Worked example: forming a slope field

WEBVTT
Kind: captions
Language: en

00:00:00.439 --> 00:00:02.144
- [Instructor] In drawing
the slope field for the

00:00:02.144 --> 00:00:05.167
differential equation, the
derivative of y with respect

00:00:05.167 --> 00:00:07.750
to x is equal to y minus 2x,

00:00:08.589 --> 00:00:10.814
I would place short line segments

00:00:10.814 --> 00:00:13.336
at select points on the xy-plane.

00:00:13.336 --> 00:00:15.176
Complete the sentences.

00:00:15.176 --> 00:00:17.617
At the point (-1,1) I would draw

00:00:17.617 --> 00:00:20.927
a short segment of slope blank,

00:00:20.927 --> 00:00:22.368
and like always pause this
video and see if you can

00:00:22.368 --> 00:00:25.437
fill out these three blanks.

00:00:25.437 --> 00:00:29.030
The short segments that
you're trying to draw

00:00:29.030 --> 00:00:31.085
to construct this slope field,

00:00:31.085 --> 00:00:34.178
you figure out their slope based on the

00:00:34.178 --> 00:00:35.729
differential equation
so you're saying when

00:00:35.729 --> 00:00:38.147
x is equal to -1 and y is equal to 1,

00:00:38.147 --> 00:00:40.938
what is the derivative
of y with respect to x,

00:00:40.938 --> 00:00:42.994
and that's what this
differential equation tells us.

00:00:42.994 --> 00:00:46.645
So for this first case, the
derivative of y with respect

00:00:46.645 --> 00:00:50.812
to x is going to be equal to
y, which is 1 - 2 times x,

00:00:52.406 --> 00:00:54.405
x is negative 1,

00:00:54.405 --> 00:00:57.088
so this is going to be -2
but you're subtracting it

00:00:57.088 --> 00:00:58.962
so it's gonna be +2,

00:00:58.962 --> 00:01:01.502
so the derivative of y with
respect to x at this point

00:01:01.502 --> 00:01:04.662
is going to be 3, so I would
draw a short line segment

00:01:04.662 --> 00:01:07.162
or a short segment of slope 3,

00:01:08.427 --> 00:01:11.329
and we keep going, at the point (0,2).

00:01:11.329 --> 00:01:15.529
Well let's see, when x is 0 and y is 2

00:01:15.529 --> 00:01:18.930
the derivative of y with
respect to x is going

00:01:18.930 --> 00:01:22.513
to be equal to y which is 2 - 2 times 0,

00:01:23.516 --> 00:01:25.959
well that's just going to be 2.

00:01:25.959 --> 00:01:28.889
And then last but not
least, for this third point

00:01:28.889 --> 00:01:31.390
the derivative of y with
respect to x is going to

00:01:31.390 --> 00:01:35.557
be equal to y which is 3
- 2 times x, x here is 2,

00:01:38.766 --> 00:01:41.683
2 times 2, 3 - 4 is equal to -1,

00:01:48.463 --> 00:01:50.722
and that's all that problem asks us to do,

00:01:50.722 --> 00:01:52.931
now if we actually had
to do it, it would look

00:01:52.931 --> 00:01:55.518
something like, I'll try
to draw it real fast,

00:01:55.518 --> 00:01:58.020
so let's see, let me make sure I have

00:01:58.020 --> 00:02:00.855
space for all of these points here,

00:02:00.855 --> 00:02:03.188
so that's my coordinate axis

00:02:07.042 --> 00:02:11.209
and I want to get the point (0,2),

00:02:14.867 --> 00:02:16.675
actually I want to go all the way to (2,3)

00:02:16.675 --> 00:02:18.730
so let me get some space here,

00:02:18.730 --> 00:02:21.147
so 1, 2, 3, and then 1, 2, 3,

00:02:25.529 --> 00:02:28.251
and then we have to go
-1 and once we might go

00:02:28.251 --> 00:02:32.256
right over here, and
so for this first one,

00:02:32.256 --> 00:02:34.567
this exercise isn't asking
us to do it but I'm just

00:02:34.567 --> 00:02:37.980
making it very clear how we
would construct the slope field,

00:02:37.980 --> 00:02:42.147
so the point (-1,1), a
short segment of slope 3,

00:02:43.891 --> 00:02:47.391
so slope 3 would look something like that,

00:02:48.684 --> 00:02:52.851
then at the point (0,2) a slope of 2,

00:02:53.738 --> 00:02:58.243
(0,2), the slope is going to
be 2 which looks something

00:02:58.243 --> 00:03:02.410
like that, and then at the point (2,3),

00:03:03.529 --> 00:03:06.495
at (2,3), a short segment of slope -1,

00:03:06.495 --> 00:03:10.662
so (2,3) a slope segment of slope -1,

00:03:11.852 --> 00:03:13.552
it would look something like that,

00:03:13.552 --> 00:03:15.385
and you would keep doing
this at more and more points,

00:03:15.385 --> 00:03:16.966
if you had a computer to do
it that's what the computer

00:03:16.966 --> 00:03:19.370
would do, and you would draw
these short line segments

00:03:19.370 --> 00:03:23.021
to indicate what the
derivative is at those points

00:03:23.021 --> 00:03:25.616
and you get a sense of,
I guess you would say

00:03:25.616 --> 00:03:29.783
the solution space for
that differential equation.


Worked example: slope field from equation

WEBVTT
Kind: captions
Language: en

00:00:00.392 --> 00:00:02.212
- [Instructor] Which
slope field is generated

00:00:02.212 --> 00:00:04.074
by the differential
equation the derivative of y

00:00:04.074 --> 00:00:08.318
with respect to x is equal to x minus y?

00:00:08.318 --> 00:00:09.775
And like always pause this video

00:00:09.775 --> 00:00:12.413
and see if you can figure
it out on your own.

00:00:12.413 --> 00:00:13.683
Well the easiest way to think

00:00:13.683 --> 00:00:15.218
about a slope field, if I was, if i needed

00:00:15.218 --> 00:00:17.679
to plot this slope field by hand,

00:00:17.679 --> 00:00:20.483
I would sample a bunch of x and y points.

00:00:20.483 --> 00:00:21.444
And then I would figure out what

00:00:21.444 --> 00:00:23.830
the derivative would
have to be at that point.

00:00:23.830 --> 00:00:25.635
And so what we can do
here, since we've already

00:00:25.635 --> 00:00:28.214
drawn some candidate slope fields for us,

00:00:28.214 --> 00:00:30.105
is figure out what we
think the slope field

00:00:30.105 --> 00:00:31.607
should be at some points and see

00:00:31.607 --> 00:00:33.866
which of these diagrams, these graphs,

00:00:33.866 --> 00:00:36.355
or these slope fields actually show that.

00:00:36.355 --> 00:00:39.850
So let's, let me make a little table here,

00:00:39.850 --> 00:00:41.350
so I'm gonna have,

00:00:43.175 --> 00:00:46.280
x, y and then the derivative

00:00:46.280 --> 00:00:48.590
of y with respect to x.

00:00:48.590 --> 00:00:50.591
And we can do it at a bunch of values,

00:00:50.591 --> 00:00:51.542
so let's think about it.

00:00:51.542 --> 00:00:54.461
Let's think about when, we're
at this point right over here.

00:00:54.461 --> 00:00:56.339
When x is two and y is two.

00:00:56.339 --> 00:00:59.517
When x is two and y is two, the derivative

00:00:59.517 --> 00:01:02.858
of y with respect to x is
going to be two minus two.

00:01:02.858 --> 00:01:05.320
It's going to be equal to zero.

00:01:05.320 --> 00:01:08.123
And just with that, let's
see, here this slope

00:01:08.123 --> 00:01:10.281
on this slope field does
not look like it's zero.

00:01:10.281 --> 00:01:12.136
This looks like it's negative one.

00:01:12.136 --> 00:01:14.345
So already I can rule this one out.

00:01:14.345 --> 00:01:16.870
This slope right over looks
like it's positive one.

00:01:16.870 --> 00:01:18.890
So I'll rule that out,
it's definitely not zero.

00:01:18.890 --> 00:01:21.243
This slope also looks like positive one.

00:01:21.243 --> 00:01:22.547
So I can rule that one out.

00:01:22.547 --> 00:01:26.003
This slope at two comma two
actually does look like zero.

00:01:26.003 --> 00:01:28.714
So I'm liking this one right over here.

00:01:28.714 --> 00:01:31.402
This slope at two comma
two looks larger than one

00:01:31.402 --> 00:01:32.815
so I can rule that out.

00:01:32.815 --> 00:01:35.136
So it was that straight forward to deduce

00:01:35.136 --> 00:01:38.487
that this choice right over here is,

00:01:38.487 --> 00:01:39.945
if any of these are going to be

00:01:39.945 --> 00:01:41.660
the accurate slope field, it's this one.

00:01:41.660 --> 00:01:43.395
But just for kicks we could keep going

00:01:43.395 --> 00:01:46.392
to verify that this is
indeed the slope field.

00:01:46.392 --> 00:01:48.262
So let's think about what happens

00:01:48.262 --> 00:01:52.429
when x is equal to a one,
whenever x is equal to y,

00:01:53.432 --> 00:01:55.704
you're gonna get the
derivative equaling zero.

