Introduction to the convolution

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:04.180
In this video, I'm going to
introduce you to the concept

00:00:04.180 --> 00:00:09.950
of the convolution, one of the
first times a mathematician's

00:00:09.950 --> 00:00:11.810
actually named something
similar to what

00:00:11.810 --> 00:00:12.590
it's actually doing.

00:00:12.590 --> 00:00:14.590
You're actually convoluting
the functions.

00:00:14.590 --> 00:00:16.970
And in this video, I'm not
going to dive into the

00:00:16.970 --> 00:00:20.830
intuition of the convolution,
because there's a lot of

00:00:20.830 --> 00:00:21.770
different ways you
can look at it.

00:00:21.770 --> 00:00:23.430
It has a lot of different
applications, and if you

00:00:23.430 --> 00:00:26.320
become an engineer really of any
kind, you're going to see

00:00:26.320 --> 00:00:29.950
the convolution in kind of a
discrete form and a continuous

00:00:29.950 --> 00:00:31.620
form, and a bunch of
different ways.

00:00:31.620 --> 00:00:35.620
But in this video I just want
to make you comfortable with

00:00:35.620 --> 00:00:39.100
the idea of a convolution,
especially in the context of

00:00:39.100 --> 00:00:41.290
taking Laplace transforms.

00:00:41.290 --> 00:00:44.760
So the convolution theorem--
well, actually, before I even

00:00:44.760 --> 00:00:46.970
go to the convolution theorem,
let me define what a

00:00:46.970 --> 00:00:48.470
convolution is.

00:00:48.470 --> 00:00:52.100
So let's say that I have
some function f of t.

00:00:52.100 --> 00:00:58.983
So if I convolute f with g-- so
this means that I'm going

00:00:58.983 --> 00:01:02.250
to take the convolution of f and
g, and this is going to be

00:01:02.250 --> 00:01:04.220
a function of t.

00:01:04.220 --> 00:01:06.610
And so far, nothing I've written
should make any sense

00:01:06.610 --> 00:01:08.790
to you, because I haven't
defined what this means.

00:01:08.790 --> 00:01:11.100
This is like those SAT problems
where they say, like,

00:01:11.100 --> 00:01:18.060
you know, a triangle b means a
plus b over 3, while you're

00:01:18.060 --> 00:01:19.840
standing on one leg or
something like that.

00:01:19.840 --> 00:01:23.000
So I need to define this
in some similar way.

00:01:23.000 --> 00:01:26.455
So let me undo this silliness
that I just wrote there.

00:01:26.455 --> 00:01:28.960
And the definition of a
convolution, we're going to do

00:01:28.960 --> 00:01:32.200
it over a-- well, there's
several definitions you'll

00:01:32.200 --> 00:01:34.140
see, but the definition we're
going to use in this, context

00:01:34.140 --> 00:01:36.150
there's actually one other
definition you'll see in the

00:01:36.150 --> 00:01:45.290
continuous case, is the integral
from 0 to t of f of t

00:01:45.290 --> 00:01:56.195
minus tau, times g of t-- let me
just write it-- sorry, it's

00:01:56.195 --> 00:02:00.400
times g of tau d tau.

00:02:00.400 --> 00:02:04.970
Now, this might seem like a very
bizarro thing to do, and

00:02:04.970 --> 00:02:07.560
you're like, Sal, how
do I even compute

00:02:07.560 --> 00:02:08.520
one of these things?

00:02:08.520 --> 00:02:11.300
And to kind of give you that
comfort, let's actually

00:02:11.300 --> 00:02:12.550
compute a convolution.

00:02:15.790 --> 00:02:19.090
Actually, it was hard to find
some functions that are very

00:02:19.090 --> 00:02:21.345
easy to analytically compute,
and you're going to find that

00:02:21.345 --> 00:02:23.790
we're going to go into a lot of
trig identities to actually

00:02:23.790 --> 00:02:24.860
compute this.

