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PROFESSOR: Hi.
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Our lesson today involves
a rather subtle difference
00:00:39.970 --> 00:00:44.400
between a curve and
a coordinate system.
00:00:44.400 --> 00:00:49.580
In other words, given a curve,
that curve has a certain shape,
00:00:49.580 --> 00:00:53.360
has a certain
position, independently
00:00:53.360 --> 00:00:55.510
of where our coordinate
axes are if we're
00:00:55.510 --> 00:00:57.750
using Cartesian coordinates,
or whether we're
00:00:57.750 --> 00:01:00.780
using other coordinate
systems, or what have you,
00:01:00.780 --> 00:01:04.129
but the equation of
the curve may very well
00:01:04.129 --> 00:01:08.630
depend on what coordinate
system we're using.
00:01:08.630 --> 00:01:11.520
The thing that we would like to
do in this particular lecture
00:01:11.520 --> 00:01:14.884
is to hit a very important
highlight that comes up-- well,
00:01:14.884 --> 00:01:17.050
you can motivate it from a
purely mathematical point
00:01:17.050 --> 00:01:19.590
of view, but physically
there's an even more
00:01:19.590 --> 00:01:21.230
natural interpretation.
00:01:21.230 --> 00:01:22.970
And basically what
the thing hinges on
00:01:22.970 --> 00:01:25.520
is this: If you're
given a curve in space--
00:01:25.520 --> 00:01:27.310
we'll start with a
curve in the plane,
00:01:27.310 --> 00:01:30.720
but it applies to
curves in space as well.
00:01:30.720 --> 00:01:34.990
Given that curve
in the plane, does
00:01:34.990 --> 00:01:38.150
that curve have certain
properties regardless
00:01:38.150 --> 00:01:41.750
of whether you know what your
coordinate system is or not?
00:01:41.750 --> 00:01:45.960
Or in still other words, can you
measure the shape of the curve,
00:01:45.960 --> 00:01:48.540
can you measure the
speed along the curve
00:01:48.540 --> 00:01:51.490
as a particle traverses
it, if you had never
00:01:51.490 --> 00:01:54.490
heard of the x- and
y-coordinate system?
00:01:54.490 --> 00:01:57.000
And what this leads
to is a new system
00:01:57.000 --> 00:02:01.070
of vectors by which we
study motion in space
00:02:01.070 --> 00:02:04.340
called tangential and
normal vectors when
00:02:04.340 --> 00:02:05.730
we're dealing in the plane.
00:02:05.730 --> 00:02:08.060
And there's a third number
called the binormal vector,
00:02:08.060 --> 00:02:10.479
which we'll talk about
later when we deal
00:02:10.479 --> 00:02:12.270
in three-dimensional space.
00:02:12.270 --> 00:02:15.860
Any rate, just for brevity, I
call this lecture "Tangential
00:02:15.860 --> 00:02:18.130
and Normal Vectors".
00:02:18.130 --> 00:02:19.990
And the idea is
something like this.
00:02:19.990 --> 00:02:24.960
We're given a curve C. Now,
given this particular curve C,
00:02:24.960 --> 00:02:27.550
it happens that we have a
Cartesian coordinate system
00:02:27.550 --> 00:02:28.450
here.
00:02:28.450 --> 00:02:30.680
And it also happens
that we prefer,
00:02:30.680 --> 00:02:32.710
at least as we've done
things in the past,
00:02:32.710 --> 00:02:35.750
we write everything in
terms of i and j components.
00:02:35.750 --> 00:02:39.060
Notice that if a person were
restricted to his universe
00:02:39.060 --> 00:02:43.700
being the curve C, i and j
have no basic meaning to him.
00:02:43.700 --> 00:02:45.740
What does have a
basic meaning to him,
00:02:45.740 --> 00:02:49.590
as he's moving along this curve,
I would imagine, would be what?
00:02:49.590 --> 00:02:52.392
What is his motion
tangential to the curve?
00:02:52.392 --> 00:02:54.850
In other words, if you want to
look at this from a calculus
00:02:54.850 --> 00:02:57.550
point of view, if this
is a smooth curve,
00:02:57.550 --> 00:03:00.680
in a sufficiently small
neighborhood of this point,
00:03:00.680 --> 00:03:04.410
you cannot distinguish between
the curve and the tangent line.
00:03:04.410 --> 00:03:06.720
And consequently,
one could interpret
00:03:06.720 --> 00:03:10.000
that at a given instance
the motion was always
00:03:10.000 --> 00:03:13.120
along the straight line
tangential to the curve.
00:03:13.120 --> 00:03:15.640
What this leads to is
the notion of inventing
00:03:15.640 --> 00:03:21.417
what we call a unit tangent
vector, which I'll call T.
00:03:21.417 --> 00:03:22.750
And what is that tangent vector?
00:03:22.750 --> 00:03:24.520
It's not a constant, mind you.
00:03:24.520 --> 00:03:27.340
It shifts with position as
you move along the curve.
00:03:27.340 --> 00:03:29.690
What is constant
is its magnitude.
00:03:29.690 --> 00:03:31.725
It has constant magnitude 1.
00:03:31.725 --> 00:03:33.350
I guess what I'm
trying to say, in sort
00:03:33.350 --> 00:03:37.790
of a surrealistic or
metamathematical way,
00:03:37.790 --> 00:03:42.460
is that T plays to a person
who's living on the curve C
00:03:42.460 --> 00:03:45.830
the same role that i
plays to a person living
00:03:45.830 --> 00:03:48.810
in our ordinary space, but
somehow or other he sees T
00:03:48.810 --> 00:03:51.700
as a constant vector as
he moves along the curve
00:03:51.700 --> 00:03:53.890
If he visualizes the curve
as being a straight line.
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He always sees it
tangential to his motion.
00:03:57.130 --> 00:04:03.440
Now, in the same way that j was
a 90-degree positive rotation
00:04:03.440 --> 00:04:06.630
of i, one would like
to mimic the i and j
00:04:06.630 --> 00:04:10.790
Cartesian coordinate system
by inventing another unit
00:04:10.790 --> 00:04:13.190
vector, which is again what?
00:04:13.190 --> 00:04:18.110
A positive 90-degree
rotation of T.