00:01:55.704 --> 00:01:57.801
And you see that here,
when you're at four, four,

00:01:57.801 --> 00:01:58.921
derivative equals zero.

00:01:58.921 --> 00:02:00.809
When it's six, six,
derivative equals zero.

00:02:00.809 --> 00:02:03.647
At negative two, negative
two, derivative equals zero.

00:02:03.647 --> 00:02:07.693
So that feels good, that this
is the right slope field.

00:02:07.693 --> 00:02:10.003
And then we could pick
other arbitrary points.

00:02:10.003 --> 00:02:13.803
Let's say when x is four, y is two,

00:02:13.803 --> 00:02:17.195
then the derivative here
should be four minus two,

00:02:17.195 --> 00:02:18.382
which is going to be two.

00:02:18.382 --> 00:02:20.977
So when x is four, y is
two, we do indeed see

00:02:20.977 --> 00:02:23.453
that the slope field is indicating a slope

00:02:23.453 --> 00:02:25.671
that looks like two right over here.

00:02:25.671 --> 00:02:27.239
And if it was the other way around,

00:02:27.239 --> 00:02:30.989
when x is let's say, x is negative four

00:02:34.151 --> 00:02:36.725
and y is negative two.

00:02:36.725 --> 00:02:39.868
The negative four, negative two.

00:02:39.868 --> 00:02:42.814
Well negative four minus negative two

00:02:42.814 --> 00:02:44.967
is going to be negative two.

00:02:44.967 --> 00:02:46.172
And you can see that right over here.

00:02:46.172 --> 00:02:47.821
Negative four, negative two, you can see

00:02:47.821 --> 00:02:48.844
the slope right over here.

00:02:48.844 --> 00:02:51.502
It's a little harder to see,
looks like negative two.

00:02:51.502 --> 00:02:54.702
So once again using even
just this first two comma two

00:02:54.702 --> 00:02:57.359
coordinates we were able to
deduce this was the choice.

00:02:57.359 --> 00:03:01.526
But it just continues to
confirm our original answer.


Worked example: exponential solution to differential equation

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.739
- [Teacher] So, we've got
the differential equation,

00:00:01.739 --> 00:00:03.701
the derivative of y with
respect to x is equal

00:00:03.701 --> 00:00:05.849
to three times y.

00:00:05.849 --> 00:00:08.075
And we want to find
the particular solution

00:00:08.075 --> 00:00:10.319
that gives us y being equal to two

00:00:10.319 --> 00:00:12.037
when x is equal to one.

00:00:12.037 --> 00:00:14.031
So, I encourage you to pause this video

00:00:14.031 --> 00:00:16.832
and see if you can figure
this out on your own.

00:00:16.832 --> 00:00:18.771
All right, now, let's
work through it together.

00:00:18.771 --> 00:00:20.210
So, some of you might
have immediately said,

00:00:20.210 --> 00:00:22.439
"Hey, this is the form of
a differential equation

00:00:22.439 --> 00:00:24.140
"where the solution is
going to be an exponential,"

00:00:24.140 --> 00:00:26.033
and you just got right to it.

00:00:26.033 --> 00:00:27.751
But, I'm not going to go straight to that,

00:00:27.751 --> 00:00:30.921
I'm just gonna recognize
that this is a separable

00:00:30.921 --> 00:00:34.234
differential equation and that
I'm gonna solve it that way.

00:00:34.234 --> 00:00:35.263
So, when I say that it's separable,

00:00:35.263 --> 00:00:37.817
that means that we can
separate the y's, dy's,

00:00:37.817 --> 00:00:41.080
on one side, and all the
x's, dx's on the other side.

00:00:41.080 --> 00:00:43.565
And so, what I can do is
if I divide both sides

00:00:43.565 --> 00:00:48.302
of this equation by y and
multiply both sides by dx,

00:00:48.302 --> 00:00:50.052
I get one over y, dy,

00:00:52.365 --> 00:00:54.115
is equal to three-dx.

00:00:56.069 --> 00:00:57.880
Now, on the left and right hand sides,

00:00:57.880 --> 00:01:00.388
I have these clean things
that I can now integrate.

00:01:00.388 --> 00:01:01.340
That's what people talk about

00:01:01.340 --> 00:01:04.219
when they say separable
differential equation.

00:01:04.219 --> 00:01:07.179
Now, here on the left,
if I wanted to write it

00:01:07.179 --> 00:01:08.782
in a fairly general form, I could write,

00:01:08.782 --> 00:01:10.906
well, the anti-derivative of one over y

00:01:10.906 --> 00:01:15.028
is gonna be the natural log
of the absolute value of y.

00:01:15.028 --> 00:01:18.255
I'm taking the anti-derivative
with respect to y, here.

00:01:18.255 --> 00:01:19.358
Now, I could add a constant,

00:01:19.358 --> 00:01:21.506
but I'm gonna add in a constant
on the right-hand side,

00:01:21.506 --> 00:01:24.072
so there's no reason to
add two arbitrary constants

00:01:24.072 --> 00:01:24.905
on both sides.

00:01:24.905 --> 00:01:26.817
I could just add one on one side.

00:01:26.817 --> 00:01:29.563
So, that is going to be equal to

00:01:29.563 --> 00:01:32.813
the anti-derivative here is going to be

00:01:34.184 --> 00:01:38.956
three-x and I'll add the
promised constant, plus c,

00:01:38.956 --> 00:01:40.373
right over there.

00:01:41.208 --> 00:01:43.681
And now, let's think
about it a little bit.

00:01:43.681 --> 00:01:46.607
Well, we can rewrite
this in exponential form.

00:01:46.607 --> 00:01:49.266
We could say, we could write,

00:01:49.266 --> 00:01:51.855
that e to the three-x plus c is equal

00:01:51.855 --> 00:01:53.573
to the natural log of y.

00:01:53.573 --> 00:01:55.941
I could write the natural of y is equal

00:01:55.941 --> 00:01:57.858
to e to three-x plus c.

00:01:59.865 --> 00:02:03.144
Now, I could rewrite this
is equal to e to three-x

00:02:03.144 --> 00:02:05.322
times e to the c.

00:02:05.322 --> 00:02:07.215
Now, e to the c is just gonna be some

00:02:07.215 --> 00:02:10.849
other arbitrary constant, which
I could still denote by c.

00:02:10.849 --> 00:02:11.917
They're are going to be different values,

00:02:11.917 --> 00:02:14.587
but we're just trying to just get a sense

00:02:14.587 --> 00:02:16.621
of what the structure of
this thing looks like.

00:02:16.621 --> 00:02:18.558
So, we could say this is
going to be some constant

00:02:18.558 --> 00:02:20.856
times e to the three-x.

00:02:20.856 --> 00:02:22.133
So, another way of thinking about it.

00:02:22.133 --> 00:02:24.444
Saying the absolute value
of y is equal to this.

00:02:24.444 --> 00:02:25.465
This isn't a function yet.

00:02:25.465 --> 00:02:26.777
We're trying to find
this function solution

00:02:26.777 --> 00:02:28.623
to this differential equation.

00:02:28.623 --> 00:02:31.294
So, this would tell us
either y is equal to c,

00:02:31.294 --> 00:02:32.942
e to the three-x,

00:02:32.942 --> 00:02:36.775
or y is equal to negative
c, e to the three-x.

00:02:38.354 --> 00:02:39.948
Well, we've kept it in general terms.

00:02:39.948 --> 00:02:40.871
I haven't put any...

00:02:40.871 --> 00:02:42.880
We don't know c is.

00:02:42.880 --> 00:02:46.383
So, what we could do, instead,
is just pick this one,

00:02:46.383 --> 00:02:48.476
and then we can solve for c

00:02:48.476 --> 00:02:50.938
assuming this one right over here.

00:02:50.938 --> 00:02:54.885
And so, we will see if we
can meet these constraints

00:02:54.885 --> 00:02:57.265
using this and it'll
essentially take the other one

00:02:57.265 --> 00:02:59.724
into consideration, whether
we're going positive

00:02:59.724 --> 00:03:01.474
or negative.

00:03:01.474 --> 00:03:03.073
So, let's do that.

00:03:03.073 --> 00:03:05.323
So, when y is equal to two,

00:03:06.782 --> 00:03:08.290
I'm now going to solve for c

00:03:08.290 --> 00:03:11.441
to find the particular
solution, x is equal to one,

00:03:11.441 --> 00:03:15.087
or when x is equal to
one, y is equal to two.

00:03:15.087 --> 00:03:16.573
So, I could write it like that,

00:03:16.573 --> 00:03:19.150
and we get two is equal

00:03:19.150 --> 00:03:20.150
to c times e

00:03:22.858 --> 00:03:25.576
to the third power, three times one.

00:03:25.576 --> 00:03:27.259
And so, to solve for
c, I could just divide

00:03:27.259 --> 00:03:29.233
both sides by e to the third,

00:03:29.233 --> 00:03:30.963
and so I could, or I
could multiply both sides

00:03:30.963 --> 00:03:32.600
times e to the negative third,

00:03:32.600 --> 00:03:35.588
and I could get two e to
the negative third power

00:03:35.588 --> 00:03:36.755
is equal to c.