00:02:24.860 --> 00:02:31.680
But if I say that f of t, if I
define f of t to be equal to

00:02:31.680 --> 00:02:36.980
the sine of t, and I define
cosine of t-- let me do it in

00:02:36.980 --> 00:02:43.810
orange-- or I define g of t to
be equal to the cosine of t.

00:02:43.810 --> 00:02:46.930
Now let's convolute
the two functions.

00:02:46.930 --> 00:02:54.310
So the convolution of f with g,
and this is going to be a

00:02:54.310 --> 00:02:59.000
function of t, it equals this.

00:02:59.000 --> 00:03:01.490
I'm just going to show you how
to apply this integral.

00:03:01.490 --> 00:03:05.040
So it equals the integral--
I'll do it in purple-- the

00:03:05.040 --> 00:03:09.120
integral from 0 to t of
f of t minus tau.

00:03:09.120 --> 00:03:10.090
This is my f of t.

00:03:10.090 --> 00:03:17.810
So it's is going to be sine of
t minus tau times g of tau.

00:03:17.810 --> 00:03:22.500
Well, this is my g of t, so
g of tau is cosine of tau,

00:03:22.500 --> 00:03:27.700
cosine of tau d tau.

00:03:27.700 --> 00:03:30.660
So that's the integral, and
now to evaluate it, we're

00:03:30.660 --> 00:03:33.000
going to have to break out
some trigonometry.

00:03:33.000 --> 00:03:34.760
So let's do that.

00:03:34.760 --> 00:03:37.100
This almost is just a very
good trigonometry and

00:03:37.100 --> 00:03:38.510
integration review.

00:03:38.510 --> 00:03:39.510
So let's evaluate this.

00:03:39.510 --> 00:03:41.470
But I wanted to evaluate this
in this video because I want

00:03:41.470 --> 00:03:43.360
to show you that this isn't some
abstract thing, that you

00:03:43.360 --> 00:03:45.730
can actually evaluate
these functions.

00:03:45.730 --> 00:03:47.835
So the first thing I want to
do-- I mean, I don't know what

00:03:47.835 --> 00:03:49.150
the antiderivative of this is.

00:03:49.150 --> 00:03:51.450
It's tempting, you see a sine
and a cosine, maybe they're

00:03:51.450 --> 00:03:52.800
the derivatives of each
other, but this is the

00:03:52.800 --> 00:03:55.830
sine of t minus tau.

00:03:55.830 --> 00:03:58.980
So let me rewrite that sine of
t minus tau, and we'll just

00:03:58.980 --> 00:04:05.210
use the trig identity, that
the sine of t minus tau is

00:04:05.210 --> 00:04:15.010
just equal to the sine of t
times the cosine of tau minus

00:04:15.010 --> 00:04:18.990
the sine of tau times
the cosine of t.

00:04:18.990 --> 00:04:21.709
And actually, I just made a
video where I go through all

00:04:21.709 --> 00:04:26.000
of these trig identities really
just to review them for

00:04:26.000 --> 00:04:28.560
myself and actually to make a
video in better quality on

00:04:28.560 --> 00:04:29.420
them as well.

00:04:29.420 --> 00:04:31.560
So if we make this subsitution,
this you'll find

00:04:31.560 --> 00:04:35.390
on the inside cover of any
trigonometry or calculus book,

00:04:35.390 --> 00:04:39.175
you get the convolution of f and
g is equal to-- I'll just

00:04:39.175 --> 00:04:42.730
write that f-star g; I'll just
write it with that-- is equal

00:04:42.730 --> 00:04:48.660
to the integral from 0 to t of,
instead of sine of t minus

00:04:48.660 --> 00:04:50.780
tau, I'm going to write this
thing right there.

00:04:50.780 --> 00:04:56.970
So I'm going to write the sine
of t times the cosine of tau

00:04:56.970 --> 00:05:04.420
minus the sine of tau times the
cosine of t, and then all

00:05:04.420 --> 00:05:06.355
of that's times the
cosine of tau.

00:05:09.450 --> 00:05:12.970
I have to be careful with my
taus and t's, and let's see, t

00:05:12.970 --> 00:05:14.770
and tau, tau and t.