00:04:18.110 --> 00:04:20.880
And we'll call that vector N.
00:04:20.880 --> 00:04:25.500
So that now we have a new
system of coordinates, T and N,
00:04:25.500 --> 00:04:27.910
new system of
variables, or vectors,
00:04:27.910 --> 00:04:31.150
whereby we can now study
motion along a curve in a very
00:04:31.150 --> 00:04:32.210
natural way.
00:04:32.210 --> 00:04:36.520
In other words, we talk about
the unit tangent direction
00:04:36.520 --> 00:04:39.410
and the unit normal direction.
00:04:39.410 --> 00:04:41.035
And we have this
thing now established.
00:04:43.950 --> 00:04:45.680
What we would like
to do is to see
00:04:45.680 --> 00:04:49.000
what happens in our study
of kinematics, motion
00:04:49.000 --> 00:04:51.950
in the plane, motion in
space, if we work in terms
00:04:51.950 --> 00:04:54.500
of tangential and
normal components now
00:04:54.500 --> 00:04:57.020
rather than in terms
of i and j components.
00:04:57.020 --> 00:04:58.940
The first thing that
we'd probably like to do
00:04:58.940 --> 00:05:01.400
is figure out how in the
world do you compute T?
00:05:01.400 --> 00:05:05.400
Well, for example, let's take
a particular application.
00:05:05.400 --> 00:05:08.740
Let's take the example that
we were dealing with last time
00:05:08.740 --> 00:05:12.170
where we had the radius
vector R-- in other the scalar
00:05:12.170 --> 00:05:14.540
function t, where
t denoted time--
00:05:14.540 --> 00:05:16.040
and we were dealing what?
00:05:16.040 --> 00:05:19.570
Motion in space, where
the radius vector
00:05:19.570 --> 00:05:22.800
R was a function of t.
00:05:22.800 --> 00:05:25.040
And remember what
we showed last time?
00:05:25.040 --> 00:05:31.480
We showed that the dR/dt, the
velocity vector, has its what?
00:05:31.480 --> 00:05:33.760
Its direction is always
tangent to the curve.
00:05:33.760 --> 00:05:35.410
We proved that last time.
00:05:35.410 --> 00:05:38.160
Well, as long as
the dR/dt is always
00:05:38.160 --> 00:05:41.220
tangent to the curve, what
prevents it from being a unit
00:05:41.220 --> 00:05:42.460
tangent vector?
00:05:42.460 --> 00:05:45.550
Well, nothing prevents it
from being a tangent vector,
00:05:45.550 --> 00:05:47.440
because it's already
tangent to the curve.
00:05:47.440 --> 00:05:50.960
All that could go wrong is
that the magnitude of the dR/dt
00:05:50.960 --> 00:05:52.120
is not 1.
00:05:52.120 --> 00:05:52.750
Well, lookit.
00:05:52.750 --> 00:05:55.300
That's, again, a very
simple point to fix up.
00:05:55.300 --> 00:05:58.690
Namely, if the magnitude
of dR/dt is not 1,
00:05:58.690 --> 00:06:01.950
suppose we divide that
vector by its magnitude.
00:06:01.950 --> 00:06:05.450
We have already seen that given
any non-zero vector, if you
00:06:05.450 --> 00:06:07.730
divide that vector
by its magnitude,
00:06:07.730 --> 00:06:12.524
you get the unit vector in the
same direction as the vector
00:06:12.524 --> 00:06:13.440
that you started with.
00:06:13.440 --> 00:06:16.510
In other words, if I take
the vector dR/dt, which
00:06:16.510 --> 00:06:19.440
is already tangential to the
curve at the given point,
00:06:19.440 --> 00:06:22.480
and I divide that by the
magnitude of the dR/dt,
00:06:22.480 --> 00:06:27.260
then I automatically get
the unit tangent vector.
00:06:27.260 --> 00:06:28.642
Is that clear?
00:06:28.642 --> 00:06:31.500
Well, since nobody says
no, I assume it is clear.
00:06:31.500 --> 00:06:32.090
Lookit.
00:06:32.090 --> 00:06:36.440
We also showed last time
that the magnitude of dR/dt
00:06:36.440 --> 00:06:38.670
is speed along the curve.
00:06:38.670 --> 00:06:43.010
Speed along the curve
happens to be called ds/dt.
00:06:43.010 --> 00:06:46.200
So, again, another name
for the unit tangent vector
00:06:46.200 --> 00:06:51.640
is dR/dt divided by ds/dt.
00:06:51.640 --> 00:06:54.030
By the chain rule,
we can cancel dt.
00:06:54.030 --> 00:06:56.280
And by the way,
notice the chain rule
00:06:56.280 --> 00:06:59.110
applies for vector functions
like this, the same
00:06:59.110 --> 00:07:01.140
as it did in part
one of our course,
00:07:01.140 --> 00:07:03.360
by virtue of what we
showed in the last unit--
00:07:03.360 --> 00:07:07.350
namely, that every formula
for derivatives that
00:07:07.350 --> 00:07:10.260
was true for scalar
functions also
00:07:10.260 --> 00:07:12.420
happens to be true for what?
00:07:12.420 --> 00:07:15.130
Vector functions of
a scalar variable.
00:07:15.130 --> 00:07:18.200
At any rate, notice then by
the chain rule, another way
00:07:18.200 --> 00:07:22.390
of saying T is that it's the
derivative of the position
00:07:22.390 --> 00:07:25.530
vector R with respect
to the arc length s.
00:07:25.530 --> 00:07:27.260
I'd like to make
one comment on this.
00:07:27.260 --> 00:07:29.100
It's important enough
so that I will also
00:07:29.100 --> 00:07:31.860
make this comment in
the notes as well when
00:07:31.860 --> 00:07:33.310
we're doing the exercises.
00:07:33.310 --> 00:07:35.420
The point is that
in many textbooks,
00:07:35.420 --> 00:07:38.970
they will define T
by saying it's dR/ds.
00:07:38.970 --> 00:07:43.200
Now, 999 times out
of 1,000-- in fact,
00:07:43.200 --> 00:07:47.010
999 times out of 998
even-- you will never
00:07:47.010 --> 00:07:50.540
be given R as a function of
arc length in the real world.
