000 --> 00:48:13.000
curl, you obtain a different
vector field.
00:48:13.000 --> 00:48:17.000
And, the way in which you would
compute the surface integral is
00:48:17.000 --> 00:48:19.000
just as with any surface
integral.
00:48:19.000 --> 00:48:23.000
You just find a formula for ndS
dot product, substitute,
00:48:23.000 --> 00:48:26.000
evaluate.
The calculation of this thing,
00:48:26.000 --> 00:48:30.000
once you've computed curl does
not remember that it was a curl.
00:48:30.000 --> 00:48:33.000
It's the same as with any other
flux integral.
00:48:33.000 --> 00:48:35.000
OK, and finally,
the last bridge,
00:48:35.000 --> 00:48:38.000
so this was between two and
three.
00:48:38.000 --> 00:48:41.000
This was between one and two.
Let me just say,
00:48:41.000 --> 00:48:45.000
there's a bridge between zero
and one,
00:48:45.000 --> 00:48:53.000
which is that if you have a
function in its gradient,
00:48:53.000 --> 00:48:57.000
well, the fundamental theorem
of calculus says that the line
00:48:57.000 --> 00:49:01.000
integral for the vector field
given by the gradient of a
00:49:01.000 --> 00:49:04.000
function is actually equal to
the change in value of a
00:49:04.000 --> 00:49:08.000
function.
That's if you have a curve
00:49:08.000 --> 00:49:11.000
bounded by P0 and P1.
So in a way, actually,
00:49:11.000 --> 00:49:15.000
each of these three theorems
relates a quantity with a
00:49:15.000 --> 00:49:19.000
certain number of integral signs
to a quantity with one more
00:49:19.000 --> 00:49:22.000
integral sign.
And, that's actually somehow a
00:49:22.000 --> 00:49:24.000
fundamental similarity between
them.
00:49:24.000 --> 00:49:27.000
But maybe it's easier to think
of them as completely different
00:49:27.000 --> 00:49:30.000
stories.
So now, with this one,
00:49:30.000 --> 00:49:36.000
we additionally have to
remember another topic is given
00:49:36.000 --> 00:49:41.000
a vector field,
F, with curl equal to zero,
00:49:41.000 --> 00:49:45.000
find the potential.
And, we've seen two methods for
00:49:45.000 --> 00:49:47.000
that, and I'm sure you remember
them.
00:49:47.000 --> 00:49:53.000
So, if not, then try to
remember them for Tuesday.
00:49:53.000 --> 00:49:54.000
OK, so anyway,
again, conceptually,
00:49:54.000 --> 00:49:56.000
we have, really,
three different kinds of
00:49:56.000 --> 00:49:59.000
integrals.
We evaluated them in completely
00:49:59.000 --> 00:50:01.000
different ways,
and we have a handful of
00:50:01.000 --> 00:50:03.000
theorems, connecting them to
each other.
00:50:03.000 --> 00:50:06.000
But, that doesn't have any
impact on how we actually
00:50:06.000 --> 00:50:08.000
compute things.