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HERBERT GROSS: Hi,
our lesson today
00:00:33.590 --> 00:00:35.390
is going to be
concerned with looking
00:00:35.390 --> 00:00:37.550
at some of the
ramifications of what
00:00:37.550 --> 00:00:41.690
it means for a complex valued
function to be analytic.
00:00:41.690 --> 00:00:44.960
And one of the side results,
I hope, of today's lesson
00:00:44.960 --> 00:00:48.830
will be to show why, if the
complex numbers had never
00:00:48.830 --> 00:00:51.680
been invented up
until this point,
00:00:51.680 --> 00:00:53.180
they would have
been invented, most
00:00:53.180 --> 00:00:58.730
likely to discuss mappings of
the xy plane into the uv plane.
00:00:58.730 --> 00:01:01.640
While I hope to make that
clearer as we go along,
00:01:01.640 --> 00:01:05.370
for the time being, let's
simply entitle today's lesson
00:01:05.370 --> 00:01:07.250
Conformal Mappings.
00:01:07.250 --> 00:01:09.110
And by way of review--
00:01:09.110 --> 00:01:10.610
and what I want to
do here, you see,
00:01:10.610 --> 00:01:14.570
is emphasize the real part
of complex variable theory.
00:01:14.570 --> 00:01:18.740
Suppose I have the usual
mapping from the xy plane
00:01:18.740 --> 00:01:23.420
into the uv plane given by u as
some function of x and y and v
00:01:23.420 --> 00:01:25.220
as some function of x and y.
00:01:25.220 --> 00:01:28.580
And you may recall this
is back much earlier
00:01:28.580 --> 00:01:31.490
in our course in
block 4, block 3,
00:01:31.490 --> 00:01:38.750
where we were labeling this
as f bar of x, y equals u,
00:01:38.750 --> 00:01:43.610
v. No necessity here of talking
about complex variables.
00:01:43.610 --> 00:01:48.530
We're mapping two space
into two space, the xy plane
00:01:48.530 --> 00:01:50.990
into the uv plane,
by the mapping f
00:01:50.990 --> 00:01:56.540
bar defined by u and v being
these functions of x and y.
00:01:56.540 --> 00:02:00.260
And what we saw was that
this particular mapping was,
00:02:00.260 --> 00:02:03.600
at least locally, meaning in a
neighborhood of a given point,
00:02:03.600 --> 00:02:08.389
in vertical, provided that the
determinant of the Jacobian
00:02:08.389 --> 00:02:10.280
matrix was not 0.
00:02:10.280 --> 00:02:14.660
In other words, that u sub
x, v sub y minus u sub y,
00:02:14.660 --> 00:02:17.330
v sub x was not equal to zero.
00:02:17.330 --> 00:02:19.130
And again, what
I want you to see
00:02:19.130 --> 00:02:20.990
is that, up until
this point, there
00:02:20.990 --> 00:02:26.030
is absolutely no knowledge that
we need of complex variables.
00:02:26.030 --> 00:02:28.520
Now, what is
interesting is this.
00:02:28.520 --> 00:02:32.810
Remember that, when we
introduced the complex numbers,
00:02:32.810 --> 00:02:37.970
we said that, graphically, their
domain would be the xy plane,
00:02:37.970 --> 00:02:40.743
and that would be called
the Argand diagram.
00:02:40.743 --> 00:02:42.410
Now, you see, coming
back to this little
00:02:42.410 --> 00:02:44.390
aside that we were
making over here,
00:02:44.390 --> 00:02:48.560
if I now view the xy plane
as being the Argand diagram,
00:02:48.560 --> 00:02:53.450
then what we're saying is that
x, y names the complex number
00:02:53.450 --> 00:02:54.770
z.
00:02:54.770 --> 00:03:01.700
u, v names the complex
number w, namely u plus iv.
00:03:01.700 --> 00:03:06.290
And since we're talking, now,
about mapping complex numbers
00:03:06.290 --> 00:03:11.450
into complex numbers, f bar
simply becomes some function f.
00:03:11.450 --> 00:03:13.970
Now, the question
that's more important
00:03:13.970 --> 00:03:18.590
is, why would one want this
particular interpretation?
00:03:18.590 --> 00:03:20.220
What particular interpretation?
00:03:20.220 --> 00:03:23.150
The particular interpretation
that I have in mind is
00:03:23.150 --> 00:03:27.530
to simply observe that, whenever
I have a change of variables--
00:03:27.530 --> 00:03:32.050
say, two equations
and two unknowns--
00:03:32.050 --> 00:03:34.030
u equals some
function of x and y,
00:03:34.030 --> 00:03:36.130
v equals some
function of x and y--
00:03:36.130 --> 00:03:41.110
what I'm saying is I can always
view this pair of equations
00:03:41.110 --> 00:03:45.820
as the single complex
variable equation that f of z
00:03:45.820 --> 00:03:49.630
equals u plus iv, that,
mechanically, by letting
00:03:49.630 --> 00:03:53.050
u be the real part and
v be the imaginary part,
00:03:53.050 --> 00:03:56.750
I can invent the complex
valued function u plus iv.
00:03:56.750 --> 00:03:58.720
By the way, this is in
much the same spirit
00:03:58.720 --> 00:04:01.630
that, when one talks
about curves in the plane,
00:04:01.630 --> 00:04:04.300
one does not have to know
anything about vectors.
00:04:04.300 --> 00:04:08.920
I can talk about the curve x
equals x of t, y equals y of t
00:04:08.920 --> 00:04:12.370
and then say, OK, let's
introduce the radius vector R
00:04:12.370 --> 00:04:16.269
and then write that R
is equal to xi plus yj,
00:04:16.269 --> 00:04:19.810
and this allows me to
take two scalar equations
00:04:19.810 --> 00:04:22.480
and write them as
one vector equation.
00:04:22.480 --> 00:04:27.580
So I can certainly take two
scalar, two real functions
00:04:27.580 --> 00:04:29.800
of two real variables
and write them
00:04:29.800 --> 00:04:32.620
in terms of the language
of complex functions.
00:04:32.620 --> 00:04:34.990
The question, of course, is
why would I want to do this?
00:04:34.990 --> 00:04:37.660
We have already seen this
in our very first lecture
00:04:37.660 --> 00:04:39.470
on complex variables.
00:04:39.470 --> 00:04:42.550
For example, we knew how to
handle things like cosine n
00:04:42.550 --> 00:04:44.440
theta without ever
having to have
00:04:44.440 --> 00:04:46.900
heard of complex variables.
00:04:46.900 --> 00:04:50.740
Yet we found that, by such
things as [INAUDIBLE] theorem,
00:04:50.740 --> 00:04:53.320
that there was some
very nice enlightenment
00:04:53.320 --> 00:04:55.660
that the structure
of complex numbers
00:04:55.660 --> 00:04:57.980
brought to real numbers.
00:04:57.980 --> 00:05:02.470
In other words, then, perhaps
by taking this pair of equations
00:05:02.470 --> 00:05:05.860
and writing them in the
language of complex variables,
00:05:05.860 --> 00:05:09.910
I might, from the vantage point
of the complex number system,
00:05:09.910 --> 00:05:13.060
be able to look down at
the real number system
00:05:13.060 --> 00:05:15.160
and see some things
very elegantly
00:05:15.160 --> 00:05:16.900
from a computational
point of view,
00:05:16.900 --> 00:05:18.520
from a philosophical
point of view,
00:05:18.520 --> 00:05:20.830
that I might not have been
able to notice otherwise.