00:07:50.540 --> 00:07:52.510
In the real world,
R is a function
00:07:52.510 --> 00:07:54.890
of some parameter, usually time.
00:07:54.890 --> 00:07:57.000
And the trouble that
happens is if you
00:07:57.000 --> 00:07:59.470
try to use this
definition, you find
00:07:59.470 --> 00:08:02.770
yourself trying to
convert things into s,
00:08:02.770 --> 00:08:05.102
and this makes sort
of a mess for you.
00:08:05.102 --> 00:08:07.310
The thing I would like to
show you-- and, by the way,
00:08:07.310 --> 00:08:10.240
this does not depend
on t standing for time.
00:08:10.240 --> 00:08:12.324
If t is any
variable-- and we show
00:08:12.324 --> 00:08:13.740
this in the notes
in the exercises
00:08:13.740 --> 00:08:17.530
again-- if t is any scalar,
if you differentiate R
00:08:17.530 --> 00:08:20.800
with respect to that scalar,
and divide that result
00:08:20.800 --> 00:08:24.100
by the magnitude of
this vector, you wind up
00:08:24.100 --> 00:08:25.740
with the unit tangent vector.
00:08:25.740 --> 00:08:28.060
In other words, in
a real-life problem,
00:08:28.060 --> 00:08:31.970
do not worry about converting
R into a function of s.
00:08:31.970 --> 00:08:35.280
Simply differentiate R
as it stands with respect
00:08:35.280 --> 00:08:38.150
to the given variable,
divide by the magnitude
00:08:38.150 --> 00:08:40.850
of the derivative,
and, presto, you
00:08:40.850 --> 00:08:42.990
have the unit tangent vector.
00:08:42.990 --> 00:08:44.970
Of course, you may
ask, if it's so simple
00:08:44.970 --> 00:08:47.860
to do what I just said, why
is it that every book defines
00:08:47.860 --> 00:08:48.940
it this way?
00:08:48.940 --> 00:08:50.410
The answer is
rather interesting,
00:08:50.410 --> 00:08:52.660
and that is, we
have just mentioned
00:08:52.660 --> 00:08:56.230
that we would like to believe
that the shape of a curve
00:08:56.230 --> 00:08:58.140
depends only on
the curve itself,
00:08:58.140 --> 00:09:00.480
not on how we parameterize it.
00:09:00.480 --> 00:09:02.680
The beauty of this
particular definition
00:09:02.680 --> 00:09:09.610
simply says the natural
parameter is arc length--
00:09:09.610 --> 00:09:11.129
namely, arc length
doesn't depend
00:09:11.129 --> 00:09:12.170
on any coordinate system.
00:09:12.170 --> 00:09:14.290
Given the curve,
start at any point
00:09:14.290 --> 00:09:16.580
you want, and you can
measure the arc length.
00:09:16.580 --> 00:09:19.720
So s is a very natural
parameter that does not depend
00:09:19.720 --> 00:09:21.360
on the coordinate system.
00:09:21.360 --> 00:09:24.050
In other words, by defining
the unit tangent vector
00:09:24.050 --> 00:09:29.890
to be dR/ds, you have a
beautiful philosophically pure
00:09:29.890 --> 00:09:33.250
mathematical definition, because
you have a definition which
00:09:33.250 --> 00:09:35.430
does not depend on
any coordinate system
00:09:35.430 --> 00:09:37.200
or any unnatural parameter.
00:09:37.200 --> 00:09:40.010
But in practice, this is the
way we compute the unit tangent
00:09:40.010 --> 00:09:40.943
vector.
00:09:40.943 --> 00:09:44.610
The question that comes up is
how do you find the vector N?
00:09:44.610 --> 00:09:47.090
And I'm going to show you the
traditional way of doing this
00:09:47.090 --> 00:09:51.210
before I jazz it up with
a more modern approach.
00:09:51.210 --> 00:09:52.910
Let's look at T over here.
00:09:52.910 --> 00:09:56.010
Let's call phi the angle
that the unit tangent vector
00:09:56.010 --> 00:09:59.520
makes with the curve here.
00:09:59.520 --> 00:10:02.210
Notice that in terms of this
diagram, since the unit tangent
00:10:02.210 --> 00:10:06.740
vector has magnitude 1,
the i component of it
00:10:06.740 --> 00:10:10.740
will be cosine phi and the j
component will be sine phi.
00:10:10.740 --> 00:10:16.280
In other words, T is equal to
cosine phi i plus sine phi j.
00:10:16.280 --> 00:10:17.630
Let's just differentiate.
00:10:17.630 --> 00:10:19.470
See, T is a function
of phi here.
00:10:19.470 --> 00:10:22.430
Let's just take the derivative
of T with respect to phi,
00:10:22.430 --> 00:10:24.340
and we get right
away-- remember, now,
00:10:24.340 --> 00:10:26.590
we're getting the mileage
out of this basic definition
00:10:26.590 --> 00:10:27.173
of derivative.
00:10:27.173 --> 00:10:28.960
That hasn't changed
since last time.
00:10:28.960 --> 00:10:31.110
We just differentiate
term by term here.
00:10:31.110 --> 00:10:35.460
We get minus sine phi
i plus cosine phi j.
00:10:35.460 --> 00:10:40.510
Right away we observe that dT
/ d phi is still a unit vector.
00:10:40.510 --> 00:10:43.300
You see, it's components are
minus sine phi and cosine phi,
00:10:43.300 --> 00:10:46.430
so its magnitude is still 1.
00:10:46.430 --> 00:10:49.150
And its slope is cosine
phi over minus sine
00:10:49.150 --> 00:10:53.240
phi, which is the negative
reciprocal of the slope here.
00:10:53.240 --> 00:10:54.820
In other words,
what this shows us,
00:10:54.820 --> 00:10:56.280
through the
traditional approach,
00:10:56.280 --> 00:10:59.010
is that whatever vector
the dT / d phi is,
00:10:59.010 --> 00:11:02.450
it's a unit vector
perpendicular to T.
00:11:02.450 --> 00:11:05.528
By the way, what that tells
us right away is that dT / d
00:11:05.528 --> 00:11:07.930
phi must be either
plus N or minus N
00:11:07.930 --> 00:11:09.260
before we go any further.
00:11:09.260 --> 00:11:10.420
Why?
