Using a line integral to find the work done by a vector field example

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.390
Let's apply what we learned in
the last video into a concrete

00:00:03.390 --> 00:00:07.810
example of the work done by a
vector field on something

00:00:07.810 --> 00:00:10.560
going through some type of
path through the field.

00:00:10.560 --> 00:00:13.400
So let's say that I
have a vector field.

00:00:13.400 --> 00:00:17.800
It's defined over r2
for the x-y plane.

00:00:17.800 --> 00:00:20.160
So it's a function of x and y.

00:00:20.160 --> 00:00:23.330
It associates a vector with
every point on the plane.

00:00:23.330 --> 00:00:31.070
And let's say my vector field
is y times the unit vector i

00:00:31.070 --> 00:00:37.240
minus x times the
unit vector j.

00:00:37.240 --> 00:00:39.980
And so you can imagine if
we were to draw-- let's

00:00:39.980 --> 00:00:42.630
draw our x- and y-axes.

00:00:42.630 --> 00:00:45.240
I'll do it over here.

00:00:45.240 --> 00:00:49.840
If we were to draw our x- and
y-axes, this associates a

00:00:49.840 --> 00:00:52.530
vector, a force vector--
let's say this is actually a

00:00:52.530 --> 00:00:57.340
force vector-- with every
point in our x-y plane.

00:00:57.340 --> 00:00:59.160
So this is x and this is y.

00:00:59.160 --> 00:01:02.805
So if we're at the point, for
example, 1, 0, what will the

00:01:02.805 --> 00:01:05.480
vector look like that's
associated with that point?

00:01:05.480 --> 00:01:12.360
Well, at 1, 0, y is 0, so this
will be 0, i minus 1, j.

00:01:12.360 --> 00:01:14.690
Minus 1, j looks like this.

00:01:14.690 --> 00:01:19.446
So minus 1, j will
look like that.

00:01:19.446 --> 00:01:22.630
At x is equal to 2-- I'm just
picking points at random, ones

00:01:22.630 --> 00:01:27.665
that'll be -- y is still 0, and
now the force vector here

00:01:27.665 --> 00:01:29.200
would be minus 2, j.

00:01:29.200 --> 00:01:32.330
So it would look
something like this.

00:01:32.330 --> 00:01:34.500
Minus 2, j.

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Something like that.

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Likewise, if we were to go
here, where y is equal to 1 and

00:01:40.110 --> 00:01:47.310
x is equal to is 0, when y is
equal to 1, we have 1, i minus

00:01:47.310 --> 00:01:49.530
0, j, so then our vector
is going to look like

00:01:49.530 --> 00:01:52.610
that at this point.

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If we're to go to 2-- you
could get the picture.

00:01:54.690 --> 00:01:55.940
You can keep plotting
these points.

00:01:55.940 --> 00:01:57.400
You just want to get a sense
of what it looks like.

00:01:57.400 --> 00:02:00.690
If you go here, the vector's
going to look like that.

00:02:00.690 --> 00:02:03.910
If you go maybe at this point
right here, the vector's

00:02:03.910 --> 00:02:05.733
going to look like that.

00:02:05.733 --> 00:02:07.850
I think you get
the general idea.

00:02:07.850 --> 00:02:09.530
I could keep filling
in the space for this

00:02:09.530 --> 00:02:11.460
entire field all over.

00:02:11.460 --> 00:02:16.140
You know, just to make it
symmetric, if I was here,

00:02:16.140 --> 00:02:19.690
the vector is going
to look like that.

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You get the idea.

00:02:20.510 --> 00:02:23.500
I could just fill in all of
the points if I had to.

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Now, in that field, I have some
particle moving, and let's say

00:02:27.890 --> 00:02:33.080
its path is described by
the curve c, and the

00:02:33.080 --> 00:02:39.120
parameterization of it is x of
t is equal to cosine of t, and

00:02:39.120 --> 00:02:45.270
y of t is equal to sine of t.

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And the path will occur from
t-- let's say, 0 is less than

00:02:50.070 --> 00:02:53.480
or equal to t is less
than or equal to 2pi.

00:02:53.480 --> 00:02:55.400
You might already recognize
what this would be.

00:02:55.400 --> 00:02:56.960
This parameterization
is essentially a

00:02:56.960 --> 00:02:59.420
counterclockwise circle.

00:02:59.420 --> 00:03:01.680
So the path that this
guy is going to go is

00:03:01.680 --> 00:03:02.730
going to start here.

00:03:02.730 --> 00:03:07.120
Well, you can imagine, t in
this case, you could almost

00:03:07.120 --> 00:03:09.980
imagine is just the angle of
the circle, but you can

00:03:09.980 --> 00:03:11.130
also imagine t is time.

00:03:11.130 --> 00:03:13.550
So at time equals 0, we're
going to be over here.

00:03:13.550 --> 00:03:18.260
Then at time of pi over 2,
we're going to have traveled a

00:03:18.260 --> 00:03:20.820
quarter of the circle to there,
so we're moving in

00:03:20.820 --> 00:03:22.020
that direction.

00:03:22.020 --> 00:03:25.690
And then at time after pi
seconds, we would have

00:03:25.690 --> 00:03:27.150
gotten right there.

00:03:27.150 --> 00:03:30.730
And then all the way after 2pi
seconds, we would have gotten

00:03:30.730 --> 00:03:32.180
all the way around the circle.

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So our path, our curve, is one
counterclockwise rotation

00:03:39.440 --> 00:03:41.150
around the circle, so to speak.

00:03:41.150 --> 00:03:45.620
So what is the work done by
this field on this curve?

00:03:45.620 --> 00:03:47.670
So the work done.

00:03:47.670 --> 00:03:52.920
So the work, we learned in the
previous video, is equal to the

00:03:52.920 --> 00:03:59.510
line integral over this contour
of our field, of our vector

00:03:59.510 --> 00:04:06.720
field, dotted with the
differential of our movement,

00:04:06.720 --> 00:04:09.950
so dotted with the differential
of our movement dr.

00:04:09.950 --> 00:04:11.790
Well, I haven't even
defined r yet.

00:04:11.790 --> 00:04:14.080
I mean, I kind of have just the
parameterization here, so we

00:04:14.080 --> 00:04:16.020
need to have a vector function.

00:04:16.020 --> 00:04:19.230
We need to have some r
that defines this path.

00:04:19.230 --> 00:04:21.360
This is just a standard
parameterization, but if I

00:04:21.360 --> 00:04:24.590
wanted to write it as a vector
function of t, we would write

00:04:24.590 --> 00:04:33.760
that r of t is equal to x of t,
which is cosine of t times i

00:04:33.760 --> 00:04:40.020
plus y of t times j, which is
just sine of t times j.

00:04:40.020 --> 00:04:44.170
And likewise, this is for 0 is
less than or equal to t, which

00:04:44.170 --> 00:04:46.740
is less than or equal to 2pi.

00:04:46.740 --> 00:04:48.230
And this are equivalent.

00:04:48.230 --> 00:04:51.210
The reason why I took the pain
of doing this is so now I can

00:04:51.210 --> 00:04:54.460
take its vector function
derivative, and can figure out

00:04:54.460 --> 00:04:56.210
its differential, and then I
can actually take the dot

00:04:56.210 --> 00:04:58.160
product with this
thing over here.

00:04:58.160 --> 00:05:01.120
So let's do all of that and
actually calculate this line

00:05:01.120 --> 00:05:03.870
integral and figure out the
work done by this field.

00:05:03.870 --> 00:05:05.420
One thing might already
pop in your mind.

00:05:05.420 --> 00:05:07.900
We're going in a
counterclockwise direction, but

00:05:07.900 --> 00:05:10.730
at every point where we're
passing through, it looks like

00:05:10.730 --> 00:05:13.540
the field is going exactly
opposite the direction

00:05:13.540 --> 00:05:13.950
of our motion.

00:05:13.950 --> 00:05:15.480
For example, here
we're moving upwards.

00:05:15.480 --> 00:05:17.350
The field is pulling
us backwards.

00:05:17.350 --> 00:05:18.970
Here we're moving
to the top left.

00:05:18.970 --> 00:05:20.550
The field is moving us
to the bottom right.

00:05:20.550 --> 00:05:22.010
Here we're moving
exactly to the left.

00:05:22.010 --> 00:05:23.590
The field is pulling
us to the right.

00:05:23.590 --> 00:05:25.900
So it looks like the field is
always doing the exact opposite

00:05:25.900 --> 00:05:26.840
of what we're trying to do.

00:05:26.840 --> 00:05:29.300
It's hindering our
ability to move.

00:05:29.300 --> 00:05:30.670
So I'll give you a
little intuition.

00:05:30.670 --> 00:05:33.600
This'll probably deal
with negative work.

00:05:33.600 --> 00:05:37.080
For example, if I lift
something off the ground,

00:05:37.080 --> 00:05:39.290
I have to apply force
to fight gravity.

00:05:39.290 --> 00:05:41.820
I'm doing positive work,
but gravity's doing

00:05:41.820 --> 00:05:42.940
negative work on that.

00:05:42.940 --> 00:05:44.930
We're just going to do the math
here just to make you

00:05:44.930 --> 00:05:46.990
comfortable with this idea, but
it's interesting to think about

00:05:46.990 --> 00:05:49.000
what's exactly going
on even here.

00:05:49.000 --> 00:05:51.490
The field is-- the field I'm
doing in that pink color,

00:05:51.490 --> 00:05:52.770
so let me stick to that.

00:05:52.770 --> 00:05:55.100
The field is pushing in that
direction, so it's always

00:05:55.100 --> 00:05:56.170
going opposite the motion.

00:05:56.170 --> 00:05:58.270
But let's just do the math to
make everything in the last

00:05:58.270 --> 00:05:59.800
video a little bit
more concrete.

00:05:59.800 --> 00:06:02.950
So a good place to start
is the derivative of our

00:06:02.950 --> 00:06:05.650
position vector function
with respect to t.

00:06:05.650 --> 00:06:11.670
So we have a dr/dt, which
we could also write

00:06:11.670 --> 00:06:14.450
as r prime of t.

00:06:14.450 --> 00:06:18.100
This is equal to the derivative
of x of t with respect to t,

00:06:18.100 --> 00:06:24.490
which is minus sine of t times
i plus the derivative of y

00:06:24.490 --> 00:06:26.070
of t with respect to t.

00:06:26.070 --> 00:06:28.720
Derivative of sine of t
is just cosine of t.

00:06:28.720 --> 00:06:31.820
Cosine of t times j.

00:06:31.820 --> 00:06:33.840
And if we want the
differential, we just multiply

00:06:33.840 --> 00:06:41.610
everything times dt, so we get
to dr is equal to-- we could

00:06:41.610 --> 00:06:42.280
write it this way.

00:06:42.280 --> 00:06:44.700
We could actually even
just put the d-- well,

00:06:44.700 --> 00:06:45.300
let me just do it.

00:06:45.300 --> 00:06:49.860
So it's minus sine of t dt--
I'm just multiplying each of

00:06:49.860 --> 00:06:53.290
these terms by dt, distributive
property-- times the unit

00:06:53.290 --> 00:06:59.420
vector i plus cosine of t dt
times the unit vector j.

00:06:59.420 --> 00:07:01.270
So we have this piece now.

00:07:01.270 --> 00:07:03.880
And now we want to take the dot
product with this over here,

00:07:03.880 --> 00:07:09.150
but let me rewrite our
vector field in terms of in

00:07:09.150 --> 00:07:11.570
terms of t, so to speak.

00:07:11.570 --> 00:07:14.030
So what's our field going to
be doing at any point t?

00:07:14.030 --> 00:07:15.810
We don't have to worry
about every point.

00:07:15.810 --> 00:07:17.760
We don't have to worry, for
example, that over here the

00:07:17.760 --> 00:07:19.900
vector field is going to be
doing something like that

00:07:19.900 --> 00:07:20.960
because that's not on our path.

00:07:20.960 --> 00:07:23.850
That force never had an
impact on the particle.

00:07:23.850 --> 00:07:26.700
We only care about what
happens along our path.

00:07:26.700 --> 00:07:32.780
So we can find a function that
we can essentially substitute y

00:07:32.780 --> 00:07:36.260
and x for, their relative
functions with respect to t,

00:07:36.260 --> 00:07:39.180
and then we'll have the force
from the field at any

00:07:39.180 --> 00:07:41.500
point or any time t.

00:07:41.500 --> 00:07:42.780
So let's do that.

00:07:42.780 --> 00:07:46.410
So this guy right here, if I
were to write it as a function

00:07:46.410 --> 00:07:51.420
of t, this is going to be equal
to y of t, right? y is a

00:07:51.420 --> 00:07:56.630
function of t, so it's sine
of t, right? that's that.

00:07:56.630 --> 00:08:05.210
Sine of t times i plus-- or
actually minus x, or x of

00:08:05.210 --> 00:08:06.310
t. x is a function of t.

00:08:06.310 --> 00:08:13.070
So minus cosine of t times j.

00:08:13.070 --> 00:08:15.080
And now all of it seems
a little bit more

00:08:15.080 --> 00:08:16.570
straightforward.

00:08:16.570 --> 00:08:20.170
If we want to find this line
integral, this line integral is

00:08:20.170 --> 00:08:22.780
going to be the same thing as
the integral-- let me pick

00:08:22.780 --> 00:08:25.050
a nice, soothing color.

00:08:25.050 --> 00:08:26.850
Maybe this is a nice one.

00:08:26.850 --> 00:08:30.010
The integral from t is
equal to 0 to t is equal

00:08:30.010 --> 00:08:35.620
to 2pi of f dot dr.

00:08:35.620 --> 00:08:39.130
Now, when you take the dot
product, you just multiply the

00:08:39.130 --> 00:08:40.830
corresponding components,
and add it up.