00:05:14.770 --> 00:05:16.490
Everything's working so far.

00:05:16.490 --> 00:05:19.810
So let's see, so
then that's dt.

00:05:19.810 --> 00:05:20.860
Oh, sorry, d tau.

00:05:20.860 --> 00:05:22.950
Let me be very careful here.

00:05:22.950 --> 00:05:26.460
Now let's distribute this
cosine of tau out,

00:05:26.460 --> 00:05:27.880
and what do we get?

00:05:27.880 --> 00:05:32.640
We get this is equal to-- so f
convoluted with g, I guess we

00:05:32.640 --> 00:05:38.710
call it f-star g, is equal to
the integral from 0 to t of

00:05:38.710 --> 00:05:43.560
sine of t times cosine of
tau times cosine of tau.

00:05:43.560 --> 00:05:44.620
I'm just distributing
this cosine of tau.

00:05:44.620 --> 00:05:51.450
So it's cosine squared of tau,
and then minus-- let's rewrite

00:05:51.450 --> 00:05:53.870
the cosine of t first, and I'm
doing that because we're

00:05:53.870 --> 00:05:55.680
integrating with
respect to tau.

00:05:55.680 --> 00:05:58.910
So I'm just going to write my
cosine of t first. So cosine

00:05:58.910 --> 00:06:09.950
of t times sine of tau times
the cosine of tau d tau.

00:06:09.950 --> 00:06:13.420
And now, since we're taking
the integral of really two

00:06:13.420 --> 00:06:15.530
things subtracting from each
other, let's just turn this

00:06:15.530 --> 00:06:17.860
into two separate integrals.

00:06:17.860 --> 00:06:24.500
So this is equal to the integral
from 0 to t, of sine

00:06:24.500 --> 00:06:32.010
of t, times the cosine squared
of tau d tau minus the

00:06:32.010 --> 00:06:40.710
integral from 0 to t of cosine
of t times sine of tau cosine

00:06:40.710 --> 00:06:43.910
of tau d tau.

00:06:43.910 --> 00:06:45.390
Now, what can we do?

00:06:45.390 --> 00:06:48.550
Well, to simplify it more,
remember, we're integrating

00:06:48.550 --> 00:06:51.330
with respect to-- let
me be careful here.

00:06:51.330 --> 00:06:53.500
We're integrating with
respect to tau.

00:06:53.500 --> 00:06:54.700
I wrote a t there.

00:06:54.700 --> 00:06:57.320
We're integrating with
respect to tau.

00:06:57.320 --> 00:06:59.580
So all of these, this
cosine of t right

00:06:59.580 --> 00:07:01.480
here, that's a constant.

00:07:01.480 --> 00:07:02.650
The sine of t is a constant.

00:07:02.650 --> 00:07:04.320
For all I know, t could
be equal to 5.

00:07:04.320 --> 00:07:06.190
It doesn't matter that one
of the boundaries of our

00:07:06.190 --> 00:07:07.520
integration is also a t.

00:07:07.520 --> 00:07:09.610
That t would be a 5,
in which case these

00:07:09.610 --> 00:07:10.730
are all just constants.

00:07:10.730 --> 00:07:13.730
We're integrating only with
respect to the tau, so if

00:07:13.730 --> 00:07:17.500
cosine of 5, that's a constant,
we can take it out

00:07:17.500 --> 00:07:17.960
of the integral.

00:07:17.960 --> 00:07:27.260
So this is equal to sine of t
times the integral from 0 to t

00:07:27.260 --> 00:07:38.360
of cosine squared of tau d tau
and then minus cosine of t--

00:07:38.360 --> 00:07:41.500
that's just a constant; I'm
bringing it out-- times the

00:07:41.500 --> 00:07:52.870
integral from 0 to t of sine
of tau cosine of tau d tau.

00:07:52.870 --> 00:07:55.270
Now, this antiderivative is
pretty straightforward.

00:07:55.270 --> 00:07:57.230
You could do u substitution.

00:07:57.230 --> 00:07:59.470
Let me do it here, instead
of doing it in our heads.