00:05:20.830 --> 00:05:23.110
Now, because this is
becoming a harangue,
00:05:23.110 --> 00:05:25.480
let's get on with
the specific details
00:05:25.480 --> 00:05:28.760
and see if this doesn't
become clear as we go along.
00:05:28.760 --> 00:05:30.430
What we've already
said is we can
00:05:30.430 --> 00:05:34.300
view this change of
variables as being
00:05:34.300 --> 00:05:38.200
the real and imaginary parts
of the complex function f of z.
00:05:38.200 --> 00:05:40.540
In other words, let's
invent the function f of z
00:05:40.540 --> 00:05:42.460
to be u plus iv.
00:05:42.460 --> 00:05:43.540
Now, the point is--
00:05:43.540 --> 00:05:46.270
let's assume, for the sake
of argument, that f of z
00:05:46.270 --> 00:05:48.610
happens to be an
analytic function.
00:05:48.610 --> 00:05:51.040
What does it mean to say
that f of z is analytic?
00:05:51.040 --> 00:05:53.140
It means that f prime exists.
00:05:53.140 --> 00:05:56.170
In particular, from our
lecture of last time,
00:05:56.170 --> 00:05:58.300
not only does f prime
exist, but if we
00:05:58.300 --> 00:06:00.010
want to write this
quite simply, it
00:06:00.010 --> 00:06:02.010
turns out to be the
partial of u with respect
00:06:02.010 --> 00:06:05.380
to x plus i times the partial
of v with respect to x.
00:06:05.380 --> 00:06:08.680
In particular, the
Cauchy-Riemann conditions hold,
00:06:08.680 --> 00:06:11.380
which means that u
sub x equals v sub y,
00:06:11.380 --> 00:06:13.720
and u sub y equals
minus v sub x.
00:06:13.720 --> 00:06:15.940
And by the way, to
reinforce what I was just
00:06:15.940 --> 00:06:20.820
saying before,
please observe that
00:06:20.820 --> 00:06:23.610
this particular
relationship does not
00:06:23.610 --> 00:06:27.060
necessitate my knowing
anything about complex numbers.
00:06:27.060 --> 00:06:30.780
Given two functions u and
v, functions of x and y,
00:06:30.780 --> 00:06:33.300
I can compute the partial
of u with respect to x.
00:06:33.300 --> 00:06:35.130
I can compute the
partial of v with respect
00:06:35.130 --> 00:06:37.620
to y, the partial of
u with respect to y,
00:06:37.620 --> 00:06:39.510
the partial of v
with respect to x
00:06:39.510 --> 00:06:41.490
and see if these
conditions hold.
00:06:41.490 --> 00:06:44.520
At any rate, if these
conditions do hold--
00:06:44.520 --> 00:06:47.430
and notice how nicely the
language of complex variables
00:06:47.430 --> 00:06:49.380
allows me to say that
these conditions hold,
00:06:49.380 --> 00:06:52.080
namely all I have to say is
that f of z is analytic--
00:06:52.080 --> 00:06:54.450
then coming back to what
the Jacobian meant--
00:06:54.450 --> 00:06:55.860
remember what the
Jacobian meant?
00:06:55.860 --> 00:07:00.870
It was u sub x v sub y
minus u sub y v sub x.
00:07:00.870 --> 00:07:04.830
Coming back to this, I can
now compute this very simply,
00:07:04.830 --> 00:07:09.300
simply by observing that
another way of saying v sub y
00:07:09.300 --> 00:07:12.060
is to replace it by u sub x.
00:07:12.060 --> 00:07:15.070
And if I do that, this term
becomes u sub x squared.
00:07:15.070 --> 00:07:19.680
If I now replace u sub
y by minus v sub x,
00:07:19.680 --> 00:07:22.210
this becomes v sub x squared.
00:07:22.210 --> 00:07:26.610
And so this expression
becomes u sub
00:07:26.610 --> 00:07:28.780
x squared plus v sub x squared.
00:07:28.780 --> 00:07:32.190
And if we now look here, notice
that, since the magnitude
00:07:32.190 --> 00:07:36.300
of a complex number is just
the positive square root
00:07:36.300 --> 00:07:39.360
of the sum of the squares of the
real and the imaginary parts,
00:07:39.360 --> 00:07:41.550
this expression
here is precisely
00:07:41.550 --> 00:07:45.090
the square of the
magnitude of f prime of z.
00:07:45.090 --> 00:07:48.600
Now, look, the only way
that this can be 0, then,
00:07:48.600 --> 00:07:50.860
is if this is 0.
00:07:50.860 --> 00:07:54.390
The only way this can
be 0 is if f prime of z
00:07:54.390 --> 00:07:57.970
is itself 0 because the
only complex number whose
00:07:57.970 --> 00:08:01.620
magnitude is 0 is 0 itself.
00:08:01.620 --> 00:08:03.870
Now, what does this
tell me, therefore?
00:08:03.870 --> 00:08:08.790
This tells me that the
system of real equations, u
00:08:08.790 --> 00:08:14.700
equals u of xy, v equals
v of xy, is invertible if,
00:08:14.700 --> 00:08:18.780
when we write the complex
function, u plus iv
00:08:18.780 --> 00:08:20.570
and call that f--
00:08:20.570 --> 00:08:25.970
if f is analytic and
f prime of z is not 0.
00:08:25.970 --> 00:08:28.860
Again, notice, I
could have stated that
00:08:28.860 --> 00:08:31.980
without the language of
complex variables at all.
00:08:31.980 --> 00:08:35.100
I could have said, look, suppose
the partial of u with respect
00:08:35.100 --> 00:08:37.890
to x equals the partial
of v with respect to y,
00:08:37.890 --> 00:08:39.659
and the partial of
u with respect to y
00:08:39.659 --> 00:08:42.030
is minus the partial
of v respect to x.
00:08:42.030 --> 00:08:45.750
And suppose that u sub x squared
plus v sub x squared is not 0.
00:08:45.750 --> 00:08:48.570
Then this will be invertible.
00:08:48.570 --> 00:08:54.330
Now, notice one
of the properties
00:08:54.330 --> 00:08:58.050
is that there are a lot
of invertible functions
00:08:58.050 --> 00:09:01.710
which do not obey these
stringent conditions.
00:09:01.710 --> 00:09:03.930
Consequently, one would
like to believe that,
00:09:03.930 --> 00:09:07.590
if we're going to introduce the
language of complex variables,
00:09:07.590 --> 00:09:10.650
that we would like to get much
more out of this than just
00:09:10.650 --> 00:09:16.650
the fact that f of z maps the
xy plane into the uv plane
00:09:16.650 --> 00:09:21.810
in a one-to-one, onto manner
as long as f prime is not 0.
00:09:21.810 --> 00:09:24.930
Let me point out that some
invertible mappings are
00:09:24.930 --> 00:09:26.290
nicer than others.
00:09:26.290 --> 00:09:28.080
Now, what do I mean by nicer?
00:09:28.080 --> 00:09:31.350
Well, let's make up
a definition here.