00:11:10.420 --> 00:11:13.830
Because we already saw that
positive N, the vector that we
00:11:13.830 --> 00:11:17.540
called N, was a positive
90-degree rotation of T.
00:11:17.540 --> 00:11:21.090
If we only knew
that dT / d phi was
00:11:21.090 --> 00:11:23.550
a positive 90-degree
rotation rather
00:11:23.550 --> 00:11:26.450
than a negative 90-degree
rotation, we'd be home free.
00:11:26.450 --> 00:11:28.720
And, again, the beauty
of trigonometry,
00:11:28.720 --> 00:11:31.590
in the non-surveyor's
sense of the word,
00:11:31.590 --> 00:11:37.360
analytically is this-- that sort
of having a premonition of what
00:11:37.360 --> 00:11:40.270
we'd like to be true,
we simply verify
00:11:40.270 --> 00:11:43.190
the trigonometric identities
that the cosine of phi
00:11:43.190 --> 00:11:46.390
plus 90 degrees
is minus sine phi,
00:11:46.390 --> 00:11:48.420
and the sine of
phi plus 90 degrees
00:11:48.420 --> 00:11:51.700
is cosine phi, so that
dT / d phi is what?
00:11:51.700 --> 00:11:54.560
It's cosine phi
plus 90 degrees i,
00:11:54.560 --> 00:11:57.340
plus sine phi plus 90 degrees j.
00:11:57.340 --> 00:12:00.640
And if we now compare
this with this,
00:12:00.640 --> 00:12:03.990
we notice that we have
exactly the same expression,
00:12:03.990 --> 00:12:08.170
except that the angle has been
increased by a positive 90
00:12:08.170 --> 00:12:09.090
degrees.
00:12:09.090 --> 00:12:14.100
In other words, dT / d phi is
a positive 90-degree rotation
00:12:14.100 --> 00:12:17.620
of T. Consequently, dT
/ d phi is the vector
00:12:17.620 --> 00:12:20.790
that was called N. OK?
00:12:20.790 --> 00:12:22.400
That's dT / d phi.
00:12:22.400 --> 00:12:24.780
Now, let's go back
to our kinematics.
00:12:24.780 --> 00:12:26.490
We have T and N now.
00:12:26.490 --> 00:12:28.770
Let's talk about our
velocity vector v,
00:12:28.770 --> 00:12:32.010
where R is still some
function of time.
00:12:32.010 --> 00:12:34.330
By definition, v is dR/dt.
00:12:34.330 --> 00:12:35.780
That isn't going to change.
00:12:35.780 --> 00:12:37.650
V was dR/dt last time.
00:12:37.650 --> 00:12:40.100
It's going to be
dR/dt this time.
00:12:40.100 --> 00:12:43.929
It's going to be dR/dt
whenever we want to use it.
00:12:43.929 --> 00:12:45.970
The only difference is
that instead of expressing
00:12:45.970 --> 00:12:48.265
this in terms of i and j,
we now want to express it
00:12:48.265 --> 00:12:51.950
in terms of T and N.
And notice that since v
00:12:51.950 --> 00:12:57.140
has as its direction the
direction of the tangent line,
00:12:57.140 --> 00:13:01.220
and as its magnitude ds/dt--
we saw that last time-- notice
00:13:01.220 --> 00:13:04.970
that in terms of T, v is just
a scalar multiple of the unit
00:13:04.970 --> 00:13:07.920
tangent vector T. And what
scalar multiple is it?
00:13:07.920 --> 00:13:09.610
It's ds/dt.
00:13:09.610 --> 00:13:10.400
All right.
00:13:10.400 --> 00:13:11.430
All that says is what?
00:13:11.430 --> 00:13:14.550
That v is the vector
in the direction of T
00:13:14.550 --> 00:13:18.500
whose magnitude is ds/dt,
which is speed along the curve.
00:13:18.500 --> 00:13:20.350
I now want to find a.
00:13:20.350 --> 00:13:21.640
a is acceleration.
00:13:21.640 --> 00:13:24.280
It's the same acceleration that
I was talking about before.
00:13:24.280 --> 00:13:26.100
It's dv/dt.
00:13:26.100 --> 00:13:28.460
The only thing that's
going to change now
00:13:28.460 --> 00:13:31.150
is I am not going to change
the acceleration vector.
00:13:31.150 --> 00:13:33.790
I am going to change how
it looks, because now I'm
00:13:33.790 --> 00:13:38.000
going to try to find it in
terms of T and N components.
00:13:38.000 --> 00:13:39.920
So what do I do here?
00:13:39.920 --> 00:13:44.440
Look at this expression for v.
ds/dt is speed along the curve.
00:13:44.440 --> 00:13:47.420
That changes from time
to time, in general.
00:13:47.420 --> 00:13:51.590
The unit tangent vector T is
also a variable function of t,
00:13:51.590 --> 00:13:53.870
unless T happens to
be a straight line
00:13:53.870 --> 00:13:57.025
through the origin-- namely,
notice that the unit tangent
00:13:57.025 --> 00:13:59.650
vector, even though it
always has unit length,
00:13:59.650 --> 00:14:03.520
changes its direction as
we move along the curve.
00:14:03.520 --> 00:14:08.190
So in other words, both of these
factors are functions of t.
00:14:08.190 --> 00:14:10.970
Consequently, to differentiate
this with respect to t,
00:14:10.970 --> 00:14:12.820
we must use the product rule.
00:14:12.820 --> 00:14:15.490
And the fact that all of
our differentiation formulas
00:14:15.490 --> 00:14:18.900
are true for vector
and scalar combinations
00:14:18.900 --> 00:14:21.610
as well as for scalars, I
now use the regular product
00:14:21.610 --> 00:14:23.340
rule-- namely,
it's the derivative
00:14:23.340 --> 00:14:26.750
of the first factor
times the second,
00:14:26.750 --> 00:14:31.200
plus the first factor times
the derivative of the second.
00:14:31.200 --> 00:14:36.650
And I now have a expressed
in terms of two vectors, T
00:14:36.650 --> 00:14:40.370
and the derivative of the
unit vector T with respect
00:14:40.370 --> 00:14:41.420
to time t.