00:07:59.470 --> 00:08:00.960
This is a complicated
problem, so we don't

00:08:00.960 --> 00:08:02.260
want to skip steps.

00:08:02.260 --> 00:08:10.260
If we said u is equal to sine of
tau, then du d tau is equal

00:08:10.260 --> 00:08:14.320
to the cosine of tau, just
the derivative of sine.

00:08:14.320 --> 00:08:18.180
Or we could write that
du is equal to the

00:08:18.180 --> 00:08:23.170
cosine of tau d tau.

00:08:28.440 --> 00:08:32.120
We'll undo the substitution
before we evaluate the

00:08:32.120 --> 00:08:33.150
endpoints here.

00:08:33.150 --> 00:08:35.520
But this was a little bit
more of a conundrum.

00:08:35.520 --> 00:08:37.370
I don't know how to take the
antiderivative of cosine

00:08:37.370 --> 00:08:38.360
squared of tau.

00:08:38.360 --> 00:08:41.270
It's not obvious what that is.

00:08:41.270 --> 00:08:43.740
So to do this, we're going
to break out some more

00:08:43.740 --> 00:08:45.600
trigonometric identities.

00:08:45.600 --> 00:08:48.250
And in a video I just recorded,
it might not be the

00:08:48.250 --> 00:08:57.830
last video in the playlist, I
showed that the cosine squared

00:08:57.830 --> 00:09:02.960
of tau-- I'm just using tau as
an example-- is equal to 1/2

00:09:02.960 --> 00:09:12.660
times 1 plus the cosine
of 2 tau.

00:09:12.660 --> 00:09:15.690
And once again, this is just a
trig identity that you'll find

00:09:15.690 --> 00:09:19.250
really in the inside cover of
probably your calculus book.

00:09:19.250 --> 00:09:22.870
So we can make this substitution
here, make this

00:09:22.870 --> 00:09:26.350
substitution right there, and
then let's see what our

00:09:26.350 --> 00:09:29.930
integrals become.

00:09:29.930 --> 00:09:33.280
So the first one over here,
let me just write it here.

00:09:33.280 --> 00:09:42.520
We get sine of t times the
integral from 0 to t of this

00:09:42.520 --> 00:09:45.060
thing here.

00:09:45.060 --> 00:09:50.055
Let me just take the 1/2 out,
to keep things simple.

00:09:50.055 --> 00:09:51.800
So I'll put the 1/2 out here.

00:09:51.800 --> 00:09:52.980
That's this 1/2.

00:09:52.980 --> 00:10:00.080
So 1 plus cosine of 2 tau and
all of that is d tau.

00:10:02.610 --> 00:10:04.530
That's this integral
right there.

00:10:04.530 --> 00:10:11.560
And then we have this integral
right here, minus cosine of t

00:10:11.560 --> 00:10:15.930
times the integral from--
let me be very clear.

00:10:15.930 --> 00:10:25.600
This is tau is equal to 0
to tau is equal to t.

00:10:25.600 --> 00:10:27.960
And then this thing right
here, I did some u

00:10:27.960 --> 00:10:28.510
subsitution.

00:10:28.510 --> 00:10:33.180
If u is equal to sine of
t, then this becomes u.

00:10:33.180 --> 00:10:36.710
And we showed that du is equal
to cosine-- sorry, u is equal

00:10:36.710 --> 00:10:38.060
to sine of tau.

00:10:38.060 --> 00:10:42.930
And then we showed that du is
equal to cosine tau d tau, so

00:10:42.930 --> 00:10:46.010
this thing right here
is equal to du.

00:10:46.010 --> 00:10:53.290
So it's u du, and let's see if
we can do anything useful now.

00:10:53.290 --> 00:11:01.210
So this integral right here, the
antiderivative of this is

00:11:01.210 --> 00:11:03.305
pretty straightforward, so
what are we going to get?

00:11:03.305 --> 00:11:05.110
Let me write this
outside part.

00:11:05.110 --> 00:11:10.060
So we have 1/2 times
the sine of t.