00:08:40.830 --> 00:08:44.150
So we take the product of the
minus sign and the sine of t--

00:08:44.150 --> 00:08:48.940
or the sine of t with the minus
sine of t dt, I get-- you're

00:08:48.940 --> 00:08:56.845
going to get minus sine squared
t dt, and then you're going

00:08:56.845 --> 00:08:59.780
to add that to-- so you're
going to have that plus.

00:08:59.780 --> 00:09:01.250
Let me write that
dt a little bit.

00:09:01.250 --> 00:09:03.310
That was a wacky-looking dt.

00:09:03.310 --> 00:09:08.270
dt, and then you're going to
have that plus these two guys

00:09:08.270 --> 00:09:09.500
multiplied by each other.

00:09:09.500 --> 00:09:12.260
So that's-- well, there's a
minus to sign here so plus.

00:09:12.260 --> 00:09:13.980
Let me just change
this to a minus.

00:09:13.980 --> 00:09:21.230
Minus cosine squared dt.

00:09:21.230 --> 00:09:23.830
And if we factor out a minus
sign and a dt, what is

00:09:23.830 --> 00:09:24.730
this going to be equal to?

00:09:24.730 --> 00:09:30.330
This is going to be equal to
the integral from 0 to 2pi of,

00:09:30.330 --> 00:09:36.010
we could say, sine squared
plus-- I want to put the t --

00:09:36.010 --> 00:09:41.130
sine squared of t plus
cosine squared of t.

00:09:41.130 --> 00:09:43.840
And actually, let me take the
minus sign out to the front.

00:09:43.840 --> 00:09:45.840
So if we just factor the
minus sign, and put a minus

00:09:45.840 --> 00:09:48.140
there, make this a plus.

00:09:48.140 --> 00:09:51.370
So the minus sign out there,
and then we factor dt out.

00:09:51.370 --> 00:09:53.880
I did a couple of steps in
there, but I think you got it.

00:09:53.880 --> 00:09:55.390
Now this is just
algebra at this point.

00:09:55.390 --> 00:09:58.140
Factoring out a minus sign,
so this becomes positive.

00:09:58.140 --> 00:09:59.800
And then you have
a dt and a dt.

00:09:59.800 --> 00:10:01.510
Factor that out,
and you get this.

00:10:01.510 --> 00:10:03.070
You could multiply this
out and you'd get what we

00:10:03.070 --> 00:10:05.830
originally have, if that
confuses you at all.

00:10:05.830 --> 00:10:08.110
And the reason why I did that:
we know what sine squared of

00:10:08.110 --> 00:10:11.190
anything plus cosine squared
of that same anything is.

00:10:11.190 --> 00:10:14.400
That falls right out of the
unit circle definition of

00:10:14.400 --> 00:10:16.850
our trig function,
so this is just 1.

00:10:16.850 --> 00:10:21.050
So our whole integral has been
reduced to the minus integral

00:10:21.050 --> 00:10:26.090
from 0 to 2pi of dt.

00:10:26.090 --> 00:10:27.910
And this is-- we have
seen this before.

00:10:27.910 --> 00:10:29.970
We can probably say that this
is of 1, if you want to

00:10:29.970 --> 00:10:31.670
put something there.

00:10:31.670 --> 00:10:35.160
Then the antiderivative of 1 is
just-- so this is just going to

00:10:35.160 --> 00:10:38.380
be equal to minus-- and that
minus sign is just the same

00:10:38.380 --> 00:10:40.510
minus sign that we're
carrying forward.

00:10:40.510 --> 00:10:43.520
The antiderivative of 1 is just
t, and we're going to evaluate

00:10:43.520 --> 00:10:47.730
it from 2pi to 0, or from 0 to
2pi, so this is equal to

00:10:47.730 --> 00:10:54.720
minus-- that minus sign right
there-- 2pi minus t

00:10:54.720 --> 00:10:56.460
at 0, so minus 0.

00:10:56.460 --> 00:10:59.180
So this is just
equal to minus 2pi.

00:10:59.180 --> 00:11:00.130
And there you have it.

00:11:00.130 --> 00:11:06.960
We figured out the work that
this field did on the particle,

00:11:06.960 --> 00:11:11.150
or whatever, whatever thing
was moving around in this

00:11:11.150 --> 00:11:12.370
counterclockwise fashion.

00:11:12.370 --> 00:11:14.620
And our intuition held up.

00:11:14.620 --> 00:11:17.820
We actually got a negative
number for the work done.

00:11:17.820 --> 00:11:21.430
And that's because, at all
times, the field was actually

00:11:21.430 --> 00:11:25.210
going exactly opposite, or was
actually opposing, the movement

00:11:25.210 --> 00:11:27.570
of, if we think of it
as a particle in its

00:11:27.570 --> 00:11:29.160
counterclockwise direction.

00:11:29.160 --> 00:11:31.910
Anyway, hopefully, you
found that helpful.


Line integrals and vector fields

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:03.110
One of the most fundamental
ideas in all of physics

00:00:03.110 --> 00:00:05.385
is the idea of work.

00:00:05.385 --> 00:00:08.450
Now when you first learn work,
you just say, oh, that's

00:00:08.450 --> 00:00:10.120
just force times distance.

00:00:10.120 --> 00:00:12.200
But then later on, when you
learn a little bit about

00:00:12.200 --> 00:00:14.770
vectors, you realize that the
force isn't always going in

00:00:14.770 --> 00:00:17.610
the same direction as
your displacement.

00:00:17.610 --> 00:00:21.450
So you learn that work is
really the magnitude, let me

00:00:21.450 --> 00:00:33.070
write this down, the magnitude
of the force, in the direction,

00:00:33.070 --> 00:00:39.460
or the component of the force
in the direction

00:00:39.460 --> 00:00:41.740
of displacement.

00:00:41.740 --> 00:00:44.206
Displacement is just distance
with some direction.

00:00:49.970 --> 00:00:55.290
Times the magnitude of the
displacement, or you could say,

00:00:55.290 --> 00:00:56.695
times the distance displaced.

00:01:00.810 --> 00:01:02.330
And the classic example.

00:01:02.330 --> 00:01:06.250
Maybe you have an ice cube,
or some type of block.

00:01:06.250 --> 00:01:08.740
I just ice so that there's
not a lot of friction.

00:01:08.740 --> 00:01:12.510
Maybe it's standing on a bigger
lake or ice or something.

00:01:12.510 --> 00:01:15.030
And maybe you're pulling on
that ice cube at an angle.

00:01:15.030 --> 00:01:17.610
Let's say, you're pulling
at an angle like that.

00:01:17.610 --> 00:01:20.820
That is my force, right there.

00:01:20.820 --> 00:01:24.080
Let's say my force is
equal to-- well, that's

00:01:24.080 --> 00:01:25.160
my force vector.

00:01:25.160 --> 00:01:33.870
Let's say the magnitude of
my force vector, let's

00:01:33.870 --> 00:01:35.310
say it's 10 newtons.

00:01:35.310 --> 00:01:37.650
And let's say the direction of
my force vector, right, any

00:01:37.650 --> 00:01:41.080
vector has to have a magnitude
and a direction, and the

00:01:41.080 --> 00:01:44.920
direction, let's say it has a
30 degree angle, let's say a 60

00:01:44.920 --> 00:01:47.770
degree angle, above horizontal.

00:01:47.770 --> 00:01:49.560
So that's the direction
I'm pulling in.

00:01:49.560 --> 00:01:52.600
And let's say I displace it.

00:01:52.600 --> 00:01:55.930
This is all review, hopefully.

00:01:55.930 --> 00:01:59.225
If you're displacing it, let's
say you displace it 5 newtons.

00:01:59.225 --> 00:02:02.570
So let's say the displacement,
that's the displacement vector

00:02:02.570 --> 00:02:10.290
right there, and the magnitude
of it is equal to 5 meters.

00:02:10.290 --> 00:02:13.460
So you've learned from the
definition of work, you can't

00:02:13.460 --> 00:02:16.940
just say, oh, I'm pulling with
10 newtons of force and

00:02:16.940 --> 00:02:18.360
I'm moving it 5 meters.

00:02:18.360 --> 00:02:22.560
You can't just multiply the 10
newtons times the 5 meters.

00:02:22.560 --> 00:02:25.660
You have to find the magnitude
of the component going in the

00:02:25.660 --> 00:02:29.050
same direction as
my displacement.

00:02:29.050 --> 00:02:31.860
So what I essentially need to
do is, the length, if you

00:02:31.860 --> 00:02:34.930
imagine the length of this
vector being 10, that's the

00:02:34.930 --> 00:02:37.750
total force, but you need to
figure out the length of the

00:02:37.750 --> 00:02:40.770
vector, that's the component of
the force, going in the same

00:02:40.770 --> 00:02:43.460
direction as my displacement.

00:02:43.460 --> 00:02:45.570
And a little simple
trigonometry, you know that

00:02:45.570 --> 00:02:53.120
this is 10 times the cosine of
60 degrees, or that's equal to,

00:02:53.120 --> 00:02:58.010
cosine of 60 degrees is 1/2, so
that's just equal to 5.

00:02:58.010 --> 00:03:00.380
So this magnitude, the
magnitude of the force going

00:03:00.380 --> 00:03:02.410
in the same direction of
the displacement in this

00:03:02.410 --> 00:03:04.810
case, is 5 newtons.

00:03:07.500 --> 00:03:09.850
And then you can
figure out the work.

00:03:09.850 --> 00:03:19.560
You could say that the work is
equal to 5 newtons times, I'll

00:03:19.560 --> 00:03:20.630
just write a dot for times.

00:03:20.630 --> 00:03:22.290
I don't want you to think
it's cross product.

00:03:22.290 --> 00:03:26.680
Times 5 meters, which is 25
newton meters, or you could

00:03:26.680 --> 00:03:31.250
even say 25 Joules of
work have been done.

00:03:31.250 --> 00:03:35.280
And this is all review of
somewhat basic physics.

00:03:35.280 --> 00:03:36.720
But just think about
what happened, here.

00:03:36.720 --> 00:03:37.430
What was the work?

00:03:37.430 --> 00:03:39.190
If I write in the abstract.

00:03:39.190 --> 00:03:42.550
The work is equal
to the 5 newtons.

00:03:42.550 --> 00:03:46.700
That was the magnitude of my
force vector, so it's the

00:03:46.700 --> 00:03:52.630
magnitude of my force vector,
times the cosine of this angle.

00:03:52.630 --> 00:03:53.860
So you know, let's
call that theta.

00:03:53.860 --> 00:03:55.010
Let's say it a
little generally.

00:03:55.010 --> 00:03:58.150
So times the cosine
of the angle.

00:03:58.150 --> 00:04:01.740
This is the amount of my force
in the direction of the

00:04:01.740 --> 00:04:04.960
displacement, the cosine of the
angle between them, times the

00:04:04.960 --> 00:04:06.800
magnitude of the displacement.

00:04:06.800 --> 00:04:12.260
So times the magnitude
of the displacement.

00:04:12.260 --> 00:04:15.560
Or if I wanted to rewrite that,
I could just write that as, the

00:04:15.560 --> 00:04:18.940
magnitude of the displacement
times the magnitude of

00:04:18.940 --> 00:04:23.400
the force times the
cosine of theta.

00:04:23.400 --> 00:04:26.760
And I've done multiple videos
of this, in the linear algebra

00:04:26.760 --> 00:04:28.880
playlist, in the physics
playlist, where I talk about

00:04:28.880 --> 00:04:31.580
the dot product and the cross
product and all of that, but

00:04:31.580 --> 00:04:40.470
this is the dot product
of the vectors d and f.

00:04:40.470 --> 00:04:43.700
So in general, if you're trying
to find the work for a constant

00:04:43.700 --> 00:04:46.730
displacement, and you have a
constant force, you just take

00:04:46.730 --> 00:04:48.530
the dot product of
those two vectors.

00:04:48.530 --> 00:04:51.330
And if the dot product is a
completely foreign concept to

00:04:51.330 --> 00:04:53.770
you, might want to watch, I
think I've made multiple, 4

00:04:53.770 --> 00:04:56.380
or 5 videos on the dot
product, and its intuition,

00:04:56.380 --> 00:04:57.420
and how it compares.

00:04:57.420 --> 00:04:59.280
But just to give you a little
bit of that intuition right

00:04:59.280 --> 00:05:03.920
here, the dot product, when
I take f dot d, or d dot f,

00:05:03.920 --> 00:05:08.440
what it's giving me is, I'm
multiplying the magnitude, well

00:05:08.440 --> 00:05:10.130
I could just read this out.

00:05:10.130 --> 00:05:13.590
But the idea of the dot product
is, take how much of this

00:05:13.590 --> 00:05:16.800
vector is going in the same
direction as this vector,

00:05:16.800 --> 00:05:18.500
in this case, this much.

00:05:18.500 --> 00:05:21.110
And then multiply
the two magnitudes.

00:05:21.110 --> 00:05:22.410
And that's what we
did right here.

00:05:22.410 --> 00:05:26.230
So the work is going to be the
force vector, dot, taking the

00:05:26.230 --> 00:05:28.980
dot part of the force vector
with the displacement vector,

00:05:28.980 --> 00:05:30.840
and this, of course,
is a scalar value.

00:05:30.840 --> 00:05:33.040
And we'll work out some
examples in the future where

00:05:33.040 --> 00:05:34.360
you'll see that that's true.

00:05:34.360 --> 00:05:39.000
So this is all review of
fairly elementary physics.

00:05:39.000 --> 00:05:42.500
Now let's take a more
complex example, but it's

00:05:42.500 --> 00:05:43.670
really the same idea.

00:05:43.670 --> 00:05:45.873
Let's define a vector field.

00:05:48.660 --> 00:05:51.371
So let's say that I have a
vector field f, and we're

00:05:51.371 --> 00:05:54.050
going to think about what
this means in a second.