00:14:41.420 --> 00:14:45.300
And somehow or other,
all that's wrong here
00:14:45.300 --> 00:14:49.120
is I would like to get this
thing expressed in terms of N.
00:14:49.120 --> 00:14:51.390
You see, when I'm working
with T and N components,
00:14:51.390 --> 00:14:54.250
I want my answer to
depend on T and N.
00:14:54.250 --> 00:14:56.710
Now, here's where I
become very shrewd.
00:14:56.710 --> 00:14:58.580
And, by the way,
this is an insight
00:14:58.580 --> 00:15:01.185
that, if you're going
to pick it up at all,
00:15:01.185 --> 00:15:04.610
you're either born with it or
you pick it up with experience.
00:15:04.610 --> 00:15:08.170
But you just have to
work with these things.
00:15:08.170 --> 00:15:10.750
There are tricks, if
you want-- I guess
00:15:10.750 --> 00:15:12.950
the novice calls them "tricks."
00:15:12.950 --> 00:15:16.920
The expert calls it "keen
analytical insight."
00:15:16.920 --> 00:15:19.760
The point is I want to get
an N out of this thing.
00:15:19.760 --> 00:15:23.760
I already know how to express
N in terms of dT / d phi.
00:15:23.760 --> 00:15:25.880
In fact, N is dT / d phi.
00:15:25.880 --> 00:15:29.560
So what I do is I
take dT/dt and say,
00:15:29.560 --> 00:15:33.780
let me write it so I can get a
dT / d phi factor out of this.
00:15:33.780 --> 00:15:37.540
I also want everything to
be in terms of arc length
00:15:37.540 --> 00:15:39.830
so I can ultimately have
an answer which doesn't
00:15:39.830 --> 00:15:41.850
depend on a coordinate system.
00:15:41.850 --> 00:15:45.360
So what I really do is
I use the chain rule
00:15:45.360 --> 00:15:49.630
to express this factor in terms
of what?-- these three factors.
00:15:49.630 --> 00:15:51.250
You see, according
to the chain rule,
00:15:51.250 --> 00:15:53.710
the d phi here cancels
the d phi here,
00:15:53.710 --> 00:15:56.710
the ds here cancels the ds
here, and all I'm saying
00:15:56.710 --> 00:16:01.770
is that dT/dt can be written
as dT / d phi times d phi / ds
00:16:01.770 --> 00:16:03.340
times ds/dt.
00:16:03.340 --> 00:16:05.410
Now we're in very
good shape, you see.
00:16:05.410 --> 00:16:12.730
dT / d phi we already know
is N. And ds/dt we already
00:16:12.730 --> 00:16:14.850
know can go with this.
00:16:14.850 --> 00:16:17.600
And the only new thing
that we have to worry about
00:16:17.600 --> 00:16:19.080
is what is d phi / ds.
00:16:21.670 --> 00:16:25.350
See, again what so often
happens, you apply logic,
00:16:25.350 --> 00:16:28.710
you get to a certain
inescapable conclusion,
00:16:28.710 --> 00:16:31.030
and then if you have
brand new terms,
00:16:31.030 --> 00:16:33.049
you have a choice between
doing what?-- saying I
00:16:33.049 --> 00:16:35.340
don't like the new terms,
I'm going to throw them away,
00:16:35.340 --> 00:16:37.700
or saying I like the result,
I had better interpret
00:16:37.700 --> 00:16:39.110
what this new term means.
00:16:39.110 --> 00:16:41.650
All I want to show you
is, is that the d phi /
00:16:41.650 --> 00:16:44.560
ds has a very natural
interpretation--
00:16:44.560 --> 00:16:46.755
namely, what is d phi / ds.
00:16:46.755 --> 00:16:48.380
Let me tell you what
it's called first.
00:16:48.380 --> 00:16:50.560
It's usually denoted by
the Greek letter kappa,
00:16:50.560 --> 00:16:52.470
and it's called curvature.
00:16:52.470 --> 00:16:56.990
Its reciprocal, 1 over kappa,
is usually denoted by rho,
00:16:56.990 --> 00:16:59.140
and it's called the
radius of curvature.
00:16:59.140 --> 00:17:01.260
And I give you plenty
of drill on the stuff.
00:17:01.260 --> 00:17:03.360
I just want to mention
what these words are now.
00:17:03.360 --> 00:17:05.500
In fact, part of
the drill is that d
00:17:05.500 --> 00:17:08.630
phi / ds is not a very
convenient thing to compute.
00:17:08.630 --> 00:17:11.130
Usually you're given y
as some function of x.
00:17:11.130 --> 00:17:14.069
And many of the drill problems
that we have in calculus
00:17:14.069 --> 00:17:17.630
ask questions like, how do you
express d phi / ds in terms
00:17:17.630 --> 00:17:19.920
of y, dy, dx, et cetera?
00:17:19.920 --> 00:17:23.390
Those are problems that we can
get into in more detail as we
00:17:23.390 --> 00:17:24.380
do the exercises.
00:17:24.380 --> 00:17:26.089
But all I wanted to
do in this lecture
00:17:26.089 --> 00:17:29.940
is to show you why d phi /
ds is such a natural thing.
00:17:29.940 --> 00:17:32.420
Look at the curve s.
00:17:32.420 --> 00:17:35.280
As you move along
this curve, notice
00:17:35.280 --> 00:17:38.890
that the change in phi
with respect to s in a way
00:17:38.890 --> 00:17:42.410
tells you how the shape
of the curve is changing.
00:17:42.410 --> 00:17:46.200
In other words, d
phi / ds measures
00:17:46.200 --> 00:17:50.330
how-- what could be a more
natural word than curvature?
00:17:50.330 --> 00:17:53.490
See, as phi changes as you
move along the curve, that's
00:17:53.490 --> 00:17:55.500
measuring how your
curvature is changing.
00:17:55.500 --> 00:17:59.900
As an extreme case, notice if
the curve were a straight line,
00:17:59.900 --> 00:18:04.760
d phi / ds would be 0, because
phi would be a constant.
00:18:04.760 --> 00:18:08.560
d phi / ds would be 0, and the
curvature of a straight line
00:18:08.560 --> 00:18:10.090
should be 0.
00:18:10.090 --> 00:18:14.260
At any rate, one defines d
phi / ds to be the curvature.