00:11:10.060 --> 00:11:13.220
And now let me take the
antiderivative of this.

00:11:13.220 --> 00:11:17.400
This is going to be tau plus
the antiderivative of this.

00:11:17.400 --> 00:11:20.465
It's going to be 1/2
sine of 2 tau.

00:11:24.870 --> 00:11:26.550
I mean, we could have done
the u substitution.

00:11:26.550 --> 00:11:29.720
we could have said u is equal to
2 tau and all of that, but

00:11:29.720 --> 00:11:31.460
I think you could do that from
recognition, and if you don't

00:11:31.460 --> 00:11:34.720
believe me, you just have to
take the derivative of this.

00:11:34.720 --> 00:11:38.230
1/2 sine of 2 tau is the
derivative of this.

00:11:38.230 --> 00:11:41.550
You multiply, you take the
derivative of the inside, so

00:11:41.550 --> 00:11:44.840
that's 2, so the 2 and the 1/2
cancel out, and the derivative

00:11:44.840 --> 00:11:47.450
of the outside, so
cosine of 2 tau.

00:11:47.450 --> 00:11:52.910
And you're going to evaluate
that from 0 to t.

00:11:52.910 --> 00:11:58.550
And then we have minus
cosine of t.

00:11:58.550 --> 00:12:01.550
When we take the antiderivative
of this-- let

00:12:01.550 --> 00:12:02.690
me do this on the side.

00:12:02.690 --> 00:12:07.540
So the integral of u du,
that's trivially easy.

00:12:07.540 --> 00:12:12.320
That's 1/2 u squared.

00:12:12.320 --> 00:12:15.940
Now, that's 1/2 u squared, but
what was u to begin with?

00:12:15.940 --> 00:12:18.610
It was sine of tau.

00:12:18.610 --> 00:12:22.970
So the antiderivative of this
thing right here is 1/2 u

00:12:22.970 --> 00:12:25.290
squared, but u is sine of tau.

00:12:25.290 --> 00:12:33.490
So it's going to be 1/2u, which
is sine of tau squared.

00:12:33.490 --> 00:12:39.130
And we're going to evaluate
that from 0 to t.

00:12:39.130 --> 00:12:41.120
And we didn't even have to do
all this u substitution.

00:12:41.120 --> 00:12:43.600
The way I often do it in my
head, I see the sine of tau,

00:12:43.600 --> 00:12:44.790
cosine of tau.

00:12:44.790 --> 00:12:47.370
if I have a function and I have
its derivative, I can

00:12:47.370 --> 00:12:50.140
treat that function just like
as if I had an x there, so

00:12:50.140 --> 00:12:53.690
it'd be sine squared of tau over
2, which is exactly what

00:12:53.690 --> 00:12:54.980
we have there.

00:12:54.980 --> 00:12:56.896
So it looks like we're
in the home stretch.

00:12:59.810 --> 00:13:03.790
We're taking the convolution of
sine of t with cosine of t.

00:13:03.790 --> 00:13:09.880
And so we get 1/2 sine of t.

00:13:09.880 --> 00:13:12.520
Now, if I evaluate this thing
at t, what do I get?

00:13:12.520 --> 00:13:18.230
I get t plus 1/2 sine
of 2t, that's when I

00:13:18.230 --> 00:13:19.800
evaluated it at t.

00:13:19.800 --> 00:13:23.160
And then from that I need to
subtract it evaluated at 0, so

00:13:23.160 --> 00:13:30.720
minus 0 minus 1/2 sine of
2 times 0, which is

00:13:30.720 --> 00:13:33.170
just sine of 0.

00:13:33.170 --> 00:13:35.930
So this part right here, this
whole thing right there, what

00:13:35.930 --> 00:13:37.460
does that simplify to?

00:13:37.460 --> 00:13:40.910
Well this is 0, sine of 0
is 0, so this is all 0.

00:13:40.910 --> 00:13:47.940
So this first integral right
there simplifies to 1/2 sine

00:13:47.940 --> 00:13:56.010
of t times t plus
1/2 sine of 2t.