00:05:54.050 --> 00:05:58.890
It's a function of x and y, and
it's equal to some scalar

00:05:58.890 --> 00:06:04.490
function of x and y times the
i-unit vector, or the

00:06:04.490 --> 00:06:08.760
horizontal unit vector, plus
some other function, scalar

00:06:08.760 --> 00:06:14.250
function of x and y, times the
vertical unit vector.

00:06:14.250 --> 00:06:15.580
So what would something
like this be?

00:06:15.580 --> 00:06:17.460
This is a vector field.

00:06:17.460 --> 00:06:20.210
This is a vector field
in 2-dimensional space.

00:06:20.210 --> 00:06:21.330
We're on the x-y plane.

00:06:31.190 --> 00:06:35.840
Or you could even say, on R2.

00:06:35.840 --> 00:06:37.690
Either way, I don't want
to get too much into

00:06:37.690 --> 00:06:39.230
the mathiness of it.

00:06:39.230 --> 00:06:40.590
But what does this do?

00:06:40.590 --> 00:06:47.270
Well, if I were to draw my x-y
plane, so that is my, again,

00:06:47.270 --> 00:06:49.070
having trouble drawing
a straight line.

00:06:49.070 --> 00:06:50.610
All right, there we go.

00:06:50.610 --> 00:06:54.050
That's my y-axis, and
that's my x-axis.

00:06:54.050 --> 00:06:56.360
I'm just drawing the first
quadrant, and but you could

00:06:56.360 --> 00:06:59.450
go negative in either
direction, if you like.

00:06:59.450 --> 00:07:01.260
What does this thing do?

00:07:01.260 --> 00:07:02.350
Well, it's essentially
saying, look.

00:07:02.350 --> 00:07:06.800
You give me any x, any y, you
give any x, y in the x-y plane,

00:07:06.800 --> 00:07:09.970
and these are going to end
up with some numbers, right?

00:07:09.970 --> 00:07:12.655
When you put x, y here, you're
going to get some value, when

00:07:12.655 --> 00:07:14.310
you put x, y here, you're
going to get some value.

00:07:14.310 --> 00:07:16.980
So you're going to get some
combination of the i-

00:07:16.980 --> 00:07:18.070
and j-unit vectors.

00:07:18.070 --> 00:07:19.770
So you're going to
get some vector.

00:07:19.770 --> 00:07:23.020
So what this does, it defines a
vector that's associated with

00:07:23.020 --> 00:07:24.810
every point on x-y plane.

00:07:24.810 --> 00:07:28.780
So you could say, if I take
this point on the x-y plane,

00:07:28.780 --> 00:07:32.480
and I would pop it into this,
I'll get something times i plus

00:07:32.480 --> 00:07:34.730
something times j, and when you
add those 2, maybe I get a

00:07:34.730 --> 00:07:37.130
vector that something
like that.

00:07:37.130 --> 00:07:38.100
And you could do that
on every point.

00:07:38.100 --> 00:07:39.190
I'm just taking random samples.

00:07:39.190 --> 00:07:41.420
Maybe when I go here,
the vector looks

00:07:41.420 --> 00:07:42.280
something like that.

00:07:42.280 --> 00:07:44.910
Maybe when I go here, the
victor looks like this.

00:07:44.910 --> 00:07:47.560
Maybe when I go here, the
vector looks like that.

00:07:47.560 --> 00:07:50.350
And maybe when I go up here,
the vector goes like that.

00:07:50.350 --> 00:07:52.320
I'm just randomly
picking points.

00:07:52.320 --> 00:07:57.090
It defines a vector on all of
the x, y coordinates where

00:07:57.090 --> 00:08:00.920
these scalar functions
are properly defined.

00:08:00.920 --> 00:08:02.370
And that's why it's
called a vector field.

00:08:02.370 --> 00:08:06.580
It defines what a potential,
maybe, force would be,

00:08:06.580 --> 00:08:11.430
or some other type of
force, at any point.

00:08:11.430 --> 00:08:14.350
At any point, if you happen
to have something there.

00:08:14.350 --> 00:08:15.900
Maybe that's what
the function is.

00:08:15.900 --> 00:08:17.750
And I could keep doing
this forever, and

00:08:17.750 --> 00:08:18.790
filling in all the gaps.

00:08:18.790 --> 00:08:19.660
But I think you get the idea.

00:08:19.660 --> 00:08:24.790
It associates a vector with
every point on x-y plane.

00:08:24.790 --> 00:08:29.010
Now, this is called a vector
field, so it probably makes a

00:08:29.010 --> 00:08:30.950
lot of sense that this could
be used to describe

00:08:30.950 --> 00:08:31.870
any type of field.

00:08:31.870 --> 00:08:33.410
It could be a
gravitation field.

00:08:33.410 --> 00:08:36.840
It could be an electric field,
it could be a magnetic field.

00:08:36.840 --> 00:08:39.630
And this could be essentially
telling you how much force

00:08:39.630 --> 00:08:43.190
there would be on some
particle in that field.

00:08:43.190 --> 00:08:44.660
That's exactly what
this would describe.

00:08:44.660 --> 00:08:48.950
Now, let's say that in this
field, I have some particle

00:08:48.950 --> 00:08:51.610
traveling on x-y plane.

00:08:51.610 --> 00:08:58.620
Let's say it starts there, and
by virtue of all of these crazy

00:08:58.620 --> 00:09:03.850
forces that are acting on it,
and maybe it's on some tracks

00:09:03.850 --> 00:09:06.900
or something, so it won't
always move exactly in the

00:09:06.900 --> 00:09:09.360
direction that the field
is trying to move it at.

00:09:09.360 --> 00:09:14.030
Let's say it moves in a path
that moves something like this.

00:09:14.030 --> 00:09:17.710
And let's say that this path,
or this curve, is defined by

00:09:17.710 --> 00:09:22.010
a position vector function.

00:09:22.010 --> 00:09:25.150
So let's say that that's
defined by r of t, which is

00:09:25.150 --> 00:09:33.780
just x of t times i plus y of
t times our unit factor j.

00:09:33.780 --> 00:09:35.130
That's r of t right there.

00:09:35.130 --> 00:09:37.730
Well, in order for this to be
a finite path, this is true

00:09:37.730 --> 00:09:42.370
before t is greater than or
equal to a, and less

00:09:42.370 --> 00:09:45.640
than or equal to b.

00:09:45.640 --> 00:09:47.830
This is the path that the
particle just happens to

00:09:47.830 --> 00:09:50.370
take, due to all of
these wacky forces.

00:09:50.370 --> 00:09:54.270
So when the particle is right
here, maybe the vector field

00:09:54.270 --> 00:09:56.960
acting on it, maybe it's
putting a force like that.

00:09:56.960 --> 00:09:59.520
But since the thing is on some
type of tracks, it moves

00:09:59.520 --> 00:10:00.400
in this direction.

00:10:00.400 --> 00:10:03.830
And then when it's here, maybe
the vector field is like that,

00:10:03.830 --> 00:10:05.740
but it moves in that direction,
because it's on some

00:10:05.740 --> 00:10:06.940
type of tracks.

00:10:06.940 --> 00:10:09.500
Now, everything I've done in
this video is to build up

00:10:09.500 --> 00:10:11.180
to a fundamental question.

00:10:11.180 --> 00:10:13.910
What was the work done on
the particle by the field?

00:10:24.960 --> 00:10:28.620
To answer that question, we
could zoom in a little bit.

00:10:28.620 --> 00:10:31.100
I'm going to zoom in on
only a little small

00:10:31.100 --> 00:10:34.710
snippet of our path.

00:10:34.710 --> 00:10:38.010
And let's try to figure out
what the work is done in a very

00:10:38.010 --> 00:10:40.470
small part of our path, because
it's constantly changing.

00:10:40.470 --> 00:10:42.190
The field is
changing direction.

00:10:42.190 --> 00:10:43.630
my object is
changing direction.

00:10:43.630 --> 00:10:47.780
So let's say when I'm here,
and let's say I move a

00:10:47.780 --> 00:10:49.740
small amount of my path.

00:10:49.740 --> 00:10:55.860
So let's say I move, this
is an infinitesimally

00:10:55.860 --> 00:10:58.500
small dr. Right?

00:10:58.500 --> 00:11:00.810
I have a differential, it's a
differential vector, infinitely

00:11:00.810 --> 00:11:02.630
small displacement.

00:11:02.630 --> 00:11:06.800
and let's say over the course
of that, the vector field is

00:11:06.800 --> 00:11:08.840
acting in this local
area, let's say it looks

00:11:08.840 --> 00:11:10.480
something like that.

00:11:10.480 --> 00:11:13.490
It's providing a force that
looks something like that.

00:11:13.490 --> 00:11:16.640
So that's the vector field in
that area, or the force

00:11:16.640 --> 00:11:18.750
directed on that particle right
when it's at that point.

00:11:18.750 --> 00:11:18.870
Right?

00:11:18.870 --> 00:11:22.420
It's an infinitesimally small
amount of time in space.

00:11:22.420 --> 00:11:24.440
You could say, OK, over that
little small point, we

00:11:24.440 --> 00:11:26.600
have this constant force.

00:11:26.600 --> 00:11:29.790
What was the work done
over this small period?

00:11:29.790 --> 00:11:32.330
You could say, what's the
small interval of work?

00:11:32.330 --> 00:11:36.120
You could say d work, or
a differential of work.

00:11:36.120 --> 00:11:38.940
Well, by the same exact logic
that we did with the simple

00:11:38.940 --> 00:11:43.810
problem, it's the magnitude of
the force in the direction of

00:11:43.810 --> 00:11:48.550
our displacement times the
magnitude of our displacement.

00:11:48.550 --> 00:11:52.800
And we know what that is, just
from this example up here.

00:11:52.800 --> 00:11:54.810
That's the dot product.

00:11:54.810 --> 00:11:58.340
It's the dot product of the
force and our super-small

00:11:58.340 --> 00:11:59.480
displacement.

00:11:59.480 --> 00:12:07.860
So that's equal to the dot
product of our force and our

00:12:07.860 --> 00:12:09.870
super-small displacement.

00:12:09.870 --> 00:12:13.240
Now, just by doing this, we're
just figuring out the work

00:12:13.240 --> 00:12:16.440
over, maybe like a really
small, super-small dr. But

00:12:16.440 --> 00:12:18.820
what we want to do, is we
want to sum them all up.

00:12:18.820 --> 00:12:21.870
We want to sum up all of the
drs to figure out the total,

00:12:21.870 --> 00:12:25.090
all of the f dot drs to figure
out the total work done.

00:12:25.090 --> 00:12:27.510
And that's where the
integral comes in.

00:12:27.510 --> 00:12:32.570
We will do a line integral
over-- I mean, you could

00:12:32.570 --> 00:12:33.910
think of it two ways.

00:12:33.910 --> 00:12:37.440
You could write just d dot w
there, but we could say, we'll

00:12:37.440 --> 00:12:42.700
do a line integral along this
curve c, could call that c

00:12:42.700 --> 00:12:46.410
or along r, whatever you
want to say it, of dw.

00:12:46.410 --> 00:12:47.800
That'll give us the total work.

00:12:47.800 --> 00:12:49.500
So let's say, work
is equal to that.

00:12:49.500 --> 00:12:54.040
Or we could also write it over
the integral, over the same

00:12:54.040 --> 00:13:00.500
curve of f of f dot dr.

00:13:00.500 --> 00:13:03.580
And this might seem like a
really, you know, gee, this

00:13:03.580 --> 00:13:05.120
is really abstract, Sal.

00:13:05.120 --> 00:13:09.220
How do we actually calculate
something like this?

00:13:09.220 --> 00:13:13.130
Especially because we have
everything parameterized

00:13:13.130 --> 00:13:14.030
in terms of t.

00:13:14.030 --> 00:13:16.130
How do we get this
in terms of t?

00:13:16.130 --> 00:13:19.710
And if you just think about
it, what is f dot r?

00:13:19.710 --> 00:13:21.030
Or what is f dot dr?

00:13:21.030 --> 00:13:23.300
Well, actually, to answer
that, let's remember

00:13:23.300 --> 00:13:25.830
what dr looked like.

00:13:25.830 --> 00:13:36.200
If you remember, dr/dt is equal
to x prime of t, I'm writing it

00:13:36.200 --> 00:13:39.120
like, I could have written dx
dt if I wanted to do, times the

00:13:39.120 --> 00:13:45.180
i-unit vector, plus y prime of
t, times the j-unit vector.

00:13:45.180 --> 00:13:49.320
And if we just wanted to dr, we
could multiply both sides, if

00:13:49.320 --> 00:13:51.850
we're being a little bit more
hand-wavy with the

00:13:51.850 --> 00:13:53.470
differentials, not
too rigorous.

00:13:53.470 --> 00:13:58.480
We'll get dr is equal to x
prime of t dt times the unit

00:13:58.480 --> 00:14:05.070
vector i plus y prime of t
times the differential dt

00:14:05.070 --> 00:14:07.280
times the unit vector j.

00:14:07.280 --> 00:14:09.070
So this is our dr right here.

00:14:12.110 --> 00:14:16.280
And remember what our
vector field was.

00:14:16.280 --> 00:14:17.440
It was this thing up here.

00:14:17.440 --> 00:14:19.590
Let me copy and paste it.

00:14:19.590 --> 00:14:21.030
And we'll see that
the dot product is

00:14:21.030 --> 00:14:23.360
actually not so crazy.

00:14:23.360 --> 00:14:26.710
So copy, and let me
paste it down here.

00:14:31.130 --> 00:14:33.820
So what's this integral
going to look like?

00:14:33.820 --> 00:14:37.600
This integral right here, that
gives the total work done by

00:14:37.600 --> 00:14:40.790
the field, on the particle,
as it moves along that path.

00:14:40.790 --> 00:14:44.090
Just super fundamental to
pretty much any serious physics

00:14:44.090 --> 00:14:47.170
that you might eventually
find yourself doing.