00:03:38.560 --> 00:03:42.252
And so let's now substitute it back in

00:03:42.252 --> 00:03:44.469
and our particular solutions is gonna be

00:03:44.469 --> 00:03:45.802
y is equal to c.

00:03:46.722 --> 00:03:49.889
C is two-e to the negative third power

00:03:51.920 --> 00:03:53.837
times e to the three-x.

00:03:55.896 --> 00:03:57.905
Now, I'm taking the product of two things

00:03:57.905 --> 00:03:58.880
with the same base.

00:03:58.880 --> 00:03:59.856
I can add the exponents.

00:03:59.856 --> 00:04:03.105
So, I could say y is equal to

00:04:03.105 --> 00:04:05.272
two times e to the three-x

00:04:08.083 --> 00:04:10.351
and I'll add the exponents

00:04:10.351 --> 00:04:13.768
to three-x minus three, and there you go.

00:04:14.867 --> 00:04:18.466
This is one way that you could
write the particular solution

00:04:18.466 --> 00:04:19.894
that meets these constraints

00:04:19.894 --> 00:04:23.311
for this separable differential equation.


Writing a differential equation

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.683
- [Instructor] Particle
moves along a straight line.

00:00:02.683 --> 00:00:05.139
Its speed is inversely proportional

00:00:05.139 --> 00:00:08.715
to the square of the
distance, S, it has traveled.

00:00:08.715 --> 00:00:11.927
Which equation describes
this relationship?

00:00:11.927 --> 00:00:13.352
So I'm not even gonna
look at these choices,

00:00:13.352 --> 00:00:15.623
and I'm just gonna try to
parse the sentence up here

00:00:15.623 --> 00:00:18.584
and see if we can come
up with an equation.

00:00:18.584 --> 00:00:22.114
So they tell us its speed
is inversely proportional,

00:00:22.114 --> 00:00:22.947
to what?

00:00:22.947 --> 00:00:26.805
To the square of the
distance, S, it has traveled.

00:00:26.805 --> 00:00:27.979
So S

00:00:27.979 --> 00:00:29.729
is equal to distance.

00:00:31.138 --> 00:00:33.055
S is equal to distance.

00:00:35.081 --> 00:00:36.958
And how would we denote speed then,

00:00:36.958 --> 00:00:38.397
if S is distance?

00:00:38.397 --> 00:00:40.725
Well speed is the rate
of change of distance

00:00:40.725 --> 00:00:42.263
with respect to time.

00:00:42.263 --> 00:00:43.813
So our speed would be

00:00:43.813 --> 00:00:46.133
the rate of distance with respect to time.

00:00:46.133 --> 00:00:48.893
The rate of change of
distance with respect to time.

00:00:48.893 --> 00:00:52.295
So this is going to be our speed.

00:00:52.295 --> 00:00:53.734
So now that we got our notation,

00:00:53.734 --> 00:00:55.073
S is the distance,

00:00:55.073 --> 00:00:58.331
the derivative of S with
respect to time is speed.

00:00:58.331 --> 00:01:01.047
We can say the speed which is d,

00:01:01.047 --> 00:01:01.880
capital S,

00:01:01.880 --> 00:01:03.105
dt,

00:01:03.105 --> 00:01:05.600
is inversely proportional.

00:01:05.600 --> 00:01:07.190
So it's inversely proportional,

00:01:07.190 --> 00:01:08.403
I wrote a proportionality constant,

00:01:08.403 --> 00:01:09.236
over what?

00:01:09.236 --> 00:01:11.320
It's inversely proportional to what?

00:01:11.320 --> 00:01:13.349
To the square of the distance!

00:01:13.349 --> 00:01:16.410
To the square of the
distance it has traveled.

00:01:16.410 --> 00:01:17.243
So there you go,

00:01:17.243 --> 00:01:18.667
this is an equation that I think

00:01:18.667 --> 00:01:20.556
is describing a differential equation,

00:01:20.556 --> 00:01:24.557
really that's describing
what we have up here.

00:01:24.557 --> 00:01:25.565
Now let's see,

00:01:25.565 --> 00:01:26.710
let's see what,

00:01:26.710 --> 00:01:29.070
which of these choices match that.

00:01:29.070 --> 00:01:31.716
Well actually this one
is exactly what we wrote.

00:01:31.716 --> 00:01:32.549
The speed,

00:01:32.549 --> 00:01:34.506
the rate of change of
distance with respect to time,

00:01:34.506 --> 00:01:38.477
is inversely proportional to
the square of the distance.

00:01:38.477 --> 00:01:40.354
Now just to make sure we
understand these other ones,

00:01:40.354 --> 00:01:42.339
let's just interpret them.

00:01:42.339 --> 00:01:44.580
This is saying that the distance,

00:01:44.580 --> 00:01:46.198
which is a function of time,

00:01:46.198 --> 00:01:49.741
is inversely proportional
to the time squared.

00:01:49.741 --> 00:01:51.207
That's not what they told us.

00:01:51.207 --> 00:01:52.771
This is saying that the distance

00:01:52.771 --> 00:01:55.533
is inversely proportional
to the distance squared.

00:01:55.533 --> 00:01:57.408
That one is especially strange.

00:01:57.408 --> 00:01:59.405
And this is saying that the
distance with respect to time,

00:01:59.405 --> 00:02:00.775
the change in distance
with respect to time,

00:02:00.775 --> 00:02:03.397
the derivative of the
distance with respect to time,

00:02:03.397 --> 00:02:05.500
dS/dt or the speed,

00:02:05.500 --> 00:02:07.633
is inversely proportional

00:02:07.633 --> 00:02:08.466
to

00:02:08.466 --> 00:02:09.684
time squared.

00:02:09.684 --> 00:02:10.659
Well that's not what they said,

00:02:10.659 --> 00:02:12.334
they said it's inversely proportional

00:02:12.334 --> 00:02:15.017
to the square of the
distance it has traveled.

00:02:15.017 --> 00:02:16.934
So we like that choice.


Worked example: range of solution curve from slope field

WEBVTT
Kind: captions
Language: en

00:00:00.155 --> 00:00:03.826
- [Instructor] If the initial
condition is zero comma six,

00:00:03.826 --> 00:00:08.288
what is the range of the solution
curve y is equal to f of x

00:00:08.288 --> 00:00:12.052
for x is greater than or equal to zero?

00:00:12.052 --> 00:00:14.858
So we have a slope field here,
for a differential equation.

00:00:14.858 --> 00:00:16.357
And we're saying, okay,
if we have a solution

00:00:16.357 --> 00:00:18.378
where the initial condition
is zero comma six,

00:00:18.378 --> 00:00:21.301
so zero comma six is
part of that solution.

00:00:21.301 --> 00:00:23.959
So let's see, zero comma six,

00:00:23.959 --> 00:00:26.709
so this is part of the solution.

00:00:26.709 --> 00:00:29.243
And we want to know the
range of the solution curve.

00:00:29.243 --> 00:00:31.411
So solution curve, you
can eyeball a little bit

00:00:31.411 --> 00:00:33.303
by looking at the slope field.

00:00:33.303 --> 00:00:34.965
So as x,

00:00:34.965 --> 00:00:36.990
remember, x is gonna be
greater than or equal to zero,

00:00:36.990 --> 00:00:39.263
so it's going to include
this point right over here.

00:00:39.263 --> 00:00:41.502
And as x increases, you
can tell from the slope,

00:00:41.502 --> 00:00:43.630
okay, y is gonna decrease,

00:00:43.630 --> 00:00:48.294
but it's gonna keep decreasing
at a slower and slower rate.

00:00:48.294 --> 00:00:50.564
And it looks like it's asymptoting

00:00:50.564 --> 00:00:53.277
towards the line y is equal to four.

00:00:53.277 --> 00:00:54.384
So it's gonna get really,

00:00:54.384 --> 00:00:55.986
as x gets larger and larger, larger,

00:00:55.986 --> 00:01:00.134
it's gonna get infinitely
close to y is equal to four,

00:01:00.134 --> 00:01:02.352
but it's not quite gonna get there.

00:01:02.352 --> 00:01:05.768
So the range, the y-values
that this is going to take on,

00:01:05.768 --> 00:01:07.380
y is going to be greater than four.

00:01:07.380 --> 00:01:09.624
It's never gonna be equal to four.

00:01:09.624 --> 00:01:13.214
So I'll do, it's going
to be greater than four.

00:01:13.214 --> 00:01:15.284
That's gonna be the
bottom end of my range.

00:01:15.284 --> 00:01:18.274
And at the top end of my
range, I will be equal to six.

00:01:18.274 --> 00:01:20.900
Six is the largest value
that I am going to take on.

00:01:20.900 --> 00:01:25.067
Another way I could have written
this is four is less than

00:01:26.045 --> 00:01:28.810
y is less than or equal to six.

00:01:28.810 --> 00:01:31.354
Either way, this is a way
of describing the range.

00:01:31.354 --> 00:01:33.963
The y-values that the
solution will take on

00:01:33.963 --> 00:01:36.659
for x being greater than or equal to zero.

00:01:36.659 --> 00:01:38.940
If they said for all x's,

00:01:38.940 --> 00:01:41.902
well, then you might have
been able to go back this way

00:01:41.902 --> 00:01:43.053
and keep going.