00:14:00.740 --> 00:14:04.280
All right, now what does this
one simplify to over here?

00:14:04.280 --> 00:14:12.191
Well, this one over here, you
have minus cosine of t.

00:14:12.191 --> 00:14:14.900
And we're going to evaluate this
whole thing at t, so you

00:14:14.900 --> 00:14:23.200
get 1/2 sine squared of t
minus 1/2 the sine of 0

00:14:23.200 --> 00:14:26.680
squared, which is just 0,
so that's just minus 0.

00:14:26.680 --> 00:14:32.180
So far, everything that we have
written simplifies to--

00:14:32.180 --> 00:14:34.550
let me multiply it all out.

00:14:34.550 --> 00:14:41.810
So I have 1/2-- let me just pick
a good color-- 1/2t sine

00:14:41.810 --> 00:14:50.580
of t-- I'm just multiplying
those out-- plus 1/4 sine of t

00:14:50.580 --> 00:14:53.110
sine of 2t.

00:14:53.110 --> 00:15:02.210
And then over here I have minus
1/2 sine squared t times

00:15:02.210 --> 00:15:03.640
cosine of t.

00:15:03.640 --> 00:15:05.910
I just took the minus cosine t
and multiplied it through here

00:15:05.910 --> 00:15:08.420
and I got that.

00:15:08.420 --> 00:15:11.200
Now, this is a valid answer,
but I suspect that we can

00:15:11.200 --> 00:15:15.000
simplify this more, maybe using
some more trigonometric

00:15:15.000 --> 00:15:16.160
identities.

00:15:16.160 --> 00:15:20.720
And this guy right there
looks ripe to simplify.

00:15:20.720 --> 00:15:26.530
And we know that the sine of
2t-- another trig identity

00:15:26.530 --> 00:15:28.790
you'll find in the inside cover
of any of your books--

00:15:28.790 --> 00:15:34.890
is 2 times the sine of t
times the cosine of t.

00:15:34.890 --> 00:15:36.930
So if you substitute that there,
what does our whole

00:15:36.930 --> 00:15:38.480
expression equal?

00:15:38.480 --> 00:15:40.705
You get this first term.

00:15:40.705 --> 00:15:42.890
Let me scroll down
a little bit.

00:15:42.890 --> 00:15:54.280
You get 1/2t times the sine of
t plus 1/4 sine of t times

00:15:54.280 --> 00:16:01.640
this thing in here, so times
2 sine of t cosine of t.

00:16:01.640 --> 00:16:04.230
Just a trig identity, nothing
more than that.

00:16:04.230 --> 00:16:09.870
And then finally I have this
minus 1/2 sine squared t

00:16:09.870 --> 00:16:11.520
cosine of t.

00:16:11.520 --> 00:16:13.590
No one ever said this was
going to be easy, but

00:16:13.590 --> 00:16:15.610
hopefully it's instructive
on some level.

00:16:15.610 --> 00:16:17.940
At least it shows you that you
didn't memorize your trig

00:16:17.940 --> 00:16:20.230
identities for nothing.

00:16:20.230 --> 00:16:22.190
So let me rewrite the whole
thing, or let me just

00:16:22.190 --> 00:16:23.420
rewrite this part.

00:16:23.420 --> 00:16:26.100
So this is equal to 1/4.

00:16:26.100 --> 00:16:29.810
Now, I have-- well let
me see, 1/4 times 2.

00:16:29.810 --> 00:16:32.040
1/4 times 2 is 1/2.

00:16:32.040 --> 00:16:35.450
And then sine squared
of t, right?

00:16:35.450 --> 00:16:39.490
This sine times this sine is
sine squared of t cosine of t.

00:16:39.490 --> 00:16:45.180
And then this one over here is
minus 1/2 sine squared of t

00:16:45.180 --> 00:16:46.930
cosine of t.

00:16:46.930 --> 00:16:50.920
And luckily for us, or lucky
for us, these cancel out.

00:16:50.920 --> 00:16:52.900
And, of course, we had this
guy out in the front.

00:16:52.900 --> 00:16:58.640
We had this 1/2t sine
t out in front.