00:14:47.170 --> 00:14:48.170
So you could say, well gee.

00:14:48.170 --> 00:14:52.420
It's going to be the integral,
let's just say from t is equal

00:14:52.420 --> 00:14:55.320
to a, to t is equal to b.

00:14:55.320 --> 00:14:58.310
Right? a is where we started
off on the path, t is equal

00:14:58.310 --> 00:14:59.790
to a to t is equal to b.

00:14:59.790 --> 00:15:01.760
You can imagine it as being
timed, as a particle

00:15:01.760 --> 00:15:03.610
moving, as time increases.

00:15:03.610 --> 00:15:07.000
And then what is f dot dr?

00:15:07.000 --> 00:15:10.640
Well, if you remember from just
what the dot product is, you

00:15:10.640 --> 00:15:15.310
can essentially just take the
product of the corresponding

00:15:15.310 --> 00:15:17.740
components of your of
vector, and add them up.

00:15:17.740 --> 00:15:20.070
So this is going to be the
integral from t equals a to t

00:15:20.070 --> 00:15:27.246
equals b, of p of p of x,
really, instead of writing x,

00:15:27.246 --> 00:15:30.740
y, it's x of t, right? x as a
function of t, y as

00:15:30.740 --> 00:15:32.350
a function of t.

00:15:32.350 --> 00:15:33.690
So that's that.

00:15:33.690 --> 00:15:37.600
Times this thing right here,
times this component, right?

00:15:37.600 --> 00:15:39.300
We're multiplying
the i-components.

00:15:39.300 --> 00:15:50.650
So times x prime of t d t, and
then that plus, we're going

00:15:50.650 --> 00:15:52.370
to do the same thing
with the q function.

00:15:52.370 --> 00:15:56.060
So this is q plus, I'll
go to another line.

00:15:56.060 --> 00:15:57.760
Hopefully you realize I could
have just kept writing, but

00:15:57.760 --> 00:15:59.020
I'm running out of space.

00:15:59.020 --> 00:16:09.960
Plus q of x of t, y of t, times
the component of our dr. Times

00:16:09.960 --> 00:16:11.900
the y-component, or
the j-component.

00:16:11.900 --> 00:16:15.530
y prime of t dt.

00:16:15.530 --> 00:16:16.620
And we're done!

00:16:16.620 --> 00:16:17.480
And we're done.

00:16:17.480 --> 00:16:19.300
This might still seem a little
bit abstract, but we're going

00:16:19.300 --> 00:16:23.020
to see in the next video,
everything is now in terms of

00:16:23.020 --> 00:16:25.480
t, so this is just a
straight-up integration,

00:16:25.480 --> 00:16:27.170
with respect to dt.

00:16:27.170 --> 00:16:30.150
If we want, we could take the
dt's outside of the equation,

00:16:30.150 --> 00:16:32.270
and it'll look a little
bit more normal for you.

00:16:32.270 --> 00:16:34.640
But this is essentially
all that we have to do.

00:16:34.640 --> 00:16:38.080
And we're going to see some
concrete examples of taking a

00:16:38.080 --> 00:16:43.230
line integral through a vector
field, or using vector

00:16:43.230 --> 00:16:45.790
functions, in the next video.


Vector valued function derivative example

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:02.830
What I want to do in this
videos is to make to

00:00:02.830 --> 00:00:06.325
parametrizations of essentially
the same curve, but we're going

00:00:06.325 --> 00:00:08.710
to go along the curve
a different rates.

00:00:08.710 --> 00:00:11.610
And hopefully we'll be able to
use that to understand, or get

00:00:11.610 --> 00:00:14.580
a better intuition, behind what
exactly it means to take a

00:00:14.580 --> 00:00:17.560
derivative of a position
vector valued function.

00:00:17.560 --> 00:00:21.080
So let's say my first
parametrization, I have

00:00:21.080 --> 00:00:25.050
x of t is equal to t.

00:00:25.050 --> 00:00:29.990
And let's say that y of t
is equal to t squared.

00:00:29.990 --> 00:00:36.570
And this is true for t is
greater than or equal to 0,

00:00:36.570 --> 00:00:41.060
and less than or equal to 2.

00:00:41.060 --> 00:00:44.020
And if I want to write this
as a position vector valued

00:00:44.020 --> 00:00:45.975
function, let me write this.

00:00:45.975 --> 00:00:50.240
x1, call that y1, and let me
write my position vector valued

00:00:50.240 --> 00:00:54.707
function; I could say r1-- I'm
numbering them because I'm

00:00:54.707 --> 00:00:56.910
going to do a different version
of this exact same curve with a

00:00:56.910 --> 00:01:00.020
slightly different
parametrization --so r1 one of

00:01:00.020 --> 00:01:05.150
t, we could say is x1 of t
times i-- the unit vector i

00:01:05.150 --> 00:01:10.300
--so we'll just say t times i
plus-- this is just x of t

00:01:10.300 --> 00:01:13.410
right here, or x1 of t; I'm
numbering them because I'll

00:01:13.410 --> 00:01:20.940
later have an x2 t --plus
t squared times j.

00:01:20.940 --> 00:01:24.940
And if I wanted to graph this,
I'm going to be very careful

00:01:24.940 --> 00:01:26.860
graphing it because I really
want to understand what the

00:01:26.860 --> 00:01:28.090
derivative means here.

00:01:31.740 --> 00:01:35.040
Try to draw it
roughly to scale.

00:01:35.040 --> 00:01:43.080
So let's say that this is
one, two, three, four.

00:01:43.080 --> 00:01:46.240
Then let me draw my x-axis.

00:01:46.240 --> 00:01:47.430
That's good enough.

00:01:47.430 --> 00:01:52.000
And my x-axis, I want it to be
roughly to scale, one and two.

00:01:52.000 --> 00:01:57.500
And so at t equals 0, both my x
and y coordinates are at 0-- or

00:01:57.500 --> 00:02:00.060
this is just going to be the 0
vector, so this is where we are

00:02:00.060 --> 00:02:04.710
a t equals 0 --at t equals 1
this is going to be one times

00:02:04.710 --> 00:02:08.800
i-- we're going to be just
like that --plus 1 times j.

00:02:08.800 --> 00:02:12.320
1 squared is j, so we're
going to be right there.

00:02:12.320 --> 00:02:16.470
And then at t is equal to 2,
we're going to be at 2i.

00:02:16.470 --> 00:02:20.410
So 2i-- you could imagine 2
times i would be this vector

00:02:20.410 --> 00:02:26.560
right there --2 times i plus
4-- 2 squared is 4 --4 times

00:02:26.560 --> 00:02:32.770
j, so plus 4 times j.

00:02:32.770 --> 00:02:35.610
If you add these two vectors
heads to tails, you're going

00:02:35.610 --> 00:02:39.370
to get a vector that's
end point is right there.

00:02:39.370 --> 00:02:41.050
The vector is going to
look something like this.

00:02:43.980 --> 00:02:46.830
So this is what, just to
make it clear what we're

00:02:46.830 --> 00:02:50.230
doing, that's r1 of 2.

00:02:50.230 --> 00:02:51.740
This is r1 of 0.

00:02:51.740 --> 00:02:53.130
This is r1 of 1.

00:02:53.130 --> 00:02:55.150
But the bottom line is the
path looks like this:

00:02:55.150 --> 00:02:57.090
it's a parabola.

00:02:57.090 --> 00:03:03.040
So the path will
look like that.

00:03:03.040 --> 00:03:05.840
Now that's in my first
parametrization of it.

00:03:05.840 --> 00:03:07.470
Actually, let me draw a
little bit more carefully.

00:03:07.470 --> 00:03:09.110
I want to get rid of this
arrows, just because I want it

00:03:09.110 --> 00:03:11.210
to be a nice clean drawing.

00:03:11.210 --> 00:03:13.185
So it's going to be a parabola.

00:03:16.250 --> 00:03:18.510
Let me get rid of that other
point, too, just because I

00:03:18.510 --> 00:03:20.550
didn't draw it exactly where
it needs to be; it needs

00:03:20.550 --> 00:03:22.170
to be right there.

00:03:22.170 --> 00:03:24.810
And my parabola, or part of
my parabola is going to

00:03:24.810 --> 00:03:27.360
look something like that.

00:03:27.360 --> 00:03:27.780
All right.

00:03:27.780 --> 00:03:28.290
Good enough.

00:03:28.290 --> 00:03:30.520
So this is the first
parametrization.

00:03:30.520 --> 00:03:32.390
Now I'm going to do this exact
same curve, but I'm going to

00:03:32.390 --> 00:03:34.160
do it slightly differently.

00:03:34.160 --> 00:03:36.660
So let's say I'll do it
in different colors.

00:03:36.660 --> 00:03:41.400
So x2 of t, let's it equals 2t.

00:03:41.400 --> 00:03:48.570
And y2 of t, let's say
it's equal to 2t squared.

00:03:48.570 --> 00:03:52.330
Or we could alternatively write
that, that's the same thing as

00:03:52.330 --> 00:03:55.290
4t squared, just phrasing
both of these guys

00:03:55.290 --> 00:03:56.340
to the second power.

00:03:59.080 --> 00:04:03.980
And then let's say instead of
going from t equals 0 to 2,

00:04:03.980 --> 00:04:10.110
we're going to go from
t goes from 0 to 1.

00:04:10.110 --> 00:04:11.460
But we're going to see,
we're going to cover

00:04:11.460 --> 00:04:13.390
the exact same path.

00:04:13.390 --> 00:04:22.220
And our second position vector
valued function, r2 of t, is

00:04:22.220 --> 00:04:29.740
going to be equal to 2t times i
plus-- I could say 2t squared

00:04:29.740 --> 00:04:33.650
--4t squared times j.

00:04:33.650 --> 00:04:37.940
And if I were to graph this guy
right here, it would look

00:04:37.940 --> 00:04:42.780
like-- let me draw my axes
again; it's going to look the

00:04:42.780 --> 00:04:46.000
same, but it's I think useful
to draw it because I'm going to

00:04:46.000 --> 00:04:49.170
draw the derivatives and
all that on it later.

00:04:49.170 --> 00:04:53.370
One, two, three, four.

00:04:53.370 --> 00:04:55.680
One, two.

00:04:55.680 --> 00:05:03.980
And then let's see what happens
when t is equal to 0-- or r of

00:05:03.980 --> 00:05:06.100
0; all these are going to be 0,
we're just going to have the

00:05:06.100 --> 00:05:11.420
zero vector; x and y are both
equal to 0 --when t is equal

00:05:11.420 --> 00:05:15.820
to 1/2 what are we
going to get here?

00:05:15.820 --> 00:05:18.530
1/2 times 2 is 1.

00:05:18.530 --> 00:05:22.300
And then we're going to
get the point 1/2 squared

00:05:22.300 --> 00:05:24.330
is 1/4 times 4 is 1.

00:05:24.330 --> 00:05:28.110
So when t is equal 1/2 we're
going to be at the point 1, 1.

00:05:28.110 --> 00:05:29.690
And when the t is equal
to 1 we're going to

00:05:29.690 --> 00:05:30.730
be at the point 2, 4.

00:05:35.570 --> 00:05:38.595
So notice the curve is
exactly, the path we go

00:05:38.595 --> 00:05:40.290
is exactly the same.

00:05:40.290 --> 00:05:42.840
But before we even do the
derivatives, these two

00:05:42.840 --> 00:05:44.310
paths are identical.

00:05:44.310 --> 00:05:45.590
I want to think
about something.

00:05:45.590 --> 00:05:51.010
Let's pretend that our
parameter, t, really is time.

00:05:51.010 --> 00:05:53.230
And that tends to be the
most common, that's

00:05:53.230 --> 00:05:55.210
why they call it t.

00:05:55.210 --> 00:05:58.260
It doesn't have to be time,
but let's say it is time.

00:05:58.260 --> 00:05:59.540
So what's happening here?

00:05:59.540 --> 00:06:03.290
In the first parametrization
when we go from 0 to 2

00:06:03.290 --> 00:06:05.680
seconds we cover this path.

00:06:05.680 --> 00:06:07.900
You can imagine after 1
second the dot moves here,

00:06:07.900 --> 00:06:08.463
then it moves there.

00:06:08.463 --> 00:06:10.310
You can imagine a dot moving
along this curve, and it

00:06:10.310 --> 00:06:12.090
takes two seconds to do so.

00:06:12.090 --> 00:06:15.580
In this situation we have a dot
moving along the same curve,

00:06:15.580 --> 00:06:19.230
but it's able to cover the same
curve in only one second; and

00:06:19.230 --> 00:06:20.530
half a second it gets here.

00:06:20.530 --> 00:06:22.390
It took this guy one
second to get here.

00:06:22.390 --> 00:06:24.810
In a one second, this guy's all
the way over here; this guy

00:06:24.810 --> 00:06:26.610
takes two seconds
to go over here.

00:06:26.610 --> 00:06:28.960
So in this second
parametrization even though the

00:06:28.960 --> 00:06:32.580
path is the same, the curves
are the same, the

00:06:32.580 --> 00:06:33.585
dot is faster.

00:06:36.580 --> 00:06:39.790
I want you to keep that in
mind when we think about

00:06:39.790 --> 00:06:43.030
the derivatives of both
of these position vector

00:06:43.030 --> 00:06:44.630
valued functions.

00:06:44.630 --> 00:06:47.860
So just remember the dot is
moving faster for every second

00:06:47.860 --> 00:06:50.616
it's getting further along the
curve than here; that's why it

00:06:50.616 --> 00:06:51.930
only took them one second.

00:06:51.930 --> 00:06:53.410
Now let's look at the
derivatives of both

00:06:53.410 --> 00:06:55.840
of these guys.

00:06:55.840 --> 00:07:03.540
So the derivative here, so if I
were to write r prime, r1 prime

00:07:03.540 --> 00:07:05.320
of t-- let me do that in a
different color, actually,

00:07:05.320 --> 00:07:11.230
already used the orange; so let
me do it in the blue

00:07:11.230 --> 00:07:14.320
--r1 prime us t.