00:18:14.260 --> 00:18:16.830
And, in fact, to play
it safely, since s
00:18:16.830 --> 00:18:18.880
can have two different
senses-- in other words,
00:18:18.880 --> 00:18:20.420
why couldn't somebody
else say why don't you
00:18:20.420 --> 00:18:22.920
go this way along the curve, I
don't know what the sense is?
00:18:22.920 --> 00:18:24.930
Usually what one
does to play it safe
00:18:24.930 --> 00:18:27.970
is we put the absolute value
signs around d phi / ds
00:18:27.970 --> 00:18:30.570
and just call the
magnitude the curvature.
00:18:30.570 --> 00:18:32.480
And the punch line
is that once I
00:18:32.480 --> 00:18:38.130
call d phi / ds the curvature,
what I wind up with is what?
00:18:38.130 --> 00:18:41.780
Just substituting in here now,
the acceleration vector is d^2
00:18:41.780 --> 00:18:47.410
s dt squared times T plus
kappa ds/dt squared times N.
00:18:47.410 --> 00:18:51.170
By the way, this entire
recipe is derived in the text.
00:18:51.170 --> 00:18:54.700
I have you do it again as a
learning exercise because I
00:18:54.700 --> 00:18:56.200
want you to practice with this.
00:18:56.200 --> 00:18:59.400
And I make additional
comments on this in the notes.
00:18:59.400 --> 00:19:02.920
The textbook makes
additional comments on it
00:19:02.920 --> 00:19:05.577
in the text, which is where
you'd expect it to be.
00:19:05.577 --> 00:19:07.160
And all I want you
to see is that this
00:19:07.160 --> 00:19:09.326
is the same acceleration
vector that we were talking
00:19:09.326 --> 00:19:11.470
about in the last
lecture, only now
00:19:11.470 --> 00:19:13.250
we're talking about
how it looks in terms
00:19:13.250 --> 00:19:15.390
of tangential and normal
components instead
00:19:15.390 --> 00:19:17.040
of i and j components.
00:19:17.040 --> 00:19:17.540
OK?
00:19:17.540 --> 00:19:19.670
And what's so good about
tangential and normal?
00:19:19.670 --> 00:19:21.810
What's so good about
tangential and normal
00:19:21.810 --> 00:19:23.850
is that you're now
moving along the curve
00:19:23.850 --> 00:19:26.290
rather than with
respect to some isolated
00:19:26.290 --> 00:19:28.950
x- and y-coordinate system.
00:19:28.950 --> 00:19:32.400
By the way, in the last unit
we showed a rather interesting
00:19:32.400 --> 00:19:38.420
result, that if T was any
vector function of the scalar x,
00:19:38.420 --> 00:19:40.940
and the magnitude
of T was a constant,
00:19:40.940 --> 00:19:44.670
then dT/dx was
perpendicular to T. That was
00:19:44.670 --> 00:19:46.710
an exercise in the last unit.
00:19:46.710 --> 00:19:49.330
Now, the interesting point
is that the modern approach
00:19:49.330 --> 00:19:51.780
to calculus says
this-- why should
00:19:51.780 --> 00:19:54.010
we single out the xy-plane?
00:19:54.010 --> 00:19:57.250
After all, you can be given a
particle moving through space,
00:19:57.250 --> 00:20:00.030
or you can be using a
different coordinate system.
00:20:00.030 --> 00:20:02.910
The natural parameter
is arc length.
00:20:02.910 --> 00:20:04.610
Consequently, the
modern approach
00:20:04.610 --> 00:20:07.310
never talks about the angle
phi or anything like this.
00:20:07.310 --> 00:20:11.220
The modern approach simply says
this-- define the unit tangent
00:20:11.220 --> 00:20:12.060
vector as before.
00:20:15.260 --> 00:20:17.220
Because the magnitude
of T is a constant,
00:20:17.220 --> 00:20:20.390
since dT/ds is already
perpendicular to T,
00:20:20.390 --> 00:20:26.270
let's define a second
vector N to be dT/ds divided
00:20:26.270 --> 00:20:27.570
by its magnitude.
00:20:27.570 --> 00:20:29.230
Again, the same old trick.
00:20:29.230 --> 00:20:30.290
What have we done here?
00:20:30.290 --> 00:20:33.900
We have simply
taken dT/ds, which
00:20:33.900 --> 00:20:36.480
we know is perpendicular
to T-- any scalar
00:20:36.480 --> 00:20:39.520
multiple of the dT/ds will
still be perpendicular to T--
00:20:39.520 --> 00:20:40.900
but now this is what?
00:20:40.900 --> 00:20:44.280
It's a unit vector because
we've divided this vector
00:20:44.280 --> 00:20:45.570
by its magnitude.
00:20:45.570 --> 00:20:47.620
Therefore, N is a unit vector.
00:20:47.620 --> 00:20:48.510
And where is it?
00:20:48.510 --> 00:20:52.520
It's perpendicular to T.
If we now cross-multiply,
00:20:52.520 --> 00:20:57.820
notice that dT/ds is equal to
the magnitude of dT/ds times
00:20:57.820 --> 00:21:01.270
N. See, just cross-multiply.
00:21:01.270 --> 00:21:06.370
I now claim that the magnitude
of dT/ds is just d. phi / ds.
00:21:06.370 --> 00:21:07.431
Now, why is that?
00:21:07.431 --> 00:21:09.180
I guess I should have
planned this better,
00:21:09.180 --> 00:21:12.910
but let me come back to the
previous board over here.
00:21:12.910 --> 00:21:16.560
Notice that since T
is a constant vector,
00:21:16.560 --> 00:21:19.250
since T is a constant
vector, how does it change?
00:21:19.250 --> 00:21:21.090
It can't change in
magnitude because it
00:21:21.090 --> 00:21:22.530
has constant magnitude.
00:21:22.530 --> 00:21:26.080
Therefore, its only change
must be due to direction alone.
00:21:26.080 --> 00:21:29.760
But the direction of
T is measured by phi.
00:21:29.760 --> 00:21:32.357
In other words, if dT/ds
is changing at all--
00:21:32.357 --> 00:21:33.940
in other words, if
this is a variable,
00:21:33.940 --> 00:21:36.190
it must be changing
only in direction,
00:21:36.190 --> 00:21:39.610
because the magnitude
of T is always 1.