00:01:43.053 --> 00:01:45.116
But they're saying the
range of the solution curve

00:01:45.116 --> 00:01:47.849
for x is greater than or equal to zero.

00:01:47.849 --> 00:01:51.779
So we won't consider those
values of x less than zero.

00:01:51.779 --> 00:01:53.932
So there you go, the curve
would look something like that.

00:01:53.932 --> 00:01:56.709
And you can see, the highest
value it takes on is six,

00:01:56.709 --> 00:01:58.265
and it actually does take on that value

00:01:58.265 --> 00:02:00.810
'cause we're including x equaling zero.

00:02:00.810 --> 00:02:03.617
And then it keeps going
down, approaching four,

00:02:03.617 --> 00:02:05.024
getting very, very close to four,

00:02:05.024 --> 00:02:07.524
but never quite equaling four.


Worked example: separable equation with an implicit solution

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.263
- [Teacher] We're given
a differential equation

00:00:01.263 --> 00:00:02.875
right over here, cosine of y plus two,

00:00:02.875 --> 00:00:04.319
this whole thing times the derivative

00:00:04.319 --> 00:00:07.140
of y with respect to x is equal to two x,

00:00:07.140 --> 00:00:09.282
and we're given that for
a particular solution,

00:00:09.282 --> 00:00:12.364
when x is equal to one, y
of one is equal to zero,

00:00:12.364 --> 00:00:15.982
and we're asked what is
x when y is equal to pi?

00:00:15.982 --> 00:00:17.410
So the first thing I like to look at

00:00:17.410 --> 00:00:20.317
when I see a differential
equation is: is it separable?

00:00:20.317 --> 00:00:23.123
Can I get all the ys and dys on one side,

00:00:23.123 --> 00:00:26.426
and can I get all the xs
and dxs on the other side?

00:00:26.426 --> 00:00:28.165
And this one seems like it is.

00:00:28.165 --> 00:00:30.748
If I multiply both sides by dx,

00:00:31.658 --> 00:00:34.986
where you can view dx
as the x differential

00:00:34.986 --> 00:00:37.177
of an infinitely small change in x,

00:00:37.177 --> 00:00:39.677
well, then you get cosine of y

00:00:41.543 --> 00:00:45.543
plus two times dy is
equal to two x times dx,

00:00:49.487 --> 00:00:51.529
so just like that I would've
been able to, all I did is

00:00:51.529 --> 00:00:54.565
I multiplied both sides of this times dx,

00:00:54.565 --> 00:00:56.545
and I was able to separate
the ys and the dys

00:00:56.545 --> 00:01:01.483
from the xs and the dxs, and
now I can integrate both sides.

00:01:01.483 --> 00:01:05.726
So, if I integrate both
sides, what am I going to get?

00:01:05.726 --> 00:01:09.893
So the antiderivative of
cosine of y with respect to y,

00:01:11.325 --> 00:01:14.566
with respect to y, is sine of y,

00:01:14.566 --> 00:01:19.250
and then the antiderivative
of 2 with respect to y is 2y,

00:01:19.250 --> 00:01:22.475
and that is going to be equal to,

00:01:22.475 --> 00:01:26.536
well, the antiderivative
of 2x with respect to x is

00:01:26.536 --> 00:01:30.442
x squared, and we can't forget
that we could say a plus

00:01:30.442 --> 00:01:32.582
a different constant on either side,

00:01:32.582 --> 00:01:36.558
but it serves our purpose just
to say plus c on one side,

00:01:36.558 --> 00:01:38.940
and so this is a general solution

00:01:38.940 --> 00:01:40.663
to the separable differential equation,

00:01:40.663 --> 00:01:43.938
and then we can find the
particular one by substituting in

00:01:43.938 --> 00:01:47.445
when x is equal to one,
y is equal to zero,

00:01:47.445 --> 00:01:50.145
so let's do that to solve for c.

00:01:50.145 --> 00:01:54.236
So we get, or when y is equal
to zero, x is equal to one,

00:01:54.236 --> 00:01:57.319
so, sine of zero plus two times zero,

00:01:59.898 --> 00:02:02.720
all I did is I substituted
in the zero for y,

00:02:02.720 --> 00:02:07.071
is equal to x squared, well now x is one,

00:02:07.071 --> 00:02:09.959
is equal to one squared plus c.

00:02:09.959 --> 00:02:12.023
Well sine of zero is zero,
two times zero is zero,

00:02:12.023 --> 00:02:13.770
all of that's just gonna be zero,

00:02:13.770 --> 00:02:17.660
so we get zero is equal to one plus c,

00:02:17.660 --> 00:02:20.023
or c is equal to negative one.

00:02:20.023 --> 00:02:21.966
So now we can write down
the particular solution

00:02:21.966 --> 00:02:24.577
to this differential equation
that meets these conditions.

00:02:24.577 --> 00:02:26.731
So we get, let me write it over here,

00:02:26.731 --> 00:02:29.398
sine of y plus two y is equal to

00:02:31.529 --> 00:02:36.475
x squared, and our constant
is negative one, so minus one.

00:02:36.475 --> 00:02:39.984
And now what is x when y is equal to pi?

00:02:39.984 --> 00:02:43.651
So, sine of pi plus two
times pi is equal to

00:02:46.017 --> 00:02:50.184
x squared minus one, see
sine of pi is equal to zero,

00:02:52.189 --> 00:02:56.131
and so we get, let's see we
can add one to both sides,

00:02:56.131 --> 00:03:00.298
and we get two pi plus
one is equal to x squared,

00:03:01.895 --> 00:03:04.943
or we could say that x is
equal to the plus or minus

00:03:04.943 --> 00:03:07.860
the square root of two pi plus one.

00:03:09.036 --> 00:03:11.774
So I would write the
plus or minus square root

00:03:11.774 --> 00:03:14.691
of two pi plus one, and we're done.


Worked example: identifying separable equations

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.337
- [Sal] Which of the differential
equations are separable?

00:00:02.337 --> 00:00:03.491
And I encourage you to pause this video

00:00:03.491 --> 00:00:06.458
and see which of these
are actually separable.

00:00:06.458 --> 00:00:08.813
Now, the way that I approach this

00:00:08.813 --> 00:00:11.735
is I try to solve for the derivative,

00:00:11.735 --> 00:00:13.956
and if when I solve for the derivative,

00:00:13.956 --> 00:00:15.206
if I get dy, dx

00:00:16.319 --> 00:00:19.003
is equal to some function of y

00:00:19.003 --> 00:00:22.087
times some other function of x

00:00:22.087 --> 00:00:23.901
then I say okay, this is separable.

00:00:23.901 --> 00:00:25.789
'Cause I could rewrite this as,

00:00:25.789 --> 00:00:27.882
I could divide both sides by g of y,

00:00:27.882 --> 00:00:29.430
and I get one over g of y,

00:00:29.430 --> 00:00:31.658
which is itself a function of y,

00:00:31.658 --> 00:00:33.325
times dy is equal to

00:00:34.829 --> 00:00:35.912
h of x dx.

00:00:37.849 --> 00:00:39.903
You would go from this first equation

00:00:39.903 --> 00:00:42.111
to the second equation just by dividing

00:00:42.111 --> 00:00:43.835
both sides by g of y

00:00:43.835 --> 00:00:45.819
and multiplying both sides by dx.

00:00:45.819 --> 00:00:47.916
And then it's clear you
have a separable equation

00:00:47.916 --> 00:00:49.931
you can integrate both sides.

00:00:49.931 --> 00:00:52.736
But the key is let's
solve for the derivative

00:00:52.736 --> 00:00:54.348
and see if we can put this in a form

00:00:54.348 --> 00:00:56.960
where we have the product
of a function of y

00:00:56.960 --> 00:00:59.349
times a function of x.

00:00:59.349 --> 00:01:01.568
So let's do it with this first one here.

00:01:01.568 --> 00:01:04.138
So let's see, if I
subtract y from both sides,

00:01:04.138 --> 00:01:05.609
I'm just trying to solve for the

00:01:05.609 --> 00:01:07.071
derivative of y with respect to x.

00:01:07.071 --> 00:01:09.154
I'm gonna get x times,

00:01:11.489 --> 00:01:12.457
I'll write y prime

00:01:12.457 --> 00:01:15.086
is the derivative of y with respect to x,

00:01:15.086 --> 00:01:17.570
is equal to 3 minus y.

00:01:17.570 --> 00:01:19.942
So I subtracted y from both sides.

00:01:19.942 --> 00:01:22.399
Let's see if I divide both sides by x,

00:01:22.399 --> 00:01:24.840
I'm gonna get the derivative of y

00:01:24.840 --> 00:01:27.037
with respect to x is equal to,

00:01:27.037 --> 00:01:28.057
actually I'm gonna write it this way.

00:01:28.057 --> 00:01:32.138
I'm gonna write it 3
minus y times 1 over x.

00:01:32.138 --> 00:01:33.416
And so it's clear I'm able to write

00:01:33.416 --> 00:01:34.457
the derivative as a product of

00:01:34.457 --> 00:01:36.734
a function of y, and a function of x.

00:01:36.734 --> 00:01:38.309
So this indeed is separable.