00:07:14.320 --> 00:07:15.650
So the is the derivative now.

00:07:15.650 --> 00:07:17.190
It's going to be, remember,
it's just the derivative of

00:07:17.190 --> 00:07:19.490
each of these times
the unit vectors.

00:07:19.490 --> 00:07:22.570
So the derivative of t with
respect to t, that's just 1.

00:07:22.570 --> 00:07:23.920
So it's 1 times i.

00:07:23.920 --> 00:07:27.940
I'll just write 1i plus-- I
didn't have to write the one

00:07:27.940 --> 00:07:31.210
there --plus the derivative
of t squared with respect

00:07:31.210 --> 00:07:36.132
to t is 2t plus 2t j.

00:07:36.132 --> 00:07:38.690
And let me take the
derivative over here.

00:07:38.690 --> 00:07:42.310
r2 prime of t.

00:07:42.310 --> 00:07:45.120
The derivative of 2t with
respect to t is 2, so

00:07:45.120 --> 00:07:50.960
2i, plus the derivative
of 4t squared is 8t.

00:07:50.960 --> 00:07:54.520
2 times 4, it is rt.

00:07:59.460 --> 00:08:00.410
Just like that.

00:08:03.180 --> 00:08:10.400
Now the question is, what do
their respective derivative

00:08:10.400 --> 00:08:12.050
vectors look like at
different points?

00:08:12.050 --> 00:08:14.680
So let's look at, I don't know,
let's see how fast they're

00:08:14.680 --> 00:08:17.350
moving when time is equal to 1.

00:08:17.350 --> 00:08:20.020
So let's take it at
a specific point.

00:08:20.020 --> 00:08:22.800
This is just the general
formula, but let's figure

00:08:22.800 --> 00:08:25.150
out what the derivative
is at a specific point.

00:08:25.150 --> 00:08:31.240
So let's take r1 when
time is equal to 1.

00:08:31.240 --> 00:08:34.110
And I want to take this
specific point on the curve,

00:08:34.110 --> 00:08:36.320
not the specific point in time.

00:08:36.320 --> 00:08:38.750
So this point on the curve
here is when time is equal

00:08:38.750 --> 00:08:41.090
to 1, you could say second.

00:08:41.090 --> 00:08:43.150
This point over here, which
is the exact corresponding

00:08:43.150 --> 00:08:46.070
point, is when time
is equal 1/2 second.

00:08:46.070 --> 00:08:50.660
So r1 of 1 is equal to--
we're taking the derivative

00:08:50.660 --> 00:08:52.950
there --is equal to 1i.

00:08:52.950 --> 00:08:54.270
It's not dependent on t at all.

00:08:54.270 --> 00:09:01.980
So it's 1i plus 2
times 1j, so plus 2j.

00:09:01.980 --> 00:09:08.720
So at this point the derivative
of our position vector function

00:09:08.720 --> 00:09:11.440
is going to be 1i plus 2j.

00:09:11.440 --> 00:09:15.150
So we can draw it like this. so
if we do 1i is like this: 1i.

00:09:15.150 --> 00:09:16.670
And then 2j.

00:09:16.670 --> 00:09:19.390
Just 2j is like that.

00:09:19.390 --> 00:09:22.130
So our derivative right there,
I'll do it in the same

00:09:22.130 --> 00:09:23.200
color that I wrote it in.

00:09:23.200 --> 00:09:25.040
It's in this green color; it's
going to look like this.

00:09:28.260 --> 00:09:31.240
And notice it looks like, at
least its direction is-- let me

00:09:31.240 --> 00:09:39.820
do it a little bit straighter
--its direction looks tangent

00:09:39.820 --> 00:09:41.900
to the curve; it's going in
the direction that my

00:09:41.900 --> 00:09:42.700
particle is moving.

00:09:42.700 --> 00:09:44.270
Remember my particle is moving
from here to there, so it's

00:09:44.270 --> 00:09:45.400
going in the direction.

00:09:45.400 --> 00:09:47.730
And I'm going to think about,
in a second, what this length

00:09:47.730 --> 00:09:49.610
of this to derivative
vector is.

00:09:49.610 --> 00:09:54.210
This right here, just to
be clear is, r1 prime.

00:09:54.210 --> 00:09:59.610
It's a vector, so it's telling
us the instantaneous change in

00:09:59.610 --> 00:10:03.540
our position vector with
respect to t, or time, when

00:10:03.540 --> 00:10:04.970
time is equal to 1 second.

00:10:04.970 --> 00:10:06.400
That's this thing right here.

00:10:06.400 --> 00:10:09.390
Now let's take the exact same
position here on our curve.

00:10:09.390 --> 00:10:11.450
But that's going to occur at a
different time for this guy.

00:10:11.450 --> 00:10:13.960
We already said it only takes
him, he's here at time

00:10:13.960 --> 00:10:15.320
is equal to 1/2 second.

00:10:15.320 --> 00:10:19.850
So let's take-- --I'll
do it in the same color

00:10:19.850 --> 00:10:23.340
--so here we have r2.

00:10:23.340 --> 00:10:26.010
We're going to evaluate it at
1/2 half because this is at

00:10:26.010 --> 00:10:27.570
time is equal 1/2 second.

00:10:27.570 --> 00:10:30.590
And this is going to be equal
to 2i-- this isn't dependent

00:10:30.590 --> 00:10:36.510
at all on time --so 2i
plus 8 times the time.

00:10:36.510 --> 00:10:38.430
So time right here is 1/2.

00:10:38.430 --> 00:10:40.650
So 8 times 1/2 is 4.

00:10:40.650 --> 00:10:42.920
So plus 4j.

00:10:42.920 --> 00:10:44.610
So what does this look like?

00:10:44.610 --> 00:10:46.220
The instantaneous
derivative here.

00:10:46.220 --> 00:10:48.640
Oh, and this is the derivative;
have to be very clear.

00:10:48.640 --> 00:10:52.130
So 2i-- let me draw some
more --so 2i maybe gets

00:10:52.130 --> 00:10:54.510
us about that far.

00:10:54.510 --> 00:10:59.350
Plus 4j will get us up
to right around there.

00:10:59.350 --> 00:11:01.800
Plus 4j is that factor.

00:11:01.800 --> 00:11:05.950
So when you add those two heads
to tails, you get this thing:

00:11:05.950 --> 00:11:10.800
you get something that-- let me
like --you get something

00:11:10.800 --> 00:11:12.065
that looks like that.

00:11:15.590 --> 00:11:18.070
I didn't draw it as neatly
as I would like to.

00:11:18.070 --> 00:11:22.370
But let's notice something:
both of these vectors are going

00:11:22.370 --> 00:11:23.820
in the exact same direction.

00:11:23.820 --> 00:11:28.550
They're both tangential to
the path, to our curve.

00:11:28.550 --> 00:11:33.340
But this vector is going, its
length, its magnitude, is

00:11:33.340 --> 00:11:36.050
much larger than this
vector's magnitude.

00:11:36.050 --> 00:11:38.610
And that makes sense because I
hinted at it when we first

00:11:38.610 --> 00:11:40.960
talked about these vector
valued position functions and

00:11:40.960 --> 00:11:46.650
their derivatives; is that the
length, you can kind of

00:11:46.650 --> 00:11:48.480
view it as the speed.

00:11:48.480 --> 00:11:52.420
The length is equal to the
speed if you imagine t being

00:11:52.420 --> 00:11:55.900
time and these parametrizations
are representing a dot

00:11:55.900 --> 00:11:57.320
moving along these curves.

00:11:57.320 --> 00:12:00.080
So in this case, the particle
only takes a second to go

00:12:00.080 --> 00:12:03.390
there, so at this point in its
path, it's moving much faster

00:12:03.390 --> 00:12:04.810
than this particle is.

00:12:04.810 --> 00:12:08.540
So if you think about it, this
vector right here, if you

00:12:08.540 --> 00:12:11.715
imagine this is a position
factor, this is velocity.

00:12:14.980 --> 00:12:18.330
Velocity is speed
plus the direction.

00:12:18.330 --> 00:12:20.630
Speed is just you know,
how fast are you going?

00:12:20.630 --> 00:12:22.930
Velocity is how fast you're
going in what direction?

00:12:22.930 --> 00:12:26.590
I'm going this fast-- and you
could calculate it using the

00:12:26.590 --> 00:12:28.260
Pythagorean Theorem, but I just
want to give you the intuition

00:12:28.260 --> 00:12:31.430
right here --I'm going that
fast in this direction.

00:12:31.430 --> 00:12:33.770
Here I'm going this fast;
I'm going even faster.

00:12:33.770 --> 00:12:36.340
That's my magnitude,
but I'm still going in

00:12:36.340 --> 00:12:37.710
the same direction.

00:12:37.710 --> 00:12:40.240
So hopefully you have a gut
feeling now of what the

00:12:40.240 --> 00:12:44.380
derivative of these position
vectors really are.


Differential of a vector valued function

WEBVTT
Kind: captions
Language: en

00:00:00.320 --> 00:00:03.070
In the last couple of videos
we saw that we can describe

00:00:03.070 --> 00:00:07.810
a curves by a position
vector-valued function.

00:00:07.810 --> 00:00:11.020
And in very general terms, it
would be the x position as a

00:00:11.020 --> 00:00:13.750
function of time times
the unit vector in the

00:00:13.750 --> 00:00:15.260
horizontal direction.

00:00:15.260 --> 00:00:19.010
Plus the y position as a
function of time times

00:00:19.010 --> 00:00:22.040
the unit victor in the
vertical direction.

00:00:22.040 --> 00:00:25.120
And this will essentially
describe this-- though, if you

00:00:25.120 --> 00:00:27.160
can imagine a particle and
let's say the parameter

00:00:27.160 --> 00:00:28.620
t represents time.

00:00:28.620 --> 00:00:32.030
It'll describe where the
particle is at any given time.

00:00:32.030 --> 00:00:34.630
And if we wanted a particular
curve we can say, well, this

00:00:34.630 --> 00:00:40.220
only applies for some curve--
we're dealing, it's r of t.

00:00:40.220 --> 00:00:44.130
And it's only applicable
between t being greater

00:00:44.130 --> 00:00:46.710
than a and less than b.

00:00:46.710 --> 00:00:48.740
And you know, that would
describe some curve

00:00:48.740 --> 00:00:50.060
in two dimensions.

00:00:50.060 --> 00:00:51.120
Just me just draw it here.

00:00:51.120 --> 00:00:54.920
This is all a review of
really, the last two videos.

00:00:54.920 --> 00:00:58.050
So this curve, it might look
something like that where this

00:00:58.050 --> 00:00:59.840
is where t is equal to a.

00:00:59.840 --> 00:01:02.610
That's where t is equal to b.

00:01:02.610 --> 00:01:06.260
And so r of a will be this
vector right here that

00:01:06.260 --> 00:01:08.440
ends at that point.

00:01:08.440 --> 00:01:11.170
And then as t or if you can
imagine the parameter being

00:01:11.170 --> 00:01:13.210
time, it doesn't have to be
time, but that's a convenient

00:01:13.210 --> 00:01:15.300
one to visualize.

00:01:15.300 --> 00:01:18.030
Each corresponding as t gets
larger and larger, we're just

00:01:18.030 --> 00:01:20.310
going to different-- we're
specifying different

00:01:20.310 --> 00:01:22.230
points on the path.

00:01:22.230 --> 00:01:24.070
We saw that two videos ago.

00:01:24.070 --> 00:01:27.470
And in the last video we
thought about, well, what does

00:01:27.470 --> 00:01:31.110
it mean to take the derivative
of a vector-valued function?

00:01:31.110 --> 00:01:34.030
And we came up with this idea
that-- and it wasn't an idea,

00:01:34.030 --> 00:01:35.550
we actually showed
it to be true.

00:01:35.550 --> 00:01:37.530
We came up with a
definition really.

00:01:37.530 --> 00:01:40.980
That the derivative-- I could
call it r prime of t-- and

00:01:40.980 --> 00:01:42.220
it's going to be a vector.

00:01:42.220 --> 00:01:43.940
The derivative of a
vector-valued function is once

00:01:43.940 --> 00:01:45.330
again going to be a derivative.

00:01:45.330 --> 00:01:49.060
But it was equal to-- the way
we defined it-- x prime of t

00:01:49.060 --> 00:01:54.690
times i plus y prime
of t times j.

00:01:54.690 --> 00:01:56.880
Or another way to write that
and I'll just write all

00:01:56.880 --> 00:01:59.160
the different ways just so
you get familiar with--

00:01:59.160 --> 00:02:04.740
dr/dt is equal to dx/dt.

00:02:04.740 --> 00:02:06.330
This is just a
standard derivative.

00:02:06.330 --> 00:02:08.650
x of t is a scalar function.

00:02:08.650 --> 00:02:12.940
So this is a standard
derivative times i

00:02:12.940 --> 00:02:17.610
plus dy/dt times j.

00:02:17.610 --> 00:02:20.680
And if we wanted to think about
the differential, one thing

00:02:20.680 --> 00:02:23.770
that we can think about-- and
whenever I do the math for

00:02:23.770 --> 00:02:25.840
the differential it's a
little bit hand wavy.

00:02:25.840 --> 00:02:26.950
I'm not being very rigorous.

00:02:26.950 --> 00:02:29.855
But if you imagine multiplying
both sides of the equation by a

00:02:29.855 --> 00:02:37.590
very small dt or this exact dt,
you would get dr is equal to--

00:02:37.590 --> 00:02:38.740
I'll just leave it like this.

00:02:38.740 --> 00:02:41.330
dx/dt times dt.

00:02:41.330 --> 00:02:43.765
I could make these cancel
out, but I'll just write

00:02:43.765 --> 00:02:45.140
it like this first.