00:09:31.350 --> 00:09:34.830
An invertible mapping
is called conformal--
00:09:34.830 --> 00:09:37.560
and maybe you can guess
what this is going to mean--
00:09:37.560 --> 00:09:39.660
if it preserves angles.
00:09:39.660 --> 00:09:41.970
Now, what do I mean
by preserving angles?
00:09:41.970 --> 00:09:43.140
What I'm saying is--
00:09:43.140 --> 00:09:47.670
let's suppose two curves meet
at a certain angle in the xy
00:09:47.670 --> 00:09:48.510
plane.
00:09:48.510 --> 00:09:51.000
When I map the xy
plane into the uv
00:09:51.000 --> 00:09:55.410
plane in a one-to-one fashion,
the two curves in the xy plane
00:09:55.410 --> 00:09:58.650
have images in the uv plane.
00:09:58.650 --> 00:10:03.690
The question that comes
up is, will the two curves
00:10:03.690 --> 00:10:06.750
intersect at the same
angle in the xy plane
00:10:06.750 --> 00:10:10.530
that their images
intersect in the uv plane?
00:10:10.530 --> 00:10:14.710
Well, heck, maybe it's easier
to do by means of an example.
00:10:14.710 --> 00:10:17.070
Let's look at the
usual linear mappings
00:10:17.070 --> 00:10:20.520
that we were talking about
when we introduced the Jacobian
00:10:20.520 --> 00:10:22.470
in double integration.
00:10:22.470 --> 00:10:25.050
Remember, one of the
problems that we tackled
00:10:25.050 --> 00:10:29.100
was looking to see how we
map, say, a parallelogram
00:10:29.100 --> 00:10:34.950
R to the unit square S by a
linear mapping of the form u
00:10:34.950 --> 00:10:38.970
equals ax plus by,
v equals cx plus dy.
00:10:38.970 --> 00:10:41.700
Notice that, even though
this map was linear,
00:10:41.700 --> 00:10:46.230
even though it was invertible,
it obviously is not conformal.
00:10:46.230 --> 00:10:48.000
Why isn't it conformal?
00:10:48.000 --> 00:10:54.630
Well, the image of the line of
the vector OA in the xy plane
00:10:54.630 --> 00:10:58.110
is O prime A prime
in the uv plane.
00:10:58.110 --> 00:11:03.000
The image of OB in the xy
plane is given by O prime B
00:11:03.000 --> 00:11:04.830
prime in the uv plane.
00:11:04.830 --> 00:11:09.000
Notice that in the xy
plane, the vectors OA and OB
00:11:09.000 --> 00:11:11.580
meet at an angle, theta,
which clearly is not
00:11:11.580 --> 00:11:17.590
90 degrees, whereas their images
intersect at a right angle.
00:11:17.590 --> 00:11:21.210
In other words, this mapping
did not preserve angles.
00:11:21.210 --> 00:11:23.640
The image of an
angle in the xy plane
00:11:23.640 --> 00:11:29.070
did not have to be the angle
of the image in the uv plane.
00:11:29.070 --> 00:11:30.930
Now, why, for
example, would it be
00:11:30.930 --> 00:11:33.600
important to want
to preserve angles?
00:11:33.600 --> 00:11:36.150
Well, among other things,
when we change variables,
00:11:36.150 --> 00:11:38.040
we sometimes don't
want to change
00:11:38.040 --> 00:11:40.420
the physical significance
of a problem.
00:11:40.420 --> 00:11:42.210
In other words, we
may be trying to solve
00:11:42.210 --> 00:11:44.100
a problem in the xy plane.
00:11:44.100 --> 00:11:47.360
For convenience, we map the
problem into the uv plane.
00:11:47.360 --> 00:11:50.070
Well, it may happen that
certain physical properties
00:11:50.070 --> 00:11:51.570
are present in the xy plane.
00:11:51.570 --> 00:11:54.030
We may be talking about
potential and force.
00:11:54.030 --> 00:11:57.870
And maybe one family of lines
intersects another family
00:11:57.870 --> 00:11:59.460
of lines at right angles.
00:11:59.460 --> 00:12:02.100
We'd like to believe
that we could
00:12:02.100 --> 00:12:05.970
have a mapping into the uv plane
where, if the two lines met
00:12:05.970 --> 00:12:08.850
at right angles in the xy
plane, their images would
00:12:08.850 --> 00:12:13.890
meet at right angles
in the uv plane.
00:12:13.890 --> 00:12:16.300
Well, let's not worry
about that right now.
00:12:16.300 --> 00:12:20.110
Let's simply emphasize the
mathematics of a situation.
00:12:20.110 --> 00:12:23.310
A conformal mapping is one
which preserves angles.
00:12:23.310 --> 00:12:26.740
And I now claim the
very interesting thing,
00:12:26.740 --> 00:12:30.660
namely, if the mapping u as
some function of x and y,
00:12:30.660 --> 00:12:33.840
v is some function of x
and y-- given that mapping,
00:12:33.840 --> 00:12:38.310
suppose I form the complex
function u plus iv equals f,
00:12:38.310 --> 00:12:41.820
and suppose that that function
turns out to be analytic,
00:12:41.820 --> 00:12:43.830
and the derivative is not 0.
00:12:43.830 --> 00:12:46.920
We just saw that that guaranteed
that the mapping would
00:12:46.920 --> 00:12:48.090
be invertible.
00:12:48.090 --> 00:12:50.340
I now claim that, in
this special case,
00:12:50.340 --> 00:12:53.220
this mapping is also conformal.
00:12:53.220 --> 00:12:55.960
That's a very beautiful
result. In other words,
00:12:55.960 --> 00:13:00.480
the mapping u equals u
of xy, v equals v is xy,
00:13:00.480 --> 00:13:03.510
is conformable as
soon as we can be sure
00:13:03.510 --> 00:13:07.980
that the function f given
by u plus iv is analytic
00:13:07.980 --> 00:13:10.290
and that f prime is not 0.
00:13:10.290 --> 00:13:12.540
Now, why is this
mapping conformal?
00:13:12.540 --> 00:13:15.480
And again, the arithmetic
of complex numbers
00:13:15.480 --> 00:13:18.580
comes in in a very
handy way over here.
00:13:18.580 --> 00:13:23.220
Namely, let's look to see what
happens under the mapping.
00:13:23.220 --> 00:13:26.130
All we're going to do now
is change the language,
00:13:26.130 --> 00:13:28.830
not from the xy plane
and the uv plane,
00:13:28.830 --> 00:13:31.230
but we are now going
to look at the xy plane
00:13:31.230 --> 00:13:35.340
as the Argand diagram,
where the point x0, y0
00:13:35.340 --> 00:13:38.500
is represented by the
complex number z0.
00:13:38.500 --> 00:13:40.140
What I'm going to
assume now is this.
00:13:40.140 --> 00:13:42.510
Let's take the point
z0, and let's take
00:13:42.510 --> 00:13:45.600
two nearby points, z1 and z2.
00:13:45.600 --> 00:13:47.700
Now, z0 has some image under f.
00:13:47.700 --> 00:13:49.860
Let's call it w0.
00:13:49.860 --> 00:13:50.730
z1 has an image.
00:13:50.730 --> 00:13:53.160
We'll call it w1
in the uv plane.
00:13:53.160 --> 00:13:56.550
And z2 has some image,
which we'll call w2.
00:13:56.550 --> 00:13:58.780
Now, what we would like
to do is the following.