00:21:39.610 --> 00:21:41.790
In other words, T cannot
change in magnitude.
00:21:41.790 --> 00:21:44.110
It must therefore change
only in direction.
00:21:44.110 --> 00:21:46.480
In other words, the
magnitude of dT/ds
00:21:46.480 --> 00:21:49.100
is the same as the
magnitude of d phi / ds.
00:21:49.100 --> 00:21:51.440
Recall that we just defined
the magnitude of d phi /
00:21:51.440 --> 00:21:55.220
ds to be kappa,
and therefore dT/ds
00:21:55.220 --> 00:22:00.210
is kappa N, the same way as
in the traditional approach.
00:22:00.210 --> 00:22:03.200
The beauty of this approach is
that we're no longer restricted
00:22:03.200 --> 00:22:05.010
to the xy-plane.
00:22:05.010 --> 00:22:06.570
We're not restricted
to any plane.
00:22:06.570 --> 00:22:08.580
We're not restricted to
any coordinate system.
00:22:08.580 --> 00:22:11.320
We can now, in fact,
generalize this to go out
00:22:11.320 --> 00:22:12.620
into three dimensions.
00:22:12.620 --> 00:22:14.487
And, in fact, some
of you will probably
00:22:14.487 --> 00:22:16.070
have enough difficulty
with what we've
00:22:16.070 --> 00:22:19.340
done so far that you won't want
to go into three dimensions.
00:22:19.340 --> 00:22:22.390
What I've done is I have
made up an optional unit that
00:22:22.390 --> 00:22:25.150
follows this one, a unit
which has no lecture.
00:22:25.150 --> 00:22:27.490
It simply has a
batch of exercises
00:22:27.490 --> 00:22:30.284
for those who have mastered
the material in this unit
00:22:30.284 --> 00:22:31.700
and would like to
see what happens
00:22:31.700 --> 00:22:33.210
in three-dimensional space.
00:22:33.210 --> 00:22:35.200
And, after all, when
you deal with real life
00:22:35.200 --> 00:22:37.450
orbit-type problems
and things like this,
00:22:37.450 --> 00:22:39.080
notice that you do
need the geometry
00:22:39.080 --> 00:22:40.950
of three-dimensional
space for this.
00:22:40.950 --> 00:22:44.027
If you so desire, you can
then do the optional unit.
00:22:44.027 --> 00:22:45.360
That's why it's called optional.
00:22:45.360 --> 00:22:46.526
You can skip it if you want.
00:22:46.526 --> 00:22:49.140
There's no loss of continuity
if you should skip it,
00:22:49.140 --> 00:22:53.230
but in that optional unit I
devote computational drill
00:22:53.230 --> 00:22:55.320
to what happens when
our curve happens
00:22:55.320 --> 00:22:58.010
to be a three-dimensional space
curve-- in other words, a curve
00:22:58.010 --> 00:22:59.520
that winds through space.
00:22:59.520 --> 00:23:03.390
Notice, by the way, that
in the same way as before,
00:23:03.390 --> 00:23:09.940
I can write R of t is x of t
i plus y of t j plus z of t k.
00:23:09.940 --> 00:23:13.120
where that is the
vector form of the curve
00:23:13.120 --> 00:23:16.190
in Cartesian coordinates
given in scalar form
00:23:16.190 --> 00:23:18.860
by the three equations,
x is some function of t,
00:23:18.860 --> 00:23:22.810
y is some function of t,
z is some function of t.
00:23:22.810 --> 00:23:26.295
Again, my claim is that
if I just take the dR/ds,
00:23:26.295 --> 00:23:28.720
I still have the
unit tangent vector.
00:23:28.720 --> 00:23:31.110
And to see that, just notice
what we're saying here.
00:23:31.110 --> 00:23:32.680
This is a space curve now.
00:23:32.680 --> 00:23:34.560
I've taken a small
segment of it.
00:23:34.560 --> 00:23:38.260
Here's R, here's R plus
delta R, so this difference
00:23:38.260 --> 00:23:41.895
is delta R. Look what
happens as you take delta R
00:23:41.895 --> 00:23:43.650
and divide it by delta s.
00:23:43.650 --> 00:23:46.070
First of all, the
direction of delta R
00:23:46.070 --> 00:23:49.090
does become the direction
of the tangent line
00:23:49.090 --> 00:23:51.450
as delta s approaches 0.
00:23:51.450 --> 00:23:54.340
So certainly we can believe
that the direction of dR/ds
00:23:54.340 --> 00:23:57.060
is going to be the
tangential direction.
00:23:57.060 --> 00:24:00.730
Also, if we invoke
the result of geometry
00:24:00.730 --> 00:24:03.030
that we talked about in
part one of our course,
00:24:03.030 --> 00:24:05.120
when we talked about
sine theta over theta
00:24:05.120 --> 00:24:08.700
as theta approaches 0,
the length of the arc
00:24:08.700 --> 00:24:14.000
is approximately the length of
the chord for small segments.
00:24:14.000 --> 00:24:15.695
So, therefore,
delta R over delta
00:24:15.695 --> 00:24:18.680
s in magnitude approaches 1.
00:24:18.680 --> 00:24:21.430
In other words, dR/ds is
still the unit tangent vector,
00:24:21.430 --> 00:24:23.480
the same as before.
00:24:23.480 --> 00:24:25.810
Again from a computational
point of view,
00:24:25.810 --> 00:24:29.950
to find dR/ds you do not
rewrite this in terms of s.
00:24:29.950 --> 00:24:31.240
You simply do what?
00:24:31.240 --> 00:24:34.050
You take dR/dt,
the same as before,
00:24:34.050 --> 00:24:38.680
divide by its magnitude, and
you automatically have dR/ds.
00:24:38.680 --> 00:24:43.430
Similarly, once T
is given, to find N,
00:24:43.430 --> 00:24:46.290
you simply differentiate
T with respect to s
00:24:46.290 --> 00:24:47.860
and divide it by its magnitude.
00:24:47.860 --> 00:24:50.440
And again notice, even
though I've written it again,
00:24:50.440 --> 00:24:52.920
if you look back to the first
third of our blackboard,
00:24:52.920 --> 00:24:54.450
this is the same
definition for N
00:24:54.450 --> 00:24:57.130
as before, because our
original definition did not
00:24:57.130 --> 00:25:00.313
specify that the curve had
to be in a particular plane.