00:02:45.140 --> 00:02:52.950
Times the unit vector i
plus dy/dt times dt.

00:02:52.950 --> 00:02:55.310
Times the unit vector j.

00:02:55.310 --> 00:02:56.510
Or we could rewrite this.

00:02:56.510 --> 00:02:58.570
And I'm just rewriting it in
all of the different ways

00:02:58.570 --> 00:03:00.040
that one can rewrite it.

00:03:00.040 --> 00:03:09.070
You could also write this as dr
is equal to x prime of t dt

00:03:09.070 --> 00:03:10.340
times the unit vector i.

00:03:10.340 --> 00:03:15.970
So this was x prime of t dt.

00:03:15.970 --> 00:03:20.010
This is x prime of t right
there times the unit vector i.

00:03:20.010 --> 00:03:22.750
Plus y prime of t.

00:03:22.750 --> 00:03:24.400
That's just that right there.

00:03:24.400 --> 00:03:26.210
Times dt.

00:03:26.210 --> 00:03:27.980
Times the unit vector j.

00:03:27.980 --> 00:03:31.000
And just to, I guess, complete
the trifecta, the other way

00:03:31.000 --> 00:03:34.720
that we could write this is
that dr is equal to-- if we

00:03:34.720 --> 00:03:37.980
just allowed these to cancel
out, then we get is equal

00:03:37.980 --> 00:03:44.700
to dx times i plus dy
times dy y times j.

00:03:44.700 --> 00:03:46.630
And that actually makes a
lot of intuitive sense.

00:03:46.630 --> 00:03:48.860
That if I look at any dr,
so let's say I look at

00:03:48.860 --> 00:03:53.325
the change between this
vector and this vector.

00:03:53.325 --> 00:03:57.300
Let's say the super small
change right there, that is our

00:03:57.300 --> 00:04:02.390
dr, and it's made up of-- it's
our dx, our change in x

00:04:02.390 --> 00:04:03.490
is that right there.

00:04:03.490 --> 00:04:06.350
You can imagine it's that right
there times-- but we're

00:04:06.350 --> 00:04:08.740
vectorizing it by multiplying
it by the unit vector in

00:04:08.740 --> 00:04:10.120
the horizontal direction.

00:04:10.120 --> 00:04:16.070
Plus dy times the unit vector
in the vertical direction.

00:04:16.070 --> 00:04:17.990
So when you multiply this
distance times the unit

00:04:17.990 --> 00:04:19.633
vector, you're essentially
getting this vector.

00:04:22.560 --> 00:04:24.840
And when you multiply this
guy-- and actually our change

00:04:24.840 --> 00:04:26.530
in y here is negative--
you're going to get

00:04:26.530 --> 00:04:28.750
this vector right here.

00:04:28.750 --> 00:04:31.390
So when you add those together
you'll get your change in

00:04:31.390 --> 00:04:33.370
your actual position vector.

00:04:33.370 --> 00:04:36.020
So that was all a little
bit of background.

00:04:36.020 --> 00:04:39.160
And this might be somewhat
useful-- a future

00:04:39.160 --> 00:04:39.800
video from now.

00:04:39.800 --> 00:04:43.160
Actually, I'm going to leave it
there because really I just

00:04:43.160 --> 00:04:44.910
wanted to introduce this
notation and get you

00:04:44.910 --> 00:04:45.720
familiar with it.

00:04:45.720 --> 00:04:48.770
In the next video, what I'm
going to do is give you a

00:04:48.770 --> 00:04:53.780
little bit more intuition
for what exactly does

00:04:53.780 --> 00:04:54.780
this thing mean?

00:04:54.780 --> 00:04:57.560
And how does it change
depending on different

00:04:57.560 --> 00:04:58.540
parameterizations.

00:04:58.540 --> 00:05:00.780
And I'll do it with two
different parameterizations

00:05:00.780 --> 00:05:03.110
for the same curve.


Derivative of a position vector valued function

WEBVTT
Kind: captions
Language: en

00:00:00.400 --> 00:00:03.100
In the last video, we hopefully
got ourselves a respectable

00:00:03.100 --> 00:00:06.760
understanding of how a
vector-valued function works,

00:00:06.760 --> 00:00:10.340
or even better, a position
vector-valued function, that

00:00:10.340 --> 00:00:13.460
is, in some ways, a
replacement for traditional

00:00:13.460 --> 00:00:16.110
parameterization to
describe a curve.

00:00:16.110 --> 00:00:18.050
And what I want to do in this
video is just get a little bit

00:00:18.050 --> 00:00:21.910
of gut sense of what it means
to take of a derivative of

00:00:21.910 --> 00:00:23.730
a vector-valued function.

00:00:23.730 --> 00:00:27.190
In this case, it'll be with
respect to our parameter t.

00:00:27.190 --> 00:00:30.510
So let me draw some
new stuff right here.

00:00:30.510 --> 00:00:36.580
So let's say I have the
vector-valued function r of t,

00:00:36.580 --> 00:00:39.450
and this is no different than
what I did in last video. x of

00:00:39.450 --> 00:00:46.830
t times unit vector i plus y of
t times the unit vector j.

00:00:46.830 --> 00:00:49.460
If we were doing it in 3
dimensions, we'd add a z of t

00:00:49.460 --> 00:00:52.510
times k, but let's keep things
relatively simple, and let's

00:00:52.510 --> 00:00:56.950
say that this describes a
curve, and let's say the curve

00:00:56.950 --> 00:01:03.630
we're dealing with, t is
between a and b, and this curve

00:01:03.630 --> 00:01:07.870
will look something like, let
me do my best effort to draw

00:01:07.870 --> 00:01:12.600
the curve, I'll just draw some
random curve here, so

00:01:12.600 --> 00:01:17.900
let's say the curve looks
something like that.

00:01:17.900 --> 00:01:19.890
This is when t is equal
to a, so it's going to

00:01:19.890 --> 00:01:21.750
go in this direction.

00:01:21.750 --> 00:01:25.620
This is when t is equal to b
right here, this is t is equal

00:01:25.620 --> 00:01:30.790
to a, so this right here would
be x of a, this right here is y

00:01:30.790 --> 00:01:36.730
of a, and similarly, this up
here, this is x of b, and

00:01:36.730 --> 00:01:40.240
this over here is y of b.

00:01:40.240 --> 00:01:45.150
Now, we saw in the last video
that the endpoints of these

00:01:45.150 --> 00:01:47.720
position vectors are what's
describing this curve.

00:01:47.720 --> 00:01:51.400
So r of a we saw in the
last video, it describes

00:01:51.400 --> 00:01:52.300
that point right there.

00:01:52.300 --> 00:01:54.270
I don't want to review
that too much.

00:01:54.270 --> 00:01:56.800
But what I want to do is
think about, what is the

00:01:56.800 --> 00:01:58.930
difference between 2 points?

00:01:58.930 --> 00:02:01.410
So let's say that we take
some random point here.

00:02:01.410 --> 00:02:03.760
Let's say, some random t here.

00:02:03.760 --> 00:02:06.780
Let's call that r of t.

00:02:06.780 --> 00:02:08.197
Well actually, I'm going to do
a different point, just because

00:02:08.197 --> 00:02:10.100
I want to make it a
little bit clearer.

00:02:10.100 --> 00:02:13.370
So let's say-- I'm going to
switch colors-- let's say

00:02:13.370 --> 00:02:17.080
that that right there
is r of some t.

00:02:17.080 --> 00:02:18.980
Some particular t, right there.

00:02:18.980 --> 00:02:22.410
That is r of t.

00:02:22.410 --> 00:02:25.050
It's going to be, you
know, a plus something.

00:02:25.050 --> 00:02:26.950
So that's some a particular t.

00:02:26.950 --> 00:02:30.080
And let's say that we want to
figure out, and let's say we

00:02:30.080 --> 00:02:31.560
increase t by a little bit.

00:02:31.560 --> 00:02:32.650
By h.

00:02:32.650 --> 00:02:37.250
So let's say that r of t plus
h, well, if we view the

00:02:37.250 --> 00:02:40.430
parameter t as time, we've
moved in forward in time by

00:02:40.430 --> 00:02:42.310
some amount, so our little
particle has moved

00:02:42.310 --> 00:02:43.180
a little bit.

00:02:43.180 --> 00:02:45.650
And let's say that
we're over here.

00:02:45.650 --> 00:02:52.530
So that is that right there,
in yellow, is r of t plus h.

00:02:52.530 --> 00:02:55.360
Just a slightly
larger value for h.

00:02:55.360 --> 00:03:00.040
Now, one question we might ask
ourselves, is how quickly is r

00:03:00.040 --> 00:03:04.650
changing with respect to t?

00:03:04.650 --> 00:03:06.380
So the first thing we might
want to say, well, what's the

00:03:06.380 --> 00:03:07.550
difference between these two?

00:03:07.550 --> 00:03:11.010
If I were to take, and I
want to visualize it.

00:03:11.010 --> 00:03:16.880
If I were to take r, the
position vector, that we get by

00:03:16.880 --> 00:03:21.080
evaluating r at t plus h, and
from that, I would

00:03:21.080 --> 00:03:22.590
subtract r of t.

00:03:25.580 --> 00:03:26.690
What do we get?

00:03:26.690 --> 00:03:29.730
Well, you might want to review
some of your vector algebra

00:03:29.730 --> 00:03:32.130
but we're essentially just
going to get this vector.

00:03:32.130 --> 00:03:34.210
Let me do it in a
nice, vibrant color.

00:03:34.210 --> 00:03:36.820
We're going to get this
vector right there, that

00:03:36.820 --> 00:03:39.320
I'm doing in magenta.

00:03:39.320 --> 00:03:44.010
So that magenta vector right
there is, let me do it, that

00:03:44.010 --> 00:03:47.080
magenta one right there,
is the vector r of t

00:03:47.080 --> 00:03:53.120
plus h minus r of t.

00:03:53.120 --> 00:03:55.110
And it should make sense,
because when you add vectors,

00:03:55.110 --> 00:03:56.050
you go heads to tails.

00:03:56.050 --> 00:04:04.060
You could alternatively write
this as r of t plus this

00:04:04.060 --> 00:04:11.490
character right here, plus r
of t plus h minus r of t.

00:04:11.490 --> 00:04:13.630
When you add two vectors,
you're adding, let me make

00:04:13.630 --> 00:04:16.180
it very clear, I'm adding
this vector to this

00:04:16.180 --> 00:04:17.350
vector right here.

00:04:17.350 --> 00:04:20.260
You put the tail of the
second vector at the

00:04:20.260 --> 00:04:21.450
head of the first.

00:04:21.450 --> 00:04:23.850
So this is the first vector,
and I put the tail of the

00:04:23.850 --> 00:04:28.000
second there, and then the sum
of those two, as we predicted,

00:04:28.000 --> 00:04:29.470
should be equal to
this last one.

00:04:29.470 --> 00:04:33.050
It should be equal
to r of t plus h.

00:04:33.050 --> 00:04:36.000
And we see that is the case,
and algebraically, you would

00:04:36.000 --> 00:04:39.240
see that obviously this
guy and that guy are

00:04:39.240 --> 00:04:40.080
going to cancel out.

00:04:40.080 --> 00:04:41.270
So hopefully that
satisfies you.

00:04:41.270 --> 00:04:42.510
And I want to be clear.

00:04:42.510 --> 00:04:45.450
This, all of a sudden, this
isn't a position vector.

00:04:45.450 --> 00:04:48.560
We're not saying that hey,
let's nail this guy's tail at

00:04:48.560 --> 00:04:51.870
the origin and use this guy to
describe a unique position.

00:04:51.870 --> 00:04:54.610
Now all of a sudden he's, it's
just kind of a pure vector.

00:04:54.610 --> 00:04:57.700
It's describing just
a change between two

00:04:57.700 --> 00:04:59.140
other position vectors.

00:04:59.140 --> 00:05:00.790
So this guy is right out here.

00:05:00.790 --> 00:05:03.930
But this vector literally
describes the change.

00:05:03.930 --> 00:05:07.070
But say we care, and how would
this look algebraically if we

00:05:07.070 --> 00:05:08.460
were to expand it like that?

00:05:08.460 --> 00:05:14.740
So this is going to be equal
to, what's r of t plus h?

00:05:14.740 --> 00:05:20.520
That's the same thing as x
of, let me do it over here.

00:05:20.520 --> 00:05:27.190
This is the same thing as x of
t plus h times the unit vector

00:05:27.190 --> 00:05:35.040
i plus y of t plus h times the
unit vector j, that's just that

00:05:35.040 --> 00:05:40.570
piece, that piece right there
is that piece, minus this

00:05:40.570 --> 00:05:44.300
piece, so minus, I'll do it in
the second line, I could have

00:05:44.300 --> 00:05:46.240
done it out here, but I'm
running out of space.

00:05:46.240 --> 00:05:52.650
Minus x of t, right r of t is
just x of t times i, plus, but

00:05:52.650 --> 00:05:55.700
I'll just distribute the minus
sign, so it's minus

00:05:55.700 --> 00:05:59.690
y of t times j.

00:05:59.690 --> 00:06:02.660
Actually let me write it,
this would be minus, let me

00:06:02.660 --> 00:06:04.130
write this way, plus this.

00:06:04.130 --> 00:06:06.200
So you realize that this
is really just this

00:06:06.200 --> 00:06:06.950
guy right here.

00:06:06.950 --> 00:06:09.200
I'm just evaluating at t.

00:06:09.200 --> 00:06:11.030
So you have x of t and y
of t, and then later we

00:06:11.030 --> 00:06:11.970
can distribute, right?

00:06:11.970 --> 00:06:14.255
If you distribute this minus
sign, you get a minus x

00:06:14.255 --> 00:06:15.920
of t and a minus y of t.