00:01:38.309 --> 00:01:39.142
And I could show you,

00:01:39.142 --> 00:01:41.420
I could multiply both sides by dx

00:01:41.420 --> 00:01:44.423
and I could divide both
sides by 3 minus y now,

00:01:44.423 --> 00:01:47.340
and I would get 1 over 3 minus y dy

00:01:48.643 --> 00:01:50.893
is equal to 1 over x dx.

00:01:52.415 --> 00:01:56.194
So clearly, this one right
over here is separable.

00:01:56.194 --> 00:01:57.288
Now let's do the second one.

00:01:57.288 --> 00:01:58.595
And I'm gonna just do the same technique,

00:01:58.595 --> 00:01:59.956
I'll do it in a slightly different colour

00:01:59.956 --> 00:02:02.622
so we don't get all of our
math all jumbled together.

00:02:02.622 --> 00:02:04.824
So in this second one,

00:02:04.824 --> 00:02:07.803
let's see, if I subtract the 2x,

00:02:07.803 --> 00:02:09.656
the 2y from both sides,

00:02:09.656 --> 00:02:11.392
so actually, lemme just do, whoops,

00:02:11.392 --> 00:02:13.569
lemme do a couple things at once.

00:02:13.569 --> 00:02:16.635
I'm gonna subtract 2x from both sides.

00:02:16.635 --> 00:02:20.581
I am going to subtract 2y from both sides.

00:02:20.581 --> 00:02:23.664
So I'm gonna subtract 2y from both sides.

00:02:23.664 --> 00:02:26.331
I'm gonna add one to both sides,

00:02:28.112 --> 00:02:29.779
so I'm gonna add one

00:02:31.249 --> 00:02:32.416
to both sides.

00:02:33.519 --> 00:02:37.097
And then what am I going
to get if I do that?

00:02:37.097 --> 00:02:37.998
This is gonna be zero,

00:02:37.998 --> 00:02:38.960
this is gonna be zero,

00:02:38.960 --> 00:02:40.486
this is gonna be zero.

00:02:40.486 --> 00:02:44.346
I'm gonna have 2 times the derivative of y

00:02:44.346 --> 00:02:47.013
with respect to x is equal to

00:02:48.085 --> 00:02:49.502
negative 2x minus

00:02:50.434 --> 00:02:51.267
2y plus 1.

00:02:53.580 --> 00:02:56.377
And now see I can divide
everything by two;

00:02:56.377 --> 00:02:58.440
I would get the derivative of y

00:02:58.440 --> 00:03:00.905
with respect to x is equal to,

00:03:00.905 --> 00:03:02.385
and actually, yeah I would get,

00:03:02.385 --> 00:03:03.872
I'm just gonna divide by two,

00:03:03.872 --> 00:03:07.111
so I'm gonna get negative x minus y

00:03:07.111 --> 00:03:09.292
and then I'm going to get

00:03:09.292 --> 00:03:10.959
plus 1/2.

00:03:11.824 --> 00:03:13.443
So it's not obvious to me

00:03:13.443 --> 00:03:15.182
how I can write this as a product

00:03:15.182 --> 00:03:18.682
of a function of x and a function of y.

00:03:19.626 --> 00:03:21.793
So this one does not feel,

00:03:22.905 --> 00:03:25.154
this one right over here is not separable.

00:03:25.154 --> 00:03:26.985
I don't know how to write this

00:03:26.985 --> 00:03:30.288
as a function of x times a function of y,

00:03:30.288 --> 00:03:31.551
so this one I'm not gonna,

00:03:31.551 --> 00:03:33.792
I'm gonna say is not separable.

00:03:33.792 --> 00:03:36.565
Now this one, they've
already written it for us

00:03:36.565 --> 00:03:39.031
as a function of x times a function of y.

00:03:39.031 --> 00:03:42.230
So this one is clearly
separable right over here

00:03:42.230 --> 00:03:44.015
and if you want me to do the separating

00:03:44.015 --> 00:03:45.729
I can rewrite this as,

00:03:45.729 --> 00:03:47.312
well this is dy dx.

00:03:48.708 --> 00:03:50.944
If I multiply both sides by dx

00:03:50.944 --> 00:03:53.581
and divide both sides
by this right over here,

00:03:53.581 --> 00:03:56.914
I would get one over y squared plus y dy

00:03:59.316 --> 00:04:02.483
is equal to x squared plus x dx.

00:04:04.406 --> 00:04:06.698
So clearly separable.

00:04:06.698 --> 00:04:09.015
Alright now this last choice,

00:04:09.015 --> 00:04:09.848
this is interesting,

00:04:09.848 --> 00:04:11.878
they've essentially
distributed the derivative

00:04:11.878 --> 00:04:12.987
right over here.

00:04:12.987 --> 00:04:15.638
So let's see, if we were to unfactor

00:04:15.638 --> 00:04:18.767
the derivative, I'm just
gonna solve for dy dx.

00:04:18.767 --> 00:04:20.849
So I'm gonna factor it out.

00:04:20.849 --> 00:04:22.432
I'm gonna get dy dx

00:04:24.306 --> 00:04:26.476
times x plus y,

00:04:26.476 --> 00:04:27.309
x plus y,

00:04:28.562 --> 00:04:30.276
is equal to x.

00:04:30.276 --> 00:04:32.953
Now if I were to divide both sides

00:04:32.953 --> 00:04:35.870
by x plus y, I'm gonna get dy dx

00:04:36.990 --> 00:04:39.823
is equal to x over x plus y.

00:04:41.124 --> 00:04:43.624
And here my algebraic tool kit

00:04:44.564 --> 00:04:46.199
of how do I separate x and y

00:04:46.199 --> 00:04:47.979
so I can write this as a function of x

00:04:47.979 --> 00:04:49.577
times a function of y,

00:04:49.577 --> 00:04:50.912
not obvious to me here.

00:04:50.912 --> 00:04:53.543
So this one is not separable.

00:04:53.543 --> 00:04:55.865
So only the first one and the third one.


The Jacobian matrix

WEBVTT
Kind: captions
Language: en

00:00:00.794 --> 00:00:01.666
- [Narrator] In the last video we were

00:00:01.666 --> 00:00:03.543
looking at this particular function.

00:00:03.543 --> 00:00:04.977
It's a very non linear function.

00:00:04.977 --> 00:00:06.960
And we were picturing
it as a transformation

00:00:06.960 --> 00:00:10.641
that takes every point x,
y in space to the point

00:00:10.641 --> 00:00:13.401
x plus sign y, y plus sign of x.

00:00:13.401 --> 00:00:16.634
And moreover, we zoomed
in on a specific point.

00:00:16.634 --> 00:00:17.731
And let me actually write down what

00:00:17.731 --> 00:00:20.958
point we zoomed in on, it was (-2,1).

00:00:20.958 --> 00:00:25.125
That's something we're gonna
want to record here (-2,1).

00:00:25.978 --> 00:00:27.948
And I added couple extra
grid lines around it

00:00:27.948 --> 00:00:30.531
just so we can see in detail
what the transformation

00:00:30.531 --> 00:00:33.803
does to points that are in the
neighborhood of that point.

00:00:33.803 --> 00:00:35.732
And over here, this
square shows the zoomed

00:00:35.732 --> 00:00:37.773
in version of that neighborhood.

00:00:37.773 --> 00:00:39.853
And what we saw is that even though the

00:00:39.853 --> 00:00:41.716
function as a whole, as a transformation,

00:00:41.716 --> 00:00:44.565
looks rather complicated,
around that one point,

00:00:44.565 --> 00:00:47.426
it looks like a linear function.

00:00:47.426 --> 00:00:49.706
It's locally linear so
what I'll show you here

00:00:49.706 --> 00:00:52.907
is what matrix is gonna
tell you the linear

00:00:52.907 --> 00:00:55.131
function that this looks like.

00:00:55.131 --> 00:00:57.730
And this is gonna be kind
of two by two matrix.

00:00:57.730 --> 00:01:00.347
I'll make a lot of room
for ourselves here.

00:01:00.347 --> 00:01:02.788
It'll be a two by two matrix and the way

00:01:02.788 --> 00:01:05.658
to think about it is to first go back

00:01:05.658 --> 00:01:07.823
to our original setup
before the transformation.

00:01:07.823 --> 00:01:10.334
And think of just a
tiny step to the right.

00:01:10.334 --> 00:01:13.576
What I'm gonna think of
as a little, partial x.

00:01:13.576 --> 00:01:16.118
A tiny step in the x direction.

00:01:16.118 --> 00:01:18.301
And what that turns into
after the transformation

00:01:18.301 --> 00:01:21.733
is gonna be some tiny
step in the output space.

00:01:21.733 --> 00:01:23.457
And here let me actually kind of draw on

00:01:23.457 --> 00:01:25.743
what that tiny step turned into.

00:01:25.743 --> 00:01:27.150
It's no longer purely in the x direction.

00:01:27.150 --> 00:01:28.486
It has some rightward component.

00:01:28.486 --> 00:01:30.357
But now also some downward component.