00:13:58.780 --> 00:14:05.940
Let's take the straight lines
that join z0 to z1, z0 to z2.
00:14:05.940 --> 00:14:10.020
Let's take the straight lines
that join w0 to w2 and w0
00:14:10.020 --> 00:14:11.580
to w2.
00:14:11.580 --> 00:14:12.420
This is a vector.
00:14:12.420 --> 00:14:14.760
We'll call that delta w1.
00:14:14.760 --> 00:14:15.550
This is a vector.
00:14:15.550 --> 00:14:17.717
In other words, we'll leave
you as a complex number.
00:14:17.717 --> 00:14:21.180
It's w2 minus w0, is delta W2.
00:14:21.180 --> 00:14:24.120
This we'll call delta z1, for
reference, and this vector
00:14:24.120 --> 00:14:26.470
we'll call delta
z2, for reference.
00:14:26.470 --> 00:14:30.180
Now, remember how one
divides two complex numbers.
00:14:30.180 --> 00:14:34.260
We divide the magnitudes, and
we subtract the arguments.
00:14:34.260 --> 00:14:40.110
Consequently, if I let theta
be the angle in reference here,
00:14:40.110 --> 00:14:44.460
notice that theta is the
angle obtained by dividing
00:14:44.460 --> 00:14:46.980
delta z2 by delta z1.
00:14:46.980 --> 00:14:50.130
Namely, if I divide
delta z2 by delta z1,
00:14:50.130 --> 00:14:51.760
I subtract the angles.
00:14:51.760 --> 00:14:53.970
If I take the angle
that delta z2 makes
00:14:53.970 --> 00:14:56.070
with the positive
x-axis, subtract
00:14:56.070 --> 00:14:59.470
from that the angle the delta z1
makes with the positive x-axis,
00:14:59.470 --> 00:15:02.310
what's left is the angle theta.
00:15:02.310 --> 00:15:08.010
Similarly, notice that
phi, this angle over here,
00:15:08.010 --> 00:15:13.350
is the argument of delta
w2 divided by delta w1.
00:15:13.350 --> 00:15:15.960
Namely, I simply
subtract the angle
00:15:15.960 --> 00:15:19.710
that delta w2 makes with
the positive u axis.
00:15:19.710 --> 00:15:21.930
I subtract from
that angle the angle
00:15:21.930 --> 00:15:27.030
that delta w1 makes with the
axis, and that result is phi.
00:15:27.030 --> 00:15:29.040
Because to divide
two complex numbers,
00:15:29.040 --> 00:15:32.180
we divide the magnitudes,
subtract the arguments.
00:15:32.180 --> 00:15:36.030
What I want to show is that,
if the complex value function
00:15:36.030 --> 00:15:39.900
f is analytic and
f prime is not 0,
00:15:39.900 --> 00:15:43.770
I want to show that
theta equals phi.
00:15:43.770 --> 00:15:45.220
Now, the way I'm
going to do this
00:15:45.220 --> 00:15:49.370
is I will assume that we're in
a very small neighborhood here
00:15:49.370 --> 00:15:51.520
so that what I'm
saying is, what?
00:15:51.520 --> 00:15:55.540
What does it mean to
say that f prime exists?
00:15:55.540 --> 00:15:59.290
Remember, f prime was
the quotient delta w
00:15:59.290 --> 00:16:00.497
divided by delta z.
00:16:00.497 --> 00:16:05.800
So for small values of delta
z, delta w divided by delta z
00:16:05.800 --> 00:16:10.030
must be approximately
f prime of z0.
00:16:10.030 --> 00:16:12.040
We talked about that last time.
00:16:12.040 --> 00:16:16.660
In particular then, delta
w2 divided by delta z2
00:16:16.660 --> 00:16:18.400
is one such ratio.
00:16:18.400 --> 00:16:22.490
Delta w1 divided by delta
z1 is another such ratio.
00:16:22.490 --> 00:16:26.230
So for small changes
in z, these two ratios
00:16:26.230 --> 00:16:28.390
should be approximately
equal, meaning
00:16:28.390 --> 00:16:31.240
that the error is negligible
when we go to the limit
00:16:31.240 --> 00:16:35.300
and that that approximate
ratio is f prime of z0.
00:16:35.300 --> 00:16:39.340
Now, looking at
this ratio, the fact
00:16:39.340 --> 00:16:41.230
that these are
approximately equal
00:16:41.230 --> 00:16:44.560
says that delta z2
divided by delta z1
00:16:44.560 --> 00:16:48.280
is approximately equal
to delta w2 divided
00:16:48.280 --> 00:16:53.820
by delta w1, noticing, by the
way, that if f prime were 0,
00:16:53.820 --> 00:16:57.460
delta w1 and delta
w2 would both be 0.
00:16:57.460 --> 00:17:00.900
And that would give me a 0 over
0 form, which is indeterminant.
00:17:00.900 --> 00:17:06.150
From a geometric point of
view, notice that f prime of 0
00:17:06.150 --> 00:17:08.430
may be viewed as a
vector in the uv plane.
00:17:08.430 --> 00:17:10.680
It has a magnitude,
and it has a direction.
00:17:10.680 --> 00:17:14.490
If f prime is 0, that
vector is just a point,
00:17:14.490 --> 00:17:18.033
and a point does not determine
a magnitude or a direction
00:17:18.033 --> 00:17:19.950
when you're talking about
this type of ratios.
00:17:19.950 --> 00:17:21.825
At least, it doesn't
determine the direction.
00:17:21.825 --> 00:17:23.760
It may determine a magnitude 0.
00:17:23.760 --> 00:17:25.829
But at any rate,
all we're saying
00:17:25.829 --> 00:17:28.680
is that, by the
analyticness of f,
00:17:28.680 --> 00:17:31.530
we now know this property here.
00:17:31.530 --> 00:17:34.650
But since these two numbers
are approximately equal,
00:17:34.650 --> 00:17:37.140
their arguments must
be approximately equal,
00:17:37.140 --> 00:17:39.810
and that says that theta is
approximately equal to phi,
00:17:39.810 --> 00:17:41.550
where, again, I want
to emphasize-- when
00:17:41.550 --> 00:17:43.620
I say approximately
equal, I mean
00:17:43.620 --> 00:17:48.450
they are equal up to errors of
second order infinitesimals,
00:17:48.450 --> 00:17:50.280
and those are the
terms that go to 0 so
00:17:50.280 --> 00:17:52.620
fast that, in the limit,
they don't appear.
00:17:52.620 --> 00:17:54.750
This does not mean that
these are almost equal.
00:17:54.750 --> 00:17:57.150
It means, in the
limit, they are equal,
00:17:57.150 --> 00:18:00.960
the usual way that I'm using
approximations whenever
00:18:00.960 --> 00:18:03.810
I've talked about linearity.
00:18:03.810 --> 00:18:08.850
Now, one application
of conformal mappings
00:18:08.850 --> 00:18:10.860
comes up when we
discuss the problem
00:18:10.860 --> 00:18:13.710
that we introduced in our
discussion of Green's theorem
00:18:13.710 --> 00:18:16.910
about boundary value,
steady state temperature.