00:25:00.313 --> 00:25:01.140
See?
00:25:01.140 --> 00:25:02.710
The same general definition.
00:25:02.710 --> 00:25:06.330
So I now have T and N.
Now what do T and N do?
00:25:06.330 --> 00:25:09.190
T and N determine a plane.
00:25:09.190 --> 00:25:12.740
It's a plane which we call the
osculating plane to the curve.
00:25:12.740 --> 00:25:15.120
That's the plane which
sort of touches the curve
00:25:15.120 --> 00:25:16.490
at that particular moment.
00:25:16.490 --> 00:25:18.690
Remember, this curve is
winding through space.
00:25:18.690 --> 00:25:22.300
And, again, this is done in
more detail in the notes.
00:25:22.300 --> 00:25:24.720
Not quite as elegantly
as going like this,
00:25:24.720 --> 00:25:26.840
but the idea is you
have this plane that's
00:25:26.840 --> 00:25:28.920
shifting along with the curve.
00:25:28.920 --> 00:25:30.830
The only thing that's
missing that causes
00:25:30.830 --> 00:25:34.240
new complications when you
deal in three-dimensional space
00:25:34.240 --> 00:25:37.500
is that in the same way that
T and N take the place of i
00:25:37.500 --> 00:25:39.910
and j in two-space,
you need something
00:25:39.910 --> 00:25:42.990
that takes the place of k
in three-dimensional space.
00:25:42.990 --> 00:25:45.170
What we do is-- again
look at the structure--
00:25:45.170 --> 00:25:49.940
we mimic how k is related to i
and j and invent a new vector
00:25:49.940 --> 00:25:53.470
called the binormal,
hence abbreviated B,
00:25:53.470 --> 00:25:57.230
which is simply defined to be
T cross N, the vector that you
00:25:57.230 --> 00:26:01.690
get by rotating the unit
vector T into the unit vector N
00:26:01.690 --> 00:26:08.220
through the smaller-- namely,
the positive 90-degree-- angle.
00:26:08.220 --> 00:26:09.610
Now what is B?
00:26:09.610 --> 00:26:12.340
B is perpendicular to both
T and N. In other words,
00:26:12.340 --> 00:26:16.410
B is a vector which is
perpendicular to the osculating
00:26:16.410 --> 00:26:17.140
plane.
00:26:17.140 --> 00:26:20.817
Since B always has a
constant magnitude,
00:26:20.817 --> 00:26:22.775
because T and N are always
perpendicular-- see,
00:26:22.775 --> 00:26:25.400
B always has magnitude
1-- the point
00:26:25.400 --> 00:26:29.570
is that dB/ds, the
magnitude of dB/ds,
00:26:29.570 --> 00:26:32.190
measures the twist of the curve.
00:26:32.190 --> 00:26:34.880
In other words, here's
this tangent plane
00:26:34.880 --> 00:26:37.840
following a point, a
particle, along the curve.
00:26:37.840 --> 00:26:39.560
And what you're
saying is how fast
00:26:39.560 --> 00:26:42.230
the direction of that
tangent plane is changing
00:26:42.230 --> 00:26:44.280
is measured by dB/ds.
00:26:44.280 --> 00:26:45.540
That is called the "twist."
00:26:45.540 --> 00:26:46.702
I call it the "twist."
00:26:46.702 --> 00:26:48.660
I put it in quotation
marks because nobody else
00:26:48.660 --> 00:26:49.720
calls it the "twist."
00:26:49.720 --> 00:26:51.580
The formal name
is the "torsion."
00:26:51.580 --> 00:26:52.080
See?
00:26:52.080 --> 00:26:53.205
This is called the torsion.
00:26:53.205 --> 00:26:55.060
I talk about that
more in the notes.
00:26:55.060 --> 00:26:56.790
The point being,
by the way, that
00:26:56.790 --> 00:27:00.000
notice that if dB/ds happens
to be 0-- in other words,
00:27:00.000 --> 00:27:02.300
if B happens to
be constant-- then
00:27:02.300 --> 00:27:03.930
the curve lies in the plane.
00:27:03.930 --> 00:27:05.580
We certainly recognize that.
00:27:05.580 --> 00:27:08.910
For example, if the curve
happens to be in the xy-plane,
00:27:08.910 --> 00:27:13.240
notice that if T
and N were i and j,
00:27:13.240 --> 00:27:17.180
i cross j would just be k,
B would then be a constant.
00:27:17.180 --> 00:27:20.620
The derivative of a constant
with respect to any variable
00:27:20.620 --> 00:27:24.030
is 0, and, therefore, when the
curve does lie in the plane,
00:27:24.030 --> 00:27:26.310
the torsion, the twist, is 0.
00:27:26.310 --> 00:27:28.920
In other words, the torsion
does for three-dimensional space
00:27:28.920 --> 00:27:32.430
what the curvature in a sense
does for two-dimensional space.
00:27:32.430 --> 00:27:34.700
At any rate, our
main aim is to get
00:27:34.700 --> 00:27:38.340
you familiar with some vector
calculus, and if in doing this
00:27:38.340 --> 00:27:40.230
we can also help
you learn how to use
00:27:40.230 --> 00:27:42.310
this stuff in some
physical applications,
00:27:42.310 --> 00:27:44.700
that happens to be
frosting on the cake.
00:27:44.700 --> 00:27:47.380
Next time, we are going
to talk about the fact
00:27:47.380 --> 00:27:50.800
that we still have to invent
additional coordinate systems,
00:27:50.800 --> 00:27:53.830
that i and j isn't enough,
T and N isn't enough.
00:27:53.830 --> 00:27:56.730
Next time we're going to show
why we need polar coordinates,
00:27:56.730 --> 00:27:58.660
but we'll worry
about that next time.
00:27:58.660 --> 00:28:02.740
Until next time then, goodbye.
00:28:02.740 --> 00:28:05.110
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publication of this video
00:28:05.110 --> 00:28:09.990
was provided by the Gabriella
and Paul Rosenbaum Foundation.
00:28:09.990 --> 00:28:14.160
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