00:01:30.357 --> 00:01:32.620
And to be able to represent
this in a nice way,

00:01:32.620 --> 00:01:35.366
what I'm gonna do is
instead of writing the

00:01:35.366 --> 00:01:37.137
entire function as something with

00:01:37.137 --> 00:01:39.081
a vector valued output, I'm gonna go ahead

00:01:39.081 --> 00:01:43.248
and represent this as a two
separate scalar value functions.

00:01:44.116 --> 00:01:48.283
I'm gonna write the scalar
value functions f1 of x, y.

00:01:50.073 --> 00:01:52.964
So I'm just giving a
name to x plus sign y.

00:01:52.964 --> 00:01:56.083
And f2 of x, y, again all I'm doing is

00:01:56.083 --> 00:02:00.348
giving a name to the functions
we already have written down.

00:02:00.348 --> 00:02:02.296
When I look at this vector, the

00:02:02.296 --> 00:02:04.334
consequence of taking a tiny d, x step

00:02:04.334 --> 00:02:06.812
in the input space that corresponds to

00:02:06.812 --> 00:02:09.423
some two d movement in the output space.

00:02:09.423 --> 00:02:11.189
And the x component of that movement.

00:02:11.189 --> 00:02:12.896
Right if I was gonna draw this out

00:02:12.896 --> 00:02:15.976
and say hey, what's the x
component of that movement.

00:02:15.976 --> 00:02:18.071
That's something we think of as a little

00:02:18.071 --> 00:02:22.238
partial change in f1, the
x component of our output.

00:02:23.179 --> 00:02:24.560
And if we divide this, if we take you know

00:02:24.560 --> 00:02:26.913
partial f1 divided by the size of that

00:02:26.913 --> 00:02:28.717
initial tiny change, it basically scales

00:02:28.717 --> 00:02:31.408
it up to be a normal sized vector.

00:02:31.408 --> 00:02:32.938
Not a tiny nudge but something that's more

00:02:32.938 --> 00:02:34.480
constant that doesn't shrink as we

00:02:34.480 --> 00:02:36.617
zoom in further and further.

00:02:36.617 --> 00:02:39.149
And then similarly the
change in the y direction,

00:02:39.149 --> 00:02:40.784
right the vertical component of that step

00:02:40.784 --> 00:02:43.185
that was still caused by the dx.

00:02:43.185 --> 00:02:45.493
Right, it's still caused by that initial

00:02:45.493 --> 00:02:46.933
step to the right, that is gonna be

00:02:46.933 --> 00:02:49.516
the tiny, partial change in f2.

00:02:50.607 --> 00:02:52.667
The y component of the output cause

00:02:52.667 --> 00:02:54.527
here we're all just
looking in the output space

00:02:54.527 --> 00:02:58.694
that was caused by a partial
change in the x direction.

00:03:00.749 --> 00:03:02.220
And again I kind of
like to think about this

00:03:02.220 --> 00:03:03.957
we're dividing by a tiny amount.

00:03:03.957 --> 00:03:07.065
This partial f2 is really
a tiny, tiny nudge.

00:03:07.065 --> 00:03:08.847
But by dividing by the size of the initial

00:03:08.847 --> 00:03:11.210
tiny nudge that caused it, we're getting

00:03:11.210 --> 00:03:12.483
something that's basically a number.

00:03:12.483 --> 00:03:13.471
Something that doesn't shrink when

00:03:13.471 --> 00:03:15.592
we consider more and
more zoomed in versions.

00:03:15.592 --> 00:03:17.517
So that, that's all what happens when

00:03:17.517 --> 00:03:19.922
we take a tiny step in the x direction.

00:03:19.922 --> 00:03:23.066
But another thing you could
do, another thing you can

00:03:23.066 --> 00:03:25.960
consider is a tiny step
in the y direction.

00:03:25.960 --> 00:03:27.835
Right cause we wanna know, hey, if

00:03:27.835 --> 00:03:31.153
you take a single step
some tiny unit upward,

00:03:31.153 --> 00:03:35.320
what does that turn into
after the transformation.

00:03:37.644 --> 00:03:41.584
And what that looks like is this vector

00:03:41.584 --> 00:03:43.398
that still has some upward component.

00:03:43.398 --> 00:03:45.328
But it also has a rightward component.

00:03:45.328 --> 00:03:46.413
And now I'm gonna write its components

00:03:46.413 --> 00:03:48.815
as the second column of the matrix.

00:03:48.815 --> 00:03:50.092
Because as we know when
you're representing

00:03:50.092 --> 00:03:52.439
a linear transformation with a matrix,

00:03:52.439 --> 00:03:54.012
the first column tells you where the first

00:03:54.012 --> 00:03:55.596
basis vector goes and the second column

00:03:55.596 --> 00:03:58.195
shows where the second basis vector goes.

00:03:58.195 --> 00:03:59.756
If that feels unfamiliar, either

00:03:59.756 --> 00:04:01.245
check out the refresher video or

00:04:01.245 --> 00:04:03.959
maybe go and look at some of
the linear algebra content.

00:04:03.959 --> 00:04:06.017
But to figure out the
coordinates of this guy,

00:04:06.017 --> 00:04:08.085
we do basically the same thing.

00:04:08.085 --> 00:04:10.671
Let's say first of all, the
change in the x direction

00:04:10.671 --> 00:04:14.682
here, the x component
of this nudge vector.

00:04:14.682 --> 00:04:18.921
That's gonna be given as a
partial change to f1, right,

00:04:18.921 --> 00:04:21.077
to the x component of the output.

00:04:21.077 --> 00:04:22.271
Here we're looking in the outputs base.

00:04:22.271 --> 00:04:25.241
We're dealing with f1, f1 and f2

00:04:25.241 --> 00:04:26.713
and we're asking what that change was

00:04:26.713 --> 00:04:30.880
that was caused by a tiny
change in the y direction.

00:04:32.140 --> 00:04:35.315
So the change in f1 caused
by some tiny step in the y

00:04:35.315 --> 00:04:38.981
direction divided by the
size of that tiny step.

00:04:38.981 --> 00:04:41.811
And then the y component
of our output here.

00:04:41.811 --> 00:04:43.891
The y component of the
step in the outputs base

00:04:43.891 --> 00:04:45.888
that was caused by the initial tiny

00:04:45.888 --> 00:04:47.876
step upward in the input space.

00:04:47.876 --> 00:04:50.376
Well that is the change of f2,

00:04:52.343 --> 00:04:55.537
second component of our
output as caused by dy.

00:04:55.537 --> 00:04:58.953
As caused by that little partial y.

00:04:58.953 --> 00:05:00.477
And of course all of this is very specific

00:05:00.477 --> 00:05:02.545
to the point that we started at right.

00:05:02.545 --> 00:05:05.447
We started at the point (-2,1).

00:05:05.447 --> 00:05:07.527
So each of these partial derivatives

00:05:07.527 --> 00:05:09.287
is something that really we're saying,

00:05:09.287 --> 00:05:12.714
don't take the function,
evaluate it at the point (2,-1),

00:05:12.714 --> 00:05:16.506
and when you evaluate each one of these

00:05:16.506 --> 00:05:18.900
at the point (2,-1)
you'll get some number.

00:05:18.900 --> 00:05:19.972
And that will give you a very

00:05:19.972 --> 00:05:21.753
concrete two by two matrix that's gonna

00:05:21.753 --> 00:05:24.065
represent the linear
transformation that this

00:05:24.065 --> 00:05:26.947
guy looks like once you've zoomed in.

00:05:26.947 --> 00:05:28.268
So this matrix here that's full of all

00:05:28.268 --> 00:05:31.582
of the partial derivatives
has a very special name.

00:05:31.582 --> 00:05:35.553
It's called as you may
have guessed, the Jacobian.

00:05:35.553 --> 00:05:38.775
Or more fully you'd call
it the Jacobian Matrix.

00:05:38.775 --> 00:05:39.980
And one way to think about it is that it

00:05:39.980 --> 00:05:43.186
carries all of the partial
differential information right.

00:05:43.186 --> 00:05:45.489
It's taking into account
both of these components

00:05:45.489 --> 00:05:48.069
of the output and both possible inputs.

00:05:48.069 --> 00:05:49.240
And giving you a kind of a grid of

00:05:49.240 --> 00:05:51.194
what all the partial derivatives are.

00:05:51.194 --> 00:05:52.260
But as I hope you see, it's much

00:05:52.260 --> 00:05:54.526
more than just a way of recording

00:05:54.526 --> 00:05:56.402
what all the partial derivatives are.

00:05:56.402 --> 00:05:57.893
There's a reason for organizing it

00:05:57.893 --> 00:05:59.786
like this in particular and it really

00:05:59.786 --> 00:06:02.439
does come down to this
idea of local linearity.

00:06:02.439 --> 00:06:04.541
If you understand that the Jacobian Matrix

00:06:04.541 --> 00:06:06.148
is fundamentally supposed to represent

00:06:06.148 --> 00:06:08.689
what a transformation
looks like when you zoom

00:06:08.689 --> 00:06:11.462
in near a specific point,
almost everything else

00:06:11.462 --> 00:06:13.999
about it will start to fall in place.

00:06:13.999 --> 00:06:14.981
And in the next video, I'll go ahead

00:06:14.981 --> 00:06:16.429
and actually compute this just to

00:06:16.429 --> 00:06:18.135
show you what the process looks like.