00:18:16.910 --> 00:18:19.170
Remember, we were talking
about a region, R,
00:18:19.170 --> 00:18:22.290
enclosed by some curve,
C. And I had a temperature
00:18:22.290 --> 00:18:26.040
distribution on
C and inside R. I
00:18:26.040 --> 00:18:29.190
knew that the temperature
satisfied Laplace's
00:18:29.190 --> 00:18:31.860
equation in R. That was called
the steady state condition.
00:18:31.860 --> 00:18:34.470
namely, the second
partial of T with respect
00:18:34.470 --> 00:18:37.350
to x plus the second partial
of T with respect to y
00:18:37.350 --> 00:18:40.170
was zero in R. I
knew what he looked
00:18:40.170 --> 00:18:44.640
like on the boundary
of R, namely on C.
00:18:44.640 --> 00:18:48.630
And now what we said was,
back in our study of Green's
00:18:48.630 --> 00:18:53.820
theorem, that this determined
a unique function, T is xy,
00:18:53.820 --> 00:18:57.060
defined in this entire
region R. And the problem
00:18:57.060 --> 00:19:01.560
is how do you determine what T
looks like in the entire region
00:19:01.560 --> 00:19:04.590
R, just from this
information alone?
00:19:04.590 --> 00:19:07.500
Notice, again, at this
point, I would never
00:19:07.500 --> 00:19:10.530
have had to have heard
of complex variables
00:19:10.530 --> 00:19:12.430
to understand this problem.
00:19:12.430 --> 00:19:13.790
How can I prove that?
00:19:13.790 --> 00:19:16.230
Hopefully, in our discussion
of Green's theorem
00:19:16.230 --> 00:19:19.950
and the exercises, you
understood this problem.
00:19:19.950 --> 00:19:23.040
Otherwise, you couldn't
have done the exercise.
00:19:23.040 --> 00:19:24.860
Well, at that stage,
we hadn't talked
00:19:24.860 --> 00:19:26.300
about analytic functions.
00:19:26.300 --> 00:19:28.670
Consequently,
that's all the proof
00:19:28.670 --> 00:19:31.280
that we need that this problem
makes perfectly good sense
00:19:31.280 --> 00:19:32.850
without complex variables.
00:19:32.850 --> 00:19:36.470
This is a real problem defined
in terms of real variables
00:19:36.470 --> 00:19:39.115
in a real world situation.
00:19:39.115 --> 00:19:40.490
The key point--
and that's what's
00:19:40.490 --> 00:19:41.960
going to be the rest
of this lecture-- is
00:19:41.960 --> 00:19:43.100
to prove that key point.
00:19:47.750 --> 00:19:50.300
See, what happens is I don't
like the problem the way
00:19:50.300 --> 00:19:51.690
it's stated over here.
00:19:51.690 --> 00:19:54.560
So I say, OK, let me make
a change of variables,
00:19:54.560 --> 00:19:56.810
the same way as I make
a change of variables
00:19:56.810 --> 00:20:00.200
in solving definite integrals
in ordinary calculus
00:20:00.200 --> 00:20:01.640
of a single real variable.
00:20:01.640 --> 00:20:03.860
I make the change of
variables, hopefully,
00:20:03.860 --> 00:20:07.010
to arrive at an integrand that
is easier for me to handle.
00:20:07.010 --> 00:20:08.820
There's no guarantee
that the new integrand
00:20:08.820 --> 00:20:11.930
grand will be any more
palatable than the old.
00:20:11.930 --> 00:20:17.300
But the idea is I say OK let
me map R from the xy plane
00:20:17.300 --> 00:20:20.090
into the uv plane
by some mapping f
00:20:20.090 --> 00:20:24.590
bar that maps e2, into e2,
two space into two space.
00:20:24.590 --> 00:20:26.990
In terms of the language
of complex variables,
00:20:26.990 --> 00:20:30.170
all I'm saying is I
can view that mapping
00:20:30.170 --> 00:20:33.325
as a complex valued function
of a complex variable.
00:20:33.325 --> 00:20:36.290
I replace f bar by f,
as I mentioned earlier
00:20:36.290 --> 00:20:37.490
in the lecture.
00:20:37.490 --> 00:20:39.960
Well, the idea is, regardless
of how I want to do that,
00:20:39.960 --> 00:20:44.300
suppose I now map
R into some region,
00:20:44.300 --> 00:20:48.620
s, in the uv plane with a
new boundary-- say, C prime.
00:20:48.620 --> 00:20:54.710
That translates my problem from
the xy plane into the uv plane.
00:20:54.710 --> 00:20:58.160
If it happens I can solve
that problem in the uv plane,
00:20:58.160 --> 00:21:00.770
the inverse mapping
then comes back
00:21:00.770 --> 00:21:04.380
to give me the solution
in the xy plane.
00:21:04.380 --> 00:21:06.050
Now, the big problem
that comes up
00:21:06.050 --> 00:21:11.360
is that, in general,
invertible transformations
00:21:11.360 --> 00:21:15.950
do not preserve
statements made in terms
00:21:15.950 --> 00:21:17.580
of a coordinate system.
00:21:17.580 --> 00:21:21.050
In other words,
for example, if u
00:21:21.050 --> 00:21:25.160
is some function of x and y and
v is some function of x and y
00:21:25.160 --> 00:21:29.150
and that gives me an invertible
mapping that maps the region
00:21:29.150 --> 00:21:33.290
R into a region S,
there is no reason
00:21:33.290 --> 00:21:37.490
to assume that, just because
this equation is obeyed in R,
00:21:37.490 --> 00:21:42.950
that t sub uu plus T sub
vv will equal 0 in s.
00:21:42.950 --> 00:21:44.690
In other words,
this is a statement
00:21:44.690 --> 00:21:47.060
that depends on the
coordinate system.
00:21:47.060 --> 00:21:49.190
It's like saying
that, when you wanted
00:21:49.190 --> 00:21:51.290
to compute the
magnitude of a vector,
00:21:51.290 --> 00:21:54.110
you just took the square root
of the sum of the squares
00:21:54.110 --> 00:21:55.460
of the components.
00:21:55.460 --> 00:21:59.150
That was only true if you were
using Cartesian coordinates.
00:21:59.150 --> 00:22:01.190
If you use polar
coordinates, you
00:22:01.190 --> 00:22:04.520
had to use a more
elaborate computational
00:22:04.520 --> 00:22:07.050
recipe for a distance function.
00:22:07.050 --> 00:22:10.460
See, the trouble is I can
map this into the uv plane,
00:22:10.460 --> 00:22:14.730
but it may happen that
Laplace's equation is not
00:22:14.730 --> 00:22:17.360
obeyed on the new region, S.
00:22:17.360 --> 00:22:19.980
But the key amazing
point is this.
00:22:19.980 --> 00:22:25.430
If I now take the mapping
induced by u and v and call
00:22:25.430 --> 00:22:28.070
that, again, u plus iv,
call that function f--
00:22:28.070 --> 00:22:33.560
in other words, if f maps the
region R in the Argand diagram
00:22:33.560 --> 00:22:37.160
interpretation of the xy plane
into the region S in the uv
00:22:37.160 --> 00:22:43.000
plane, where f is you plus iv,
and f is analytic and f prime
00:22:43.000 --> 00:22:50.180
is not 0, then the amazing thing
is that this is equal to 0 in R
00:22:50.180 --> 00:22:51.120
if and only if--
00:22:51.120 --> 00:22:53.120
I might as well put this
in here because it does
00:22:53.120 --> 00:22:55.340
go both ways by invertibility--
00:22:55.340 --> 00:22:58.220
this is obeyed in s.