00:06:15.920 --> 00:06:19.380
And in vector addition, you
might need a little review on

00:06:19.380 --> 00:06:20.920
this if you haven't seen it in
a while, you know that you can

00:06:20.920 --> 00:06:22.990
just add the corresponding
components.

00:06:22.990 --> 00:06:24.450
You can add the x-components
and you can add

00:06:24.450 --> 00:06:25.750
the y-components.

00:06:25.750 --> 00:06:28.790
So this is going to be equal
to, let me rewrite it over

00:06:28.790 --> 00:06:31.810
here, because I think I'm going
to need some space later on.

00:06:31.810 --> 00:06:34.510
So let me rewrite it over here.

00:06:34.510 --> 00:06:42.930
So I have r of t plus h minus r
of t is equal to, and I'm just

00:06:42.930 --> 00:06:47.100
going to group the x- and the
y-components, this is equal to

00:06:47.100 --> 00:06:49.920
the x-components added
together, but this is a

00:06:49.920 --> 00:06:51.400
negative, so we're going to
subtract this guy

00:06:51.400 --> 00:06:52.100
from that guy.

00:06:52.100 --> 00:07:02.400
So x of t plus h minus x of t,
and then all of that times our

00:07:02.400 --> 00:07:09.840
unit vector in the x-direction,
and then we'll have plus y of t

00:07:09.840 --> 00:07:16.900
plus h minus y of t times a
unit vector the j-direction,

00:07:16.900 --> 00:07:19.490
I'm just rearranging things
right now, and this will tell

00:07:19.490 --> 00:07:25.870
us what is our change between
any 2 r's for given

00:07:25.870 --> 00:07:26.610
change in distance.

00:07:26.610 --> 00:07:29.140
And our change in distance
here is h between any

00:07:29.140 --> 00:07:31.000
2 position vectors.

00:07:31.000 --> 00:07:33.500
Now, what I set out at the
beginning of this video, I

00:07:33.500 --> 00:07:36.100
said, well, I wanted to figure
out the change, and we're going

00:07:36.100 --> 00:07:37.990
to think about the
instantaneous change

00:07:37.990 --> 00:07:39.790
with respect to t.

00:07:39.790 --> 00:07:42.920
So I want to see, well,
how much did this change

00:07:42.920 --> 00:07:46.570
over a period of h?

00:07:46.570 --> 00:07:48.750
Instead of writing h we could
have written delta t, it

00:07:48.750 --> 00:07:50.010
would've been the same thing.

00:07:50.010 --> 00:07:52.570
So I want to divide this by h.

00:07:52.570 --> 00:07:55.410
So I want to say, look.

00:07:55.410 --> 00:07:58.200
My vectors changed this much,
but I want to say it's

00:07:58.200 --> 00:07:59.310
over a period of h.

00:07:59.310 --> 00:08:01.250
And this is analogous
to when we do slope.

00:08:01.250 --> 00:08:05.090
We say rise over run, over
delta y, or change in

00:08:05.090 --> 00:08:06.290
y, over change in x.

00:08:06.290 --> 00:08:09.160
This is kind of the change in
our function per change in x.

00:08:09.160 --> 00:08:11.960
Let's just divide everything,
or I shouldn't say change

00:08:11.960 --> 00:08:13.470
x, per change in t.

00:08:13.470 --> 00:08:17.100
So here, our change
in t is h, right?

00:08:17.100 --> 00:08:21.290
The difference between t plus h
and t is just going to be h.

00:08:21.290 --> 00:08:25.650
And so we're going to
divide everything by h.

00:08:25.650 --> 00:08:28.750
When you multiply a vector by
some scalar, or divide it by

00:08:28.750 --> 00:08:31.070
some scalar, or you're just the
taking each of its components

00:08:31.070 --> 00:08:34.140
and multiplying or dividing
by that scalar, and we

00:08:34.140 --> 00:08:35.660
get that right there.

00:08:35.660 --> 00:08:39.540
So this, for any finite
difference right here, h,

00:08:39.540 --> 00:08:43.300
this'll tell us how much
our vector changes per h.

00:08:43.300 --> 00:08:46.220
But if we want to find the
instantaneous change, right,

00:08:46.220 --> 00:08:49.230
just like what we did when we
first learned differential

00:08:49.230 --> 00:08:50.520
calculus, we said, ok.

00:08:50.520 --> 00:08:52.670
This is kind of
analogous to a slope.

00:08:52.670 --> 00:08:55.520
This would be good, this would
work out well for us, if the

00:08:55.520 --> 00:09:02.220
path under question looked
something like this.

00:09:02.220 --> 00:09:04.920
If it was a linear path.

00:09:04.920 --> 00:09:07.240
If our path looked
something like this.

00:09:07.240 --> 00:09:09.800
We could just calculate this,
and we'll essentially have the

00:09:09.800 --> 00:09:13.080
average change in our position
vectors, so you could imagine,

00:09:13.080 --> 00:09:15.880
2 position vectors,
that's one of them.

00:09:15.880 --> 00:09:18.050
Well, actually, they'd
all be parallel.

00:09:18.050 --> 00:09:24.470
Well, the position vectors,
they don't have to be parallel.

00:09:24.470 --> 00:09:26.740
They could be like that.

00:09:26.740 --> 00:09:29.340
And then, this would just
describe the change between

00:09:29.340 --> 00:09:32.940
these 2 per h, or how quickly
are the position vectors

00:09:32.940 --> 00:09:36.230
changing per our change
in our parameter, right?

00:09:36.230 --> 00:09:40.530
This is, the h, you could also
consider, is kind of a delta t.

00:09:40.530 --> 00:09:44.050
Sometimes people find the
h simpler, or sometimes

00:09:44.050 --> 00:09:44.900
they find the delta t.

00:09:44.900 --> 00:09:47.230
But anyway, I'm concerned
with the instantaneous.

00:09:47.230 --> 00:09:49.140
We're dealing with curves,
we're dealing with calculus.

00:09:49.140 --> 00:09:50.870
This would have been OK
if we were just in an

00:09:50.870 --> 00:09:52.670
algebraic, linear world.

00:09:52.670 --> 00:09:53.850
So what do we do?

00:09:53.850 --> 00:09:58.080
Well maybe, we can just take
the limit as h approaches 0.

00:09:58.080 --> 00:09:59.760
Let me scroll this over.

00:09:59.760 --> 00:10:03.700
So let's just take the limit,
let me do this in a nice

00:10:03.700 --> 00:10:07.910
vibrant color, let's take, I'm
running out of colors, the

00:10:07.910 --> 00:10:12.170
limit as h approaches 0
of both sides of this.

00:10:12.170 --> 00:10:16.440
So here, too, I'm going to take
the limit as h approaches 0,

00:10:16.440 --> 00:10:18.520
and here, too, I'm going to
take the limit as

00:10:18.520 --> 00:10:20.380
h approaches 0.

00:10:20.380 --> 00:10:22.970
So I just want to say, well,
what happens, how much do I

00:10:22.970 --> 00:10:27.866
change per a change in my
parameter t, but what's kind of

00:10:27.866 --> 00:10:30.480
the instantaneous change, as
the difference gets smaller

00:10:30.480 --> 00:10:31.280
and smaller and smaller?

00:10:31.280 --> 00:10:33.870
This is exactly what we first
learned when we learned about

00:10:33.870 --> 00:10:36.810
instantaneous slope, or
instantaneous velocity, or

00:10:36.810 --> 00:10:38.330
slope of a tangent line.

00:10:38.330 --> 00:10:41.270
Well, this thing looks a little
bit undefined to me, right now.

00:10:41.270 --> 00:10:45.260
We haven't defined limits for
vector-valued functions, we

00:10:45.260 --> 00:10:47.820
haven't defined derivatives
for vector-valued functions.

00:10:47.820 --> 00:10:50.250
But lucky for us, all of
this stuff here looks

00:10:50.250 --> 00:10:51.160
pretty familiar.

00:10:51.160 --> 00:10:54.400
This is actually the
definition of our derivative.

00:10:54.400 --> 00:10:56.310
And these are scalar-valued
functions right here.

00:10:56.310 --> 00:10:58.340
They're multiplied by vectors,
in order for us to get

00:10:58.340 --> 00:10:59.880
vector-valued functions.

00:10:59.880 --> 00:11:02.100
But this right here, by
definition, this is the

00:11:02.100 --> 00:11:04.330
derivative, this
is x prime of t.

00:11:04.330 --> 00:11:08.020
Or this is dx dt.

00:11:08.020 --> 00:11:12.280
This right here is y
prime of t, or we could

00:11:12.280 --> 00:11:14.950
write that as dy dt.

00:11:14.950 --> 00:11:17.520
So all of a sudden we can
define, we can say, and I'm

00:11:17.520 --> 00:11:19.220
being a little hand-wavy here,
but I want to give you the

00:11:19.220 --> 00:11:20.590
intuition, more than anything.

00:11:20.590 --> 00:11:23.890
We can say that the derivative,
we can call this expression

00:11:23.890 --> 00:11:30.100
right here, as the derivative
of my vector-valued function r

00:11:30.100 --> 00:11:35.440
with respect to t, or we could
call it dr dt, notice I keep

00:11:35.440 --> 00:11:37.170
the vector signs there.

00:11:37.170 --> 00:11:44.470
This is its derivative, and all
it's going to be equal to, r

00:11:44.470 --> 00:11:47.700
prime of t, is going to be
equal to, well, this is just

00:11:47.700 --> 00:11:50.450
the derivative of x with
respect to t, is equal to x

00:11:50.450 --> 00:11:54.230
prime of t times the x-unit
vector, the horizontal unit

00:11:54.230 --> 00:12:01.170
vector, plus y prime of t,
times the y-unit vector, times

00:12:01.170 --> 00:12:04.240
j, the unit vector in the
horizontal direction.

00:12:04.240 --> 00:12:06.800
That's a pretty nice
and simple outcome.

00:12:06.800 --> 00:12:09.340
But the hard thing may be
to a kind of visualize

00:12:09.340 --> 00:12:10.730
what it represents.

00:12:10.730 --> 00:12:14.490
So if we think about what
happens, let me draw a big

00:12:14.490 --> 00:12:19.730
graph, just to get the
visualization in a healthy way.

00:12:19.730 --> 00:12:22.310
So let's say my curve looks
something like this.

00:12:22.310 --> 00:12:23.700
That's my curve.

00:12:23.700 --> 00:12:26.860
And let's say that this is,
we want to figure out the

00:12:26.860 --> 00:12:29.210
instantaneous change at
this point right here.

00:12:29.210 --> 00:12:32.550
So that is r of t.

00:12:32.550 --> 00:12:35.520
And then if we take r of t plus
h, we saw this already, you

00:12:35.520 --> 00:12:39.740
know, t plus h might be
something like right there.

00:12:39.740 --> 00:12:43.090
So this is r of t plus h.

00:12:43.090 --> 00:12:46.290
Right now, the difference
between these two, and this is

00:12:46.290 --> 00:12:49.250
just the numerator when you
take the difference, or how

00:12:49.250 --> 00:12:55.170
fast we're changing from this
vector to that vector in terms

00:12:55.170 --> 00:12:57.140
of t, and it's hard
to visualize here.

00:12:57.140 --> 00:12:58.960
And I'm going to do a whole
video so we can think about

00:12:58.960 --> 00:13:00.160
the magnitudes here.

00:13:00.160 --> 00:13:02.200
That might be some vector.

00:13:02.200 --> 00:13:03.560
Well, the difference
between these two is

00:13:03.560 --> 00:13:05.070
just going to be that.

00:13:05.070 --> 00:13:07.470
But then when you divide it by
h, it's going to be a larger

00:13:07.470 --> 00:13:10.840
vector, right, if we assume
that h is a small number.

00:13:10.840 --> 00:13:12.410
Let's say h is less than one.

00:13:12.410 --> 00:13:14.410
We're going to get a
larger vector, right?

00:13:14.410 --> 00:13:17.100
But this is kind of the average
change over this time.

00:13:17.100 --> 00:13:22.910
But as h gets smaller and
smaller and smaller, this r

00:13:22.910 --> 00:13:26.100
prime of t is going to, its
direction is going to be

00:13:26.100 --> 00:13:28.090
tangential to the curve.

00:13:28.090 --> 00:13:29.900
And I think you can
visualize that, right?

00:13:29.900 --> 00:13:33.320
As these two guys get closer
and closer and closer, the dr's

00:13:33.320 --> 00:13:37.790
get smaller, so the change, the
dr, the difference between the

00:13:37.790 --> 00:13:41.220
two, the delta r's, get smaller
and smaller, you can imagine if

00:13:41.220 --> 00:13:43.790
h was even smaller, if
it was right here.

00:13:43.790 --> 00:13:45.510
Then all of a sudden, the
difference between those two

00:13:45.510 --> 00:13:47.795
vectors is getting smaller.

00:13:47.795 --> 00:13:50.500
And it's getting more and more
tangential to the curve.

00:13:53.770 --> 00:13:56.800
But then we're also dividing by
a smaller h, so the actual

00:13:56.800 --> 00:14:00.670
derivative, as the limit of h
approaches 0, it might be you

00:14:00.670 --> 00:14:02.120
know, maybe it's even a
bigger number there.

00:14:02.120 --> 00:14:04.420
And actually, the magnitude
of this vector, it's a

00:14:04.420 --> 00:14:05.720
little hard to visualize.

00:14:05.720 --> 00:14:09.510
It's going to be dependent our
parameterization for the curve.

00:14:09.510 --> 00:14:11.990
it's not dependent on
the shape of the curve.

00:14:11.990 --> 00:14:15.170
The direction of this vector is
dependent on the shape of this

00:14:15.170 --> 00:14:17.860
curve, and the direction, so
the direction, this will

00:14:17.860 --> 00:14:22.460
be tangent to the curve.

00:14:22.460 --> 00:14:26.830
Or you could imagine that this
vector is on the tangent

00:14:26.830 --> 00:14:28.220
line to the curve.