00:06:18.135 --> 00:06:19.383
And how the result we get kind of

00:06:19.383 --> 00:06:20.760
matches with the picture we're

00:06:20.760 --> 00:06:21.593
looking at, see you then.


Local linearity for a multivariable function

WEBVTT
Kind: captions
Language: en

00:00:00.262 --> 00:00:01.420
- [Tutor] So, a lot of the concepts

00:00:01.420 --> 00:00:03.784
that you learn about in
multi-variable calculus,

00:00:03.784 --> 00:00:06.082
are really all about
ideas that you originally

00:00:06.082 --> 00:00:08.313
might've learned in linear algebra,

00:00:08.313 --> 00:00:11.812
and then transferring those to
apply to nonlinear problems.

00:00:11.812 --> 00:00:14.349
So for example, I'm gonna
give you a function,

00:00:14.349 --> 00:00:18.494
some kind of function that
takes in a 2D vector, xy,

00:00:18.494 --> 00:00:21.806
and it's also going to
spit out a 2D vector,

00:00:21.806 --> 00:00:23.879
and the specific one I have in mind,

00:00:23.879 --> 00:00:28.046
this is just kind of
arbitrary, is x plus sin of y,

00:00:29.729 --> 00:00:31.751
and then, because I'm
a sucker for symmetry,

00:00:31.751 --> 00:00:34.501
I'm gonna make it y plus sin of x.

00:00:34.501 --> 00:00:36.677
Though of course, this could
be any arbitrary function,

00:00:36.677 --> 00:00:39.108
you don't need that kind of symmetry.

00:00:39.108 --> 00:00:40.997
So, in the last video, I
gave a little refresher

00:00:40.997 --> 00:00:43.229
on how to think about
linear transformations

00:00:43.229 --> 00:00:45.056
and ideas from linear algebra,

00:00:45.056 --> 00:00:47.964
and how you encode a linear
transformation using a matrix,

00:00:47.964 --> 00:00:50.409
and kind of visualize it, I use this grid,

00:00:50.409 --> 00:00:53.247
and here, I wanna show what
this function looks like

00:00:53.247 --> 00:00:55.413
as a transformation of space.

00:00:55.413 --> 00:00:57.092
As in, I'm gonna tell the computer,

00:00:57.092 --> 00:01:00.080
take every single point
on this blue grid here,

00:01:00.080 --> 00:01:02.187
and if that point is xy, I want you

00:01:02.187 --> 00:01:04.956
to move it over to the
point x plus sin of y,

00:01:04.956 --> 00:01:06.642
y plus sin of x.

00:01:06.642 --> 00:01:09.914
And here's what that looks like.

00:01:09.914 --> 00:01:13.046
Alright, so things get
really wavy, really curly,

00:01:13.046 --> 00:01:15.459
this is not at all a
linear transformation,

00:01:15.459 --> 00:01:17.301
all of the lines don't remain lines,

00:01:17.301 --> 00:01:19.832
they're no longer nice grid
lines that are parallel

00:01:19.832 --> 00:01:22.058
and evenly spaced, in some sense,

00:01:22.058 --> 00:01:24.031
there is much, much more information

00:01:24.031 --> 00:01:25.654
that goes into nonlinear functions

00:01:25.654 --> 00:01:27.810
than into linear functions.

00:01:27.810 --> 00:01:29.383
And because this is rather complicated,

00:01:29.383 --> 00:01:31.591
I think it might be easy
to see what's going on

00:01:31.591 --> 00:01:34.224
if we just focus on a
single individual point.

00:01:34.224 --> 00:01:38.032
So let me look at a point
like, hmm, let's say,

00:01:38.032 --> 00:01:40.615
pi/2 and zero, okay?

00:01:43.051 --> 00:01:46.164
So if that's what I'm
plugging in, x is pi/2,

00:01:46.164 --> 00:01:50.495
so at the top here, x
stays the same, it's pi/2,

00:01:50.495 --> 00:01:52.684
and then sin of y would be sin of zero,

00:01:52.684 --> 00:01:57.035
so that x component is gonna
completely stay the same.

00:01:57.035 --> 00:02:00.747
And then for the bottom,
y, or y is also zero,

00:02:00.747 --> 00:02:04.391
plus sin of x, sin of
pi/2 is one, and just,

00:02:04.391 --> 00:02:06.755
I'll go ahead and write sin of pi/2,

00:02:06.755 --> 00:02:10.922
sin of pi/2, but you can think
of that as just being one.

00:02:14.080 --> 00:02:16.054
So what that means on the
transformation over here,

00:02:16.054 --> 00:02:18.419
is if we look at the point
that's at pi/2, zero,

00:02:18.419 --> 00:02:21.098
and pi/2 is a little about 1.5,

00:02:21.098 --> 00:02:22.612
so that's gonna be around here,

00:02:22.612 --> 00:02:25.807
we expect it to move
to the point pi/2, one.

00:02:25.807 --> 00:02:28.785
So it should just move
vertically by one unit,

00:02:28.785 --> 00:02:30.725
and if you just focus on that one point

00:02:30.725 --> 00:02:33.624
during the transformation,
notice that's exactly

00:02:33.624 --> 00:02:37.207
what happens, it just
moves vertically one point.

00:02:37.207 --> 00:02:38.525
And of course, things
are quite complicated

00:02:38.525 --> 00:02:40.237
because every point is doing that, right?

00:02:40.237 --> 00:02:41.789
The computer's taking in every point

00:02:41.789 --> 00:02:43.580
and moving it to where it should go.

00:02:43.580 --> 00:02:45.201
So, after having given the refresher

00:02:45.201 --> 00:02:46.981
on thinking about linear transformations

00:02:46.981 --> 00:02:49.613
and encoding them with matrices last time,

00:02:49.613 --> 00:02:52.763
something like this might
feel completely intractable.

00:02:52.763 --> 00:02:54.729
You certainly have to
store much more information

00:02:54.729 --> 00:02:58.510
than just four numbers to
record where everything goes.

00:02:58.510 --> 00:03:00.304
But this function has a nice property,

00:03:00.304 --> 00:03:01.865
a property that we deal with all the time

00:03:01.865 --> 00:03:03.570
in multi-variable calculus.

00:03:03.570 --> 00:03:07.119
It's what we call locally linear.

00:03:07.119 --> 00:03:08.369
Locally linear.

00:03:10.236 --> 00:03:13.855
And what that means is if I
was to take our initial setup,

00:03:13.855 --> 00:03:15.569
and then zoom in on a given point,

00:03:15.569 --> 00:03:18.101
so I'm gonna zoom in around
this point on the left here,

00:03:18.101 --> 00:03:20.229
and this box, kind of in the upper right,

00:03:20.229 --> 00:03:22.380
just shows the zoomed-in version of that.

00:03:22.380 --> 00:03:24.880
And first of all, I'm gonna
add some more grid lines,

00:03:24.880 --> 00:03:26.507
so they're really very close grid lines,

00:03:26.507 --> 00:03:28.421
we can see from the zoomed-out picture,

00:03:28.421 --> 00:03:30.481
but this just makes it so
that when we're zoomed in,

00:03:30.481 --> 00:03:32.954
we can see a little bit
more of what's going on.

00:03:32.954 --> 00:03:34.956
And now, when I play the animation,

00:03:34.956 --> 00:03:37.642
I'm gonna have this yellow
box that's doing the zooming

00:03:37.642 --> 00:03:41.309
follow the point at its center,
so this box will be moving,

00:03:41.309 --> 00:03:44.099
and we're always just gonna
look at what it zoomed in on,

00:03:44.099 --> 00:03:44.932
'kay?

00:03:44.932 --> 00:03:47.693
So it's gonna be following
what's going on around that point

00:03:47.693 --> 00:03:49.633
during the transformation.

00:03:49.633 --> 00:03:52.424
And we can see, inside
this zoomed version,

00:03:52.424 --> 00:03:55.118
it's still not linear, the
lines get a little bit curved,

00:03:55.118 --> 00:03:57.671
but this looks a lot more
like a linear function,

00:03:57.671 --> 00:03:59.416
it looks a lot more like the grid lines

00:03:59.416 --> 00:04:01.643
that started off horizontal and vertical,

00:04:01.643 --> 00:04:04.230
are remaining parallel and evenly spaced.

00:04:04.230 --> 00:04:07.096
And in fact, let's say
I zoom in even further,

00:04:07.096 --> 00:04:09.542
to an even smaller yellow
box here, and again,

00:04:09.542 --> 00:04:12.016
I'm gonna add in some more
grid lines right around it,

00:04:12.016 --> 00:04:13.373
so they're very, very densely packed.

00:04:13.373 --> 00:04:16.405
And this is purely an artifact
of visualizing things,

00:04:16.405 --> 00:04:18.634
I could choose to put
points or lines or anything

00:04:18.634 --> 00:04:21.224
wherever I want, and I just
think showing the grid lines

00:04:21.224 --> 00:04:23.066
and only the grid lines
and where they move,

00:04:23.066 --> 00:04:25.366
gives sort of a feel for
what the function is doing.