00:22:58.220 --> 00:23:02.480
In other words, a
conformal mapping
00:23:02.480 --> 00:23:05.420
preserves Laplace's equation.
00:23:05.420 --> 00:23:06.720
What does that mean?
00:23:06.720 --> 00:23:07.620
It means this.
00:23:07.620 --> 00:23:10.370
Let's suppose that f is a
conformal mapping-- namely,
00:23:10.370 --> 00:23:13.730
it's analytic, and its
derivative is never 0.
00:23:13.730 --> 00:23:17.800
I'm given that T equals
T sub 0 of xy on C
00:23:17.800 --> 00:23:21.670
and that it satisfies
Laplace's equation in R. I now
00:23:21.670 --> 00:23:23.260
make the mapping f.
00:23:23.260 --> 00:23:27.490
Since f is invertible, it
carries the closed curve C
00:23:27.490 --> 00:23:29.860
into a closed curve C prime.
00:23:29.860 --> 00:23:33.760
And the interior of c is carried
into the interior of C prime,
00:23:33.760 --> 00:23:36.820
which is S. And that
gives me a new problem,
00:23:36.820 --> 00:23:40.870
namely T is some
function of u and v
00:23:40.870 --> 00:23:44.020
in the uv plane on C
prime, and it still
00:23:44.020 --> 00:23:46.510
satisfies Laplace's equation.
00:23:46.510 --> 00:23:50.380
Suppose it happens that,
because of the geometry
00:23:50.380 --> 00:23:54.270
here, I can solve this
problem in the uv plane.
00:23:54.270 --> 00:23:57.450
If I can do that, I
simply find what t of uv
00:23:57.450 --> 00:24:01.930
looks like in S. Remembering
that u is u of xy,
00:24:01.930 --> 00:24:04.680
v is v of xy, I plug this in.
00:24:04.680 --> 00:24:07.200
This gives me t in
terms of x and y,
00:24:07.200 --> 00:24:11.250
and that would be a solution in
the region R because, you see,
00:24:11.250 --> 00:24:13.570
Laplace's equation is obeyed.
00:24:13.570 --> 00:24:15.090
You see the key point is what?
00:24:15.090 --> 00:24:18.360
That conformal mappings
preserve the solution
00:24:18.360 --> 00:24:20.340
of Laplace's
equation, and that is
00:24:20.340 --> 00:24:22.920
one of the very
important applications
00:24:22.920 --> 00:24:27.160
of conformal mappings in
the study of the real world.
00:24:27.160 --> 00:24:28.830
And again notice I
could have defined
00:24:28.830 --> 00:24:32.430
conformal without any
reference to complex numbers,
00:24:32.430 --> 00:24:34.530
just in terms of
preserving angles,
00:24:34.530 --> 00:24:38.190
and then have invented what I
mean by an analytic function
00:24:38.190 --> 00:24:41.760
by studying the geometry,
inventing the Argand diagram,
00:24:41.760 --> 00:24:42.420
et cetera.
00:24:42.420 --> 00:24:45.060
But why not take advantage
of the structure which
00:24:45.060 --> 00:24:46.140
already exists?
00:24:46.140 --> 00:24:47.700
Well, at any rate,
what I would like
00:24:47.700 --> 00:24:50.190
to do for the finale
for today's lesson
00:24:50.190 --> 00:24:52.368
is to prove this
particular result.
00:24:52.368 --> 00:24:54.660
And the reason I would like
to prove that is that, once
00:24:54.660 --> 00:24:57.570
and for all, this should
review the chain rule
00:24:57.570 --> 00:24:59.430
for real variables.
00:24:59.430 --> 00:25:02.790
It should show you how
the chain rule is used
00:25:02.790 --> 00:25:04.920
and what happens with
ordinary transformations
00:25:04.920 --> 00:25:06.850
from an algebraic point of view.
00:25:06.850 --> 00:25:09.990
And finally, because
the proof never makes
00:25:09.990 --> 00:25:15.410
use of complex numbers directly
but only properties of u and v,
00:25:15.410 --> 00:25:17.300
where u and v are the
real imaginary parts
00:25:17.300 --> 00:25:19.220
of a complex function,
I think that this
00:25:19.220 --> 00:25:22.640
should psychologically eliminate
the traumatic experience
00:25:22.640 --> 00:25:26.330
that complex valued functions
have no real application.
00:25:26.330 --> 00:25:28.230
You see, the thing I
want to do is this.
00:25:28.230 --> 00:25:31.760
I want to compute
T sub xx plus T sub
00:25:31.760 --> 00:25:35.420
yy, given the
transformation that u
00:25:35.420 --> 00:25:39.530
is some function of x and y and
v is some function of x and y.
00:25:39.530 --> 00:25:42.830
No assumptions about the
mapping being invertible yet.
00:25:42.830 --> 00:25:47.090
All I'm going to assume is
that u and v are continuously
00:25:47.090 --> 00:25:49.460
differentiable
functions of x and y
00:25:49.460 --> 00:25:50.960
so I can make
whatever manipulations
00:25:50.960 --> 00:25:52.100
I want with these.
00:25:52.100 --> 00:25:56.150
And what I would like to do is
now compute the Laplacian T sub
00:25:56.150 --> 00:26:00.800
xx plus T sub yy as it would
look in terms of u and v.
00:26:00.800 --> 00:26:03.020
And the first thing I
point out, quite simply,
00:26:03.020 --> 00:26:06.050
by the chain rule, is
to take the partial of T
00:26:06.050 --> 00:26:07.280
with respect to x.
00:26:07.280 --> 00:26:08.390
I simply do not what?
00:26:08.390 --> 00:26:10.100
Take the partial
of T with respect
00:26:10.100 --> 00:26:12.140
to u times the partial
of u with respect
00:26:12.140 --> 00:26:14.360
to x, plus the partial
of T with respect
00:26:14.360 --> 00:26:16.970
to v times the partial
of u with respect x.
00:26:16.970 --> 00:26:20.120
It's as if the us
and the vs cancel.
00:26:20.120 --> 00:26:22.760
This is the contribution
of T sub x due to u,
00:26:22.760 --> 00:26:25.640
contribution of T sub x
due to v. I add them up
00:26:25.640 --> 00:26:27.380
because you and v
are independent.
00:26:27.380 --> 00:26:30.770
I hope by now you remember
this almost automatically.
00:26:30.770 --> 00:26:34.730
Then, I want the partial
of this with respect to x.
00:26:34.730 --> 00:26:37.460
That means I want the partial
of this expression with respect
00:26:37.460 --> 00:26:38.360
to x.
00:26:38.360 --> 00:26:41.090
The partial of a sum is
the sum of the partials.
00:26:41.090 --> 00:26:43.800
That brings me
from here to here.
00:26:43.800 --> 00:26:45.560
Each of these is a product.
00:26:45.560 --> 00:26:49.790
The partial of a product, I have
to use the product rule for.
00:26:49.790 --> 00:26:50.510
That's what?
00:26:50.510 --> 00:26:53.420
The first times the partial
of the second with respect
00:26:53.420 --> 00:26:55.760
to x plus the
partial of the first,
00:26:55.760 --> 00:26:59.060
T sub u, with respect
to x times the second.