00:14:28.220 --> 00:14:30.960
The magnitude of it is a little
bit hard to understand.

00:14:30.960 --> 00:14:33.050
I'll try to give you a
little bit of intuition on

00:14:33.050 --> 00:14:35.060
that in the next video.

00:14:35.060 --> 00:14:37.380
But this is what I want you to
understand right now, because

00:14:37.380 --> 00:14:40.990
we're going to be able to use
this in the future, when we do

00:14:40.990 --> 00:14:44.400
the line integral over
vector-valued functions.


Position vector valued functions

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.940
All of the work we've been
doing so far with line

00:00:02.940 --> 00:00:05.940
integrals has been with
scalar functions or

00:00:05.940 --> 00:00:07.420
scalar-valued functions.

00:00:07.420 --> 00:00:11.020
And when I say that, that just
means you give me an x and a y

00:00:11.020 --> 00:00:14.130
and you evaluate the function
at that x and y, and you

00:00:14.130 --> 00:00:15.640
get another scalar value.

00:00:15.640 --> 00:00:16.710
You get just a number.

00:00:16.710 --> 00:00:18.620
You don't get a vector.

00:00:18.620 --> 00:00:21.940
So what I want to do in this
video is start to get ourselves

00:00:21.940 --> 00:00:24.860
warmed up with regards to
vectors so that we can

00:00:24.860 --> 00:00:27.220
understand what it means to
take a line integral with

00:00:27.220 --> 00:00:29.030
vector-valued functions.

00:00:29.030 --> 00:00:31.910
So let me write down some
vector-valued functions.

00:00:31.910 --> 00:00:35.052
Actually, even a better place
to start, let me draw a curve

00:00:35.052 --> 00:00:36.290
or let me describe a curve.

00:00:36.290 --> 00:00:38.440
So let's put that little
f of x, y to the side.

00:00:38.440 --> 00:00:39.820
We can ignore it for now.

00:00:39.820 --> 00:00:45.525
Let's say I have some curve c
and it's described, it can be

00:00:45.525 --> 00:00:50.720
parameterized-- I can't say
that word-- as let's say, x is

00:00:50.720 --> 00:00:57.590
equal to x of t, y is equal
to some function y of t.

00:00:57.590 --> 00:01:02.440
And let's say that this is
valid for t is between a and b.

00:01:02.440 --> 00:01:06.280
So t is greater than or
equal to a and then,

00:01:06.280 --> 00:01:08.240
less than or equal to b.

00:01:08.240 --> 00:01:11.990
So if I were to just draw
this on-- let me see-- I

00:01:11.990 --> 00:01:13.320
could draw it like this.

00:01:13.320 --> 00:01:16.270
I'm staying very
abstract right now.

00:01:16.270 --> 00:01:18.740
This is not a very
specific example.

00:01:18.740 --> 00:01:19.910
This is the x-axis.

00:01:19.910 --> 00:01:21.360
This is the y-axis.

00:01:21.360 --> 00:01:26.150
My curve-- let's say this
is when t is equal to a.

00:01:26.150 --> 00:01:30.750
And then the curve might
do something like this.

00:01:30.750 --> 00:01:31.640
I don't know what it does.

00:01:31.640 --> 00:01:32.920
Let's say it's over there.

00:01:32.920 --> 00:01:35.510
This is t is equal to b.

00:01:35.510 --> 00:01:40.430
This actual point right
here will be x of b.

00:01:40.430 --> 00:01:41.560
That would be the x-coordinate.

00:01:41.560 --> 00:01:46.270
You evaluate this function
at b and y of b.

00:01:46.270 --> 00:01:48.610
And this is, of course,
when t is equal to a.

00:01:48.610 --> 00:01:53.370
The actual coordinate in r2 on
the Cartesian coordinates will

00:01:53.370 --> 00:01:57.140
be x of a, which is
this right here.

00:01:57.140 --> 00:02:01.840
And then, y of a, which
is that right there.

00:02:01.840 --> 00:02:03.050
And we've seen that before.

00:02:03.050 --> 00:02:05.770
That's just a standard way
of describing a parametric

00:02:05.770 --> 00:02:09.650
equation or curve using
2 parametric equations.

00:02:09.650 --> 00:02:13.350
What I want to do now is
describe this same exact curve

00:02:13.350 --> 00:02:16.340
using a vector-valued function.

00:02:16.340 --> 00:02:19.970
So if I define a vector-valued
function-- and if you don't

00:02:19.970 --> 00:02:21.230
remember what those are, we'll
have a little bit

00:02:21.230 --> 00:02:21.910
of review here.

00:02:21.910 --> 00:02:25.180
Let me say I have a
vector-valued function, r,

00:02:25.180 --> 00:02:28.150
and I'll put a little
vector arrow on top of it.

00:02:28.150 --> 00:02:30.670
And a lot of textbooks, they'll
just bold it and they'll

00:02:30.670 --> 00:02:32.940
leave scalar-valued
functions unbolded.

00:02:32.940 --> 00:02:35.120
But it's hard to draw bold,
so I'll put a little

00:02:35.120 --> 00:02:36.550
vector on top.

00:02:36.550 --> 00:02:39.220
And let's say that r
is a function of t.

00:02:39.220 --> 00:02:41.045
And these are going to
be position vectors.

00:02:46.800 --> 00:02:49.930
And I'm specifying that
because, in general, when

00:02:49.930 --> 00:02:54.170
someone talks about a vector,
this vector and this vector

00:02:54.170 --> 00:02:55.320
are considered equivalent.

00:02:55.320 --> 00:02:57.990
As long as they have the same
magnitude and direction, no one

00:02:57.990 --> 00:03:00.550
really cares about what their
start and end points are as

00:03:00.550 --> 00:03:02.490
long as their direction's
the same and their

00:03:02.490 --> 00:03:03.710
length is the same.

00:03:03.710 --> 00:03:05.520
But when you talk about
position vectors you're saying

00:03:05.520 --> 00:03:10.155
no, these vectors are all going
to start at 0, at the origin.

00:03:10.155 --> 00:03:12.440
And when you say it's a
position vector, you're

00:03:12.440 --> 00:03:16.000
implicitly saying this is
specifying a unique position.

00:03:16.000 --> 00:03:17.710
In this case, it's going
to be in two-dimensional

00:03:17.710 --> 00:03:19.550
space, but it could be in
three-dimensional space.

00:03:19.550 --> 00:03:22.260
Or really, even four, five,
whatever-- n dimensional space.

00:03:22.260 --> 00:03:24.010
So when you say it's a position
vector, you're literally

00:03:24.010 --> 00:03:27.520
saying, OK, this vector
literally specifies

00:03:27.520 --> 00:03:29.580
that point in space.

00:03:29.580 --> 00:03:35.560
So let's see if we can describe
this curve as a position

00:03:35.560 --> 00:03:38.050
vector-valued function.

00:03:38.050 --> 00:03:39.110
So we could say r of t.

00:03:39.110 --> 00:03:41.750
Let me switch back
to that pink color.

00:03:41.750 --> 00:03:43.020
This can stay in green.

00:03:43.020 --> 00:03:48.750
Is equal to x of t times the
unit vector in the x direction.

00:03:48.750 --> 00:03:51.400
The unit vector gets a little
caret on top-- a little hat.

00:03:51.400 --> 00:03:52.750
That's like the arrow for it.

00:03:52.750 --> 00:03:54.600
That just says it's
a unit vector.

00:03:54.600 --> 00:04:00.870
Plus y of t times j.

00:04:00.870 --> 00:04:02.970
If I was dealing with a curve
in three dimensions I would

00:04:02.970 --> 00:04:04.735
have plus z of t times k.

00:04:04.735 --> 00:04:07.270
But we're dealing with two
dimensions right here.

00:04:07.270 --> 00:04:09.450
And so the way this works is
you're just taking your-- well,

00:04:09.450 --> 00:04:14.690
for any t and still, we're
going to have t is greater

00:04:14.690 --> 00:04:17.610
than or equal to a and then,
less than or equal to b.

00:04:17.610 --> 00:04:19.810
And this is the exact
same thing as that.

00:04:19.810 --> 00:04:21.105
Let me just redraw it.

00:04:21.105 --> 00:04:24.070
So let me draw our coordinates.

00:04:24.070 --> 00:04:28.400
Our coordinates right
here, our axes.

00:04:28.400 --> 00:04:31.410
So that's the y-axis and
this is the x-axis.

00:04:31.410 --> 00:04:34.765
So when you evaluate r of a,
that's our starting point.

00:04:34.765 --> 00:04:36.180
So let me do that.

00:04:36.180 --> 00:04:40.140
So r of a-- maybe I'll
do it right over here.

00:04:40.140 --> 00:04:43.860
Our position vector-valued
function evaluated at t is

00:04:43.860 --> 00:04:50.390
equal to a, is going to be
equal to x of a times our unit

00:04:50.390 --> 00:04:52.040
vector in the x direction.

00:04:52.040 --> 00:04:56.460
Plus y of a times our unit
vector in the vertical

00:04:56.460 --> 00:04:57.865
direction, or in
the y direction.

00:04:57.865 --> 00:04:59.500
And what's that
going to look like?

00:04:59.500 --> 00:05:04.830
Well, x of a is this thing
right here, so it's x of

00:05:04.830 --> 00:05:07.220
a times a unit vector.

00:05:07.220 --> 00:05:09.860
You know, maybe the unit
vector is this long.

00:05:09.860 --> 00:05:12.180
It has length 1, so now we're
just going to have a length

00:05:12.180 --> 00:05:14.070
of x of a in that direction.

00:05:14.070 --> 00:05:15.520
And then, same thing in y of a.

00:05:15.520 --> 00:05:18.240
It's going to be y of a
length in that direction.

00:05:18.240 --> 00:05:20.830
But the bottom line, this
vector right here-- if you add

00:05:20.830 --> 00:05:25.910
these scaled values of these
two unit vectors, you're going

00:05:25.910 --> 00:05:28.460
to get r of a looking
something like this.

00:05:28.460 --> 00:05:32.070
it's going to be a vector that
looks something like that.

00:05:32.070 --> 00:05:33.270
Just like that.

00:05:33.270 --> 00:05:34.770
It's a position vector.

00:05:34.770 --> 00:05:37.330
That's why we're nailing it
at the origin, but drawing

00:05:37.330 --> 00:05:38.840
it in standard position.

00:05:38.840 --> 00:05:44.450
And that right there is r of a.

00:05:44.450 --> 00:05:46.980
Now what happens if a
increases a little bit?

00:05:46.980 --> 00:05:49.590
What is r of a plus
a little bit?

00:05:49.590 --> 00:05:51.680
And I don't know, we could
call that r of a plus

00:05:51.680 --> 00:05:54.305
delta or r of a plus h.

00:05:54.305 --> 00:05:56.355
I'll do it in a
different color.

00:05:56.355 --> 00:05:59.850
Let's say we increase a
a little bit. r of a

00:05:59.850 --> 00:06:02.070
plus some small h.

00:06:02.070 --> 00:06:06.060
Well, that's just going to
be x of a plus h times

00:06:06.060 --> 00:06:07.710
a unit vector i.

00:06:07.710 --> 00:06:11.820
Plus y times a plus h
times the unit vector j.

00:06:11.820 --> 00:06:13.230
And what's that
going to look like?

00:06:13.230 --> 00:06:14.730
Well, we're going to go
a little bit further

00:06:14.730 --> 00:06:15.370
down the curve.

00:06:15.370 --> 00:06:17.100
That's like saying the
coordinate x of a plus

00:06:17.100 --> 00:06:18.980
h and y plus a plus h.

00:06:18.980 --> 00:06:20.590
I might be that
point right there.

00:06:20.590 --> 00:06:25.640
So it'll be a new unit vector.

00:06:25.640 --> 00:06:28.300
Sorry, it'll be a new
vector-- position vector--

00:06:28.300 --> 00:06:29.070
not a unit vector.

00:06:29.070 --> 00:06:30.940
These don't necessarily
have length 1.

00:06:30.940 --> 00:06:33.470
That might be right here.

00:06:33.470 --> 00:06:35.590
Let me do that same
color as this.

00:06:35.590 --> 00:06:39.970
So it might be just like that.

00:06:39.970 --> 00:06:44.930
So that right here
is r of a plus h.

00:06:44.930 --> 00:06:48.270
So you see, as you keep
increasing you value of t until

00:06:48.270 --> 00:06:53.950
you get to b, these position
vectors-- we're going to keep

00:06:53.950 --> 00:06:55.860
specifying points
along this curve.

00:06:55.860 --> 00:06:59.160
So the curve-- let me draw the
curve in a different color.

00:06:59.160 --> 00:07:01.230
The curve looks
something like this.

00:07:01.230 --> 00:07:04.150
It's meant to look exactly like
the curve that I have up here.

00:07:04.150 --> 00:07:08.080
And for example, r of b
is going to be a vector

00:07:08.080 --> 00:07:09.180
that looks like this.

00:07:09.180 --> 00:07:12.680
It's going to be a vector
that looks like that.

00:07:12.680 --> 00:07:16.120
I want to draw it
relatively straight.

00:07:16.120 --> 00:07:22.310
That vector right
there is r of b.

00:07:22.310 --> 00:07:24.740
So hopefully you realize that,
look, these position vectors

00:07:24.740 --> 00:07:27.910
really are specifying the same
points on this curve as this

00:07:27.910 --> 00:07:30.880
original, I guess, straight
up parameterization that

00:07:30.880 --> 00:07:33.070
we did for this curve.

00:07:33.070 --> 00:07:34.880
And I just wanted to that as a
little bit of review because

00:07:34.880 --> 00:07:37.730
we're now going to break in
into the idea of actually

00:07:37.730 --> 00:07:41.860
taking a derivative of this
vector-valued function.

00:07:41.860 --> 00:07:44.210
And I'll do that in
the next video.