00:04:25.366 --> 00:04:28.613
So this time, when I play
it, and that zooming-in box

00:04:28.613 --> 00:04:31.949
kind of tracks the point
that we're looking at,

00:04:31.949 --> 00:04:35.143
as it goes, the neighborhood around it,

00:04:35.143 --> 00:04:37.567
all of the points around
it, really, really do look

00:04:37.567 --> 00:04:39.434
like a linear function.

00:04:39.434 --> 00:04:41.411
And the more you zoom
in, the more it looks

00:04:41.411 --> 00:04:44.133
precisely like a certain linear function.

00:04:44.133 --> 00:04:46.688
Oh, I guess I should've
written an r over here.

00:04:46.688 --> 00:04:47.893
Locally linear.

00:04:47.893 --> 00:04:50.401
So this raises the question,
if we're looking around

00:04:50.401 --> 00:04:54.568
some specific point, which I'll
call x nought and y nought,

00:04:56.224 --> 00:04:58.711
this should correspond in some way

00:04:58.711 --> 00:05:01.256
to the linear transformation
that it looks like around it.

00:05:01.256 --> 00:05:03.305
There should be some kind of matrix,

00:05:03.305 --> 00:05:05.997
some two-by-two matrix, that represents

00:05:05.997 --> 00:05:08.747
the linear transformation
that this function,

00:05:08.747 --> 00:05:11.365
this much more complicated
function, looks like

00:05:11.365 --> 00:05:13.074
around that point.

00:05:13.074 --> 00:05:15.728
So this idea of zooming in
is what we mean by local,

00:05:15.728 --> 00:05:17.390
and in the next video, I'm gonna show you

00:05:17.390 --> 00:05:20.693
what this matrix looks like in
terms of partial derivatives

00:05:20.693 --> 00:05:21.967
for our original function.

00:05:21.967 --> 00:05:22.800
See you then!


Jacobian prerequisite knowledge

WEBVTT
Kind: captions
Language: en

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- Hello, everyone. In
these next few videos,

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I'm going to be talking
about something called,

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the Jacobian,

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and more specifically,
it's the Jacobian matrix,

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or sometimes the associated determinant,

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and here, I just want to talk

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about some of the background
knowledge that I'm assuming,

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because to understand the Jacobian,

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you do have to have a little bit

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of a background in linear
algebra, and in particular,

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I want to make sure that
everyone here understands

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how to think about matrices
as transformations of space,

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When I say transformations,

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here, let me just get
kind of a matrix on here.

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I'll call it two one
and negative three one.

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You'll see why I'm coloring
it like this in just a moment.

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When I say, how to think about this

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as a transformation of space,

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I mean, you can multiply a matrix

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by some kind of two-dimensional vector,

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some kind of two-dimensional x y,

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and this is going to give us
a new two-dimensional vector.

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This is going to bring us
to, let's see in this case,

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it'll be,

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I'll write kind of two
one negative three one,

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where what it gives us

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is two x plus negative three times y,

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and then, one x plus one times y.

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Right? This is a new
two-dimensional vector

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somewhere else in space,

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and even if you know how to compute it,

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there's still room for a
deeper geometric understanding

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of what it actually means
to take a vector x y

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to the vector two x plus negative three y

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and one x plus one y.

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There's also still a deeper understanding

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in what we mean when we call
this a linear transformation,

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a linear transformation.

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What I'm going to do is just show you

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what this particular
transformation looks like

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on the left here, where every single point

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on this blue grid, I'm
going to tell the computer,

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"Hey, if that point was x
y, I want you to take it

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to two x plus negative
three y, one x plus one y.

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Here's what it looks like.

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Let me just kind of play it out here.

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All of the points in space move,

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and you end up in some final state here.

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There are a couple
important things to note.

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First of all, all of the grid lines

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remain parallel and evenly spaced,

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and they're still lines.

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They didn't get curved in some way,

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and that's very, very special.

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That is the geometric
way that you can think

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about this term, this idea
of a linear transformation.

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I kind of like to think about
it that lines stay lines,

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and in particular the grid lines here,

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the ones that started off

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as kind of vertical and horizontal,

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they still remain parallel,

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and they still remain evenly spaced.

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The other thing to notice here is

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I have these two vectors highlighted:

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the green vector and the red vector.

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These are the ones that started off,

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if we kind of back things up,

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these are the ones that started off

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as the basis vectors, right?

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Let me kind of make a
little bit more room here.

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The green vector is one zero,

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one in the x-direction,
zero in the y-direction,

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and then that red vertical vector here,

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is zero one,

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zero one.

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If we notice where they land,

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under this transformation,

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when the matrix is multiplied

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by every single vector in space,

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the place where the green vector lands,

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the one that started off as one zero,

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has coordinates two one,

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and that corresponds very
directly with the fact

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that the first column of
our matrix is two one.

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Then, similarly, over
here, the second vector,

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the one that started off as zero one,

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ends up at the coordinates
negative three one,

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and that's what corresponds with the fact

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that the next column
is negative three one.

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It's actually relatively simple to see

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why that's going to be true.

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Here, I'll go ahead and
multiply this matrix

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that we had that was ...

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See now it's kind of easy to remember

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what the matrix is, right?

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I can just kind of read it off here

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as two one negative three one,

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but just to see why it's
actually taking the basis vectors

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to the columns like this,

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when you do the
multiplication by one zero,

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notice how it's going to take us to,

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so it's two times one, that'll be two,

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and then, negative three times zero,

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so that'll just be zero,

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and over here, it's one times one,

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so that's one, and then, one times zero,

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so again we're adding zero.

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The only terms that actually matter

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because of the zero down here,

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was everything in that first column.

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Similarly, if we take that same matrix,

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two one negative three one,

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and we multiply it by zero one over here,

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by the second basis vector,

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what you're going to get,

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is two times zero, so zero,

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plus that element in that second column,

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and then, one times zero, so another zero,

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plus one times one, plus that one.

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Again, it's kind of like
that zero knocks out

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all of the terms in other columns.

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Then, like I said, geometrically,

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the meaning of a linear transformation

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is that grid lines remain
parallel and evenly spaced.

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When you start to think
about it a little bit,

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if you can know where
the screen vector lands

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and where the spread vector lands,

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that's going to lock into place

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where the entire grid has to go.

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Let me show you what I mean
and how this corresponds with,

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maybe, a different
definition that you've heard

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for what linear transformation means.

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If we have some kind of function L,

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and it's going to take in a
vector and spit out a vector,

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it's said to be linear if
it satisfies the property

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that when you take a
constant times a vector,

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what it produces is
that same constant times

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whatever would have
happened if you applied

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that transformation to the
vector, not scaled, right,

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so here you're applying
that transformation

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to a scaled vector,

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and evidently, that's the same

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as scaling the
transformation of the vector.

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Similarly, second property of linearity

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is that if you add two vectors,

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it doesn't really matter if you add them

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before or after the transformation.

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If you take the sum of the vectors

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then apply the transformation,

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that's the same as first
applying the transformation

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to each one separately,

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and then adding up the results.

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One of the most important consequences

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of this formal definition of linearity,

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is that it means if you take your function

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and apply it to some vector x y,

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I can split up that vector

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as x times the first basis vector,

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x times one zero

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plus y,

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let's see y, times that
second basis vector,

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zero one,

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and because of these two
properties of linearity,

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if I can split it up like this,

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it doesn't matter if I
do the scaling and adding

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before the transformation,

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or if I do that scaling and adding

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after the transformation,

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and say that it's x times
whatever the transformed version

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of one zero is,

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and I'll show you
geometrically what this means

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in just a moment, but
I kind of want to get

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all the algebra on the screen,

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plus y times the transformed version

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of zero one,

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zero one.

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To be concrete, let's actually put in

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a value for x and y here,

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and try to think about that
specific vector geometrically.

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Maybe I'll put in something
like vector two one.

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If we look over on the grid,

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we're going to be focusing
on the point that's over here

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at two one, and this particular point.

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I'm going to play the transformation,

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and I want you to follow this
point to see where it lands,

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and it's going to end up over here.

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Okay, so, in terms of the old grid, right,

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the original one that we started with,

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it's now at the point one three.

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This is where we've ended up,

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but importantly, I want you
to notice how it's still

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two times that green vector
plus one times that red vector.

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It's satisfying that
property that it's still

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x times whatever the transformed version

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of that first basis vector is,

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plus y times the transformed version

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of that second basis vector.

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That's all just a little overview,

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and the upshot,

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the main thing I want you
to remember from all of this

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is when you have some kind of matrix,

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you can think of it as a
transformation of space

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that keeps grid lines
parallel and evenly spaced.

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That's a very special
kind of transformation.

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That is a very restrictive
property to have

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on a function from 2-D
points to other 2-D points.

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The convenient way to encode that,

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is that the landing spot
for that first basis vector,

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the one that started off
one unit to the right,

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is represented with the
first column of the matrix,

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and the landing spot for
the second basis vector,

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the one that was pointing one unit up,

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is encoded with that second column.

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If this feels totally unfamiliar,

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or you want to learn more about this,

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it's something that I've made
other videos on in the past,

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but in terms of understanding
the Jacobian matrix,

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where we're going with this,

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and kind of getting a
geometric feel for it,

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that short overview that I gave

00:09:16.554 --> 00:09:19.457
should be enough to get us going.

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With that, I will see you next video.