00:26:59.060 --> 00:27:03.440
Similarly, this term
here becomes this?
00:27:03.440 --> 00:27:06.020
The first times the
partial of the second plus
00:27:06.020 --> 00:27:08.750
the partial of the
first times the second.
00:27:08.750 --> 00:27:11.750
And the key point is that
I put these in parentheses
00:27:11.750 --> 00:27:14.390
here to emphasize the
fact that these are still
00:27:14.390 --> 00:27:16.160
single functions.
00:27:16.160 --> 00:27:20.270
Consequently, what I can now
do is apply the chain rule
00:27:20.270 --> 00:27:22.850
to each of these
expressions again.
00:27:22.850 --> 00:27:25.730
Don't be thrown off by the
subscripts u and v. Think
00:27:25.730 --> 00:27:27.680
of the whole thing
in parentheses
00:27:27.680 --> 00:27:29.990
as being some
function of u and v.
00:27:29.990 --> 00:27:32.540
To take the partial of what's
in parentheses with respect
00:27:32.540 --> 00:27:36.210
to x, I take the partial
first with respect to u times
00:27:36.210 --> 00:27:38.420
the partial of u with
respect to x, then
00:27:38.420 --> 00:27:40.910
the partial with respect
to v times the partial of v
00:27:40.910 --> 00:27:41.990
respect to x--
00:27:41.990 --> 00:27:43.820
in other words,
leaving the details,
00:27:43.820 --> 00:27:45.680
again, for you to review.
00:27:45.680 --> 00:27:49.805
This expression
here becomes this.
00:27:49.805 --> 00:27:51.470
This expression is what?
00:27:51.470 --> 00:27:53.640
The partial of this
with respect to u
00:27:53.640 --> 00:27:55.850
times the partial
of u respect to x,
00:27:55.850 --> 00:27:59.080
the partial of this with
respect to v times the partial
00:27:59.080 --> 00:28:00.620
of v respect to x.
00:28:00.620 --> 00:28:07.380
And now taking these
expressions and replacing these
00:28:07.380 --> 00:28:09.940
by this in our
previous expression.
00:28:09.940 --> 00:28:11.790
And, again, leaving
the details to you,
00:28:11.790 --> 00:28:15.000
I wind up with the
fairly complicated result
00:28:15.000 --> 00:28:18.330
that, in terms of the partials
with respect to u and v,
00:28:18.330 --> 00:28:24.990
t sub xx is this fairly
messy expression.
00:28:24.990 --> 00:28:28.800
By the way, I do not have to
do this whole thing over again
00:28:28.800 --> 00:28:31.170
to find the second
partial of T with respect
00:28:31.170 --> 00:28:37.563
to y because this derivation
is symmetric in x and y.
00:28:37.563 --> 00:28:38.980
And if you don't
believe this, you
00:28:38.980 --> 00:28:41.050
can do the thing
over as an exercise
00:28:41.050 --> 00:28:42.280
and see what does happen.
00:28:42.280 --> 00:28:44.320
I claim all I have
to do now to get
00:28:44.320 --> 00:28:46.510
what the second partial
of T with respect to y
00:28:46.510 --> 00:28:49.030
looks like is to go
through this result.
00:28:49.030 --> 00:28:52.060
And every place I see
an x I replace it by a y
00:28:52.060 --> 00:28:53.950
because I'm just taking
partials with respect
00:28:53.950 --> 00:28:55.840
to y rather than
with respect to x.
00:28:55.840 --> 00:28:57.530
Everything else stays the same.
00:28:57.530 --> 00:29:00.460
So I now wind up with
this expression here.
00:29:00.460 --> 00:29:02.380
And now the interesting
thing happens.
00:29:02.380 --> 00:29:04.730
I just add these
two expressions.
00:29:04.730 --> 00:29:07.180
And I get-- well, I
guess the technical word
00:29:07.180 --> 00:29:09.220
for it is a "mess."
00:29:09.220 --> 00:29:12.610
I get T sub xx plus T sub yy--
00:29:12.610 --> 00:29:15.640
actually involves
five different terms.
00:29:15.640 --> 00:29:18.910
There's a second partial
of T with respect to u.
00:29:18.910 --> 00:29:21.580
And the coefficient of that,
you see, would be what?
00:29:21.580 --> 00:29:23.370
It's u sub of x squared here.
00:29:23.370 --> 00:29:24.920
It's u sub of y squared here.
00:29:24.920 --> 00:29:26.440
So the coefficient
of that is u sub
00:29:26.440 --> 00:29:28.662
x squared plus u sub y squared.
00:29:28.662 --> 00:29:30.370
Again, without going
through the details,
00:29:30.370 --> 00:29:33.940
I get a second partial of
T with respect to v term
00:29:33.940 --> 00:29:38.590
and that, multiplied by v sub
x squared plus v sub y squared.
00:29:38.590 --> 00:29:40.960
Observing that
the mixed partials
00:29:40.960 --> 00:29:45.670
are equal by continuity, I
can combine the T sub uv terms
00:29:45.670 --> 00:29:48.340
and get that the
coefficient is twice u sub
00:29:48.340 --> 00:29:51.370
x, v sub x plus
u sub y, v sub y.
00:29:51.370 --> 00:29:55.810
There's a term involving T sub
u, whose coefficient u sub xx
00:29:55.810 --> 00:29:58.960
plus u sub yy and
a term involving
00:29:58.960 --> 00:30:00.850
T sub v, the partial
of T with with respect
00:30:00.850 --> 00:30:05.020
to v, whose coefficient
is v sub xx plus v sub yy.
00:30:05.020 --> 00:30:09.490
And so far, I've imposed
no conditions on u and v
00:30:09.490 --> 00:30:12.820
other than that u and v were
continuously differentiable
00:30:12.820 --> 00:30:14.360
functions of x and y.
00:30:14.360 --> 00:30:17.170
And I hope that what this
proves to you conclusively
00:30:17.170 --> 00:30:19.340
is that, when you
translate Laplace's
00:30:19.340 --> 00:30:25.150
equation into an arbitrary
uv coordinate system,
00:30:25.150 --> 00:30:29.440
you do not get just a T
sub uu and T sub vv term.
00:30:29.440 --> 00:30:31.070
There are five terms.
00:30:31.070 --> 00:30:33.910
And if you're lucky, some
of them happen to drop out.
00:30:33.910 --> 00:30:36.100
And by the way,
one hint here is,
00:30:36.100 --> 00:30:38.770
as you may remember from
our lecture of last time,
00:30:38.770 --> 00:30:41.440
it happened that if
u and v were the real
00:30:41.440 --> 00:30:44.090
and the imaginary parts
of an analytic function
00:30:44.090 --> 00:30:46.480
or, without using the language
of analytic functions,
00:30:46.480 --> 00:30:51.520
if u sub x equaled v sub y
and u sub y was minus v sub x,
00:30:51.520 --> 00:30:55.990
it turned out that u sub
xx plus u sub yy was 0
00:30:55.990 --> 00:30:59.080
and v sub xx plus v sub yy is 0.
00:30:59.080 --> 00:31:01.240
So in that special
case, notice that we
00:31:01.240 --> 00:31:04.960
have the good luck that both of
these two terms would vanish.
00:31:04.960 --> 00:31:08.470
But in general, it's not true
that for arbitrary functions, u
00:31:08.470 --> 00:31:12.020
and v, that they satisfy the
Cauchy-Riemann conditions.
00:31:12.020 --> 00:31:13.990
So now we're going
to invoke what we
00:31:13.990 --> 00:31:15.940
know about conformal mappings.
00:31:15.940 --> 00:31:17.230
Now we say, look.
00:31:17.230 --> 00:31:23.120
Let's form the complex valued
function, u plus iv, where
00:31:23.120 --> 00:31:25.760
u and v are as given over here.
00:31:25.760 --> 00:31:29.920
If it turns out that
the function, f, defined
00:31:29.920 --> 00:31:32.890
by u plus iv is
analytic, then what
00:31:32.890 --> 00:31:35.140
do we already know from before?
00:31:35.140 --> 00:31:40.090
We know from before that f
prime is u sub x plus iv sub x,
00:31:40.090 --> 00:31:45.580
that u sub x is v sub y, that
u sub y is minus v sub x.
00:31:45.580 --> 00:31:49.480
We know that the square of
the magnitude of f prime
00:31:49.480 --> 00:31:52.670
is u sub x squared
plus v sub x squared.
00:31:52.670 --> 00:31:57.760
By the way, since v sub x is
just the negative of u sub y,
00:31:57.760 --> 00:32:00.620
v sub x squared is
u sub of y squared.
00:32:00.620 --> 00:32:04.470
So this can be written
in this equivalent form.
00:32:04.470 --> 00:32:08.190
Similarly, since u sub
x is equal to v sub y,
00:32:08.190 --> 00:32:10.860
u sub x squared is equal
to v sub y squared,
00:32:10.860 --> 00:32:12.720
so these are three
different forms
00:32:12.720 --> 00:32:16.650
for expressing the square
of the magnitude of f prime.
00:32:16.650 --> 00:32:18.900
We also knew that,
if this was analytic,
00:32:18.900 --> 00:32:25.260
that u sub xx plus u sub yy and
v sub xx plus v sub yy is 0.
00:32:25.260 --> 00:32:27.240
By the way, notice
that what this does
00:32:27.240 --> 00:32:30.180
is that this is giving us a
hold on all of the coefficients
00:32:30.180 --> 00:32:31.600
that we had over here.
00:32:31.600 --> 00:32:34.800
In fact, it seems that the
only thing left to worry about
00:32:34.800 --> 00:32:40.470
is what is u sub x, v sub
x plus u sub y, v sub y.
00:32:40.470 --> 00:32:41.430
Well, look at.
00:32:41.430 --> 00:32:46.650
u sub x, v sub x plus u sub
y, v sub y is simply this.
00:32:46.650 --> 00:32:52.290
Notice that another name
for u sub of x is v sub y.
00:32:52.290 --> 00:32:58.630
And another name for u
sub of y is minus v sub x.
00:32:58.630 --> 00:33:01.855
Consequently, this
expression is just v sub y v
00:33:01.855 --> 00:33:04.780
sub x minus v sub s v sub y.
00:33:04.780 --> 00:33:07.770
These are numbers, so
it's communitative here.
00:33:07.770 --> 00:33:10.450
In other words, this is just
this, so when you subtract them
00:33:10.450 --> 00:33:11.920
the result is 0.
00:33:11.920 --> 00:33:15.220
That means, by the way,
that, under the assumption
00:33:15.220 --> 00:33:18.190
that u plus iv is
analytic, then we
00:33:18.190 --> 00:33:22.840
can say that u sub x v sub
x plus u sub y v sub y is 0,
00:33:22.840 --> 00:33:25.840
so the T sub uv term drops out.
00:33:25.840 --> 00:33:28.510
To make a long story
short, all of our terms
00:33:28.510 --> 00:33:33.776
drop out except for the terms
that involve t sub uu and t sub
00:33:33.776 --> 00:33:38.560
vv, where what we showed was
that their coefficients were
00:33:38.560 --> 00:33:42.070
simply the square of the
magnitude of F prime of z.
00:33:42.070 --> 00:33:48.170
So in the special case that the
mapping u plus iv is conformal,
00:33:48.170 --> 00:33:53.180
we have the remarkable result
that the Laplacian is changed
00:33:53.180 --> 00:33:58.670
only by a non-negative factor
as we go from the xy plane
00:33:58.670 --> 00:34:00.020
to the uv plane.
00:34:00.020 --> 00:34:04.273
In particular, assuming
that f prime of z is not 0--
00:34:04.273 --> 00:34:06.440
and notice that that's the
condition for the mapping
00:34:06.440 --> 00:34:07.630
to be conformal.
00:34:07.630 --> 00:34:12.020
If f prime of z is not 0,
notice that if this is not 0,
00:34:12.020 --> 00:34:15.110
this expression can
be zero if and only
00:34:15.110 --> 00:34:17.360
if this expression is 0.
00:34:17.360 --> 00:34:23.570
And that's precisely the result
that we needed when we talked
00:34:23.570 --> 00:34:27.560
about the fact that Laplace's
equation was preserved
00:34:27.560 --> 00:34:29.719
by a common formal
mapping, and that
00:34:29.719 --> 00:34:32.630
is one of the most
important reasons as to why
00:34:32.630 --> 00:34:35.750
many branches of
physics, for example, use
00:34:35.750 --> 00:34:37.489
the theory of
complex variables--
00:34:37.489 --> 00:34:41.090
is that the derivative in
terms of complex variables
00:34:41.090 --> 00:34:43.580
gives us a very nice
language and some very
00:34:43.580 --> 00:34:47.389
nice computational holds
that, technically speaking,
00:34:47.389 --> 00:34:50.023
we could have done without.
00:34:50.023 --> 00:34:51.440
We could technically
have speaking
00:34:51.440 --> 00:34:53.565
have done this all in terms
of the language of real
00:34:53.565 --> 00:34:56.570
valued functions and constructed
a very artificial language
00:34:56.570 --> 00:34:58.190
a very unnatural language.
00:34:58.190 --> 00:35:02.210
But notice that, because the
calculus of complex variables
00:35:02.210 --> 00:35:05.210
mimics that of
real variables, we
00:35:05.210 --> 00:35:08.360
wind up with a very natural
language from which we
00:35:08.360 --> 00:35:11.120
can operate, and this gives
us a tremendous vantage
00:35:11.120 --> 00:35:13.590
point over the real world.
00:35:13.590 --> 00:35:16.140
In other words, the complex
numbers are not only real,
00:35:16.140 --> 00:35:19.430
but they are an advantage point
over the so-called real number
00:35:19.430 --> 00:35:19.970
world.
00:35:19.970 --> 00:35:24.260
At any rate, hopefully the
exercises in today's assignment
00:35:24.260 --> 00:35:26.510
will help clarify this further.
00:35:26.510 --> 00:35:27.980
Next time, we will
look at, still,
00:35:27.980 --> 00:35:31.370
another aspect of the
value of complex variables
00:35:31.370 --> 00:35:35.090
in the study of real
variable theory.
00:35:35.090 --> 00:35:39.480
But at any rate, until
next time, goodbye.
00:35:39.480 --> 00:35:41.880
Funding for the
publication of this video
00:35:41.880 --> 00:35:46.740
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:35:46.740 --> 00:35:50.910
Help OCW continue to provide
free and open access to MIT
00:35:50.910 --> 00:35:56.345
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