00:16:58.640 --> 00:17:01.900
Now, this guy cancels with this
guy, and all we're left

00:17:01.900 --> 00:17:05.450
with, through this whole hairy
problem, and this is pretty

00:17:05.450 --> 00:17:11.180
satisfying, is 1/2t sine of t.

00:17:11.180 --> 00:17:16.450
So we just showed you that the
convolution-- if I define--

00:17:16.450 --> 00:17:17.869
let me write our result.

00:17:17.869 --> 00:17:23.260
I feel like writing this
in stone because

00:17:23.260 --> 00:17:25.119
this was so much work.

00:17:25.119 --> 00:17:32.800
But if we write that f of t is
equal to sine of t, and g of t

00:17:32.800 --> 00:17:35.870
is equal to cosine of t, I
just showed you that the

00:17:35.870 --> 00:17:41.040
convolution of f with g, which
is a function of t, which is

00:17:41.040 --> 00:17:48.950
defined as the integral from 0
to t of f of t minus tau times

00:17:48.950 --> 00:17:54.180
g of tau d tau, which was equal
to-- and I'll switch

00:17:54.180 --> 00:17:59.400
colors here-- which was equal to
the integral from 0 to t of

00:17:59.400 --> 00:18:08.470
sine of t minus tau times g of
tau d tau, that all of this

00:18:08.470 --> 00:18:12.180
mess, all of this convolution,
it all equals-- and this is

00:18:12.180 --> 00:18:21.230
pretty satisfying-- it all
equals 1/2t sine of t.

00:18:21.230 --> 00:18:23.400
And the whole reason why I went
through all of this mess

00:18:23.400 --> 00:18:26.680
and kind of bringing out the
neurons that had the trig

00:18:26.680 --> 00:18:29.280
identities memorized or having
to reproof them or whatever

00:18:29.280 --> 00:18:32.310
else is to just show you that
this convolution, it is

00:18:32.310 --> 00:18:34.740
convoluted and it seems a little
bit bizarre, but you

00:18:34.740 --> 00:18:39.100
really can take the convolutions
of actual

00:18:39.100 --> 00:18:41.390
functions and get an
actual answer.

00:18:41.390 --> 00:18:44.250
So the convolution of sine
of t with cosine of t is

00:18:44.250 --> 00:18:46.660
1/2t sine of t.

00:18:46.660 --> 00:18:49.060
So, hopefully, you have a little
of intuition of-- well,

00:18:49.060 --> 00:18:52.370
not intuition, but you at least
have a little bit of

00:18:52.370 --> 00:18:56.620
hands-on understanding of how
the convolution can be

00:18:56.620 --> 00:18:57.870
calculated.


Trigonometry identity review/fun

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.010
i've already made a handful
of videos that covers what

00:00:03.010 --> 00:00:04.990
I'm going to cover, the
trigonometric identities I'm

00:00:04.990 --> 00:00:06.480
going to cover in this video.

00:00:06.480 --> 00:00:09.330
The reason why I'm doing it is
that I'm in need of review

00:00:09.330 --> 00:00:12.220
myself because I was doing some
calculus problems that required

00:00:12.220 --> 00:00:15.150
me to know this, and I have
better recording software now

00:00:15.150 --> 00:00:18.550
so I thought two birds with one
stone, let me rerecord a

00:00:18.550 --> 00:00:22.290
video and kind of refresh
things in my own mind.

00:00:22.290 --> 00:00:25.080
So the trig identities that I'm
going to assume that we know

00:00:25.080 --> 00:00:26.730
because I've already made
videos on them and they're a

00:00:26.730 --> 00:00:32.060
little bit involved to remember
or to prove, are that the sine

00:00:32.060 --> 00:00:38.980
of a plus b is equal to the
sine of a times the cosine

00:00:38.980 --> 00:00:47.750
of b plus the sine of b
times the cosine of a.

00:00:47.750 --> 00:00:50.670
That's the first one, I assume,
going into this video we know.

00:00:50.670 --> 00:00:53.310
And then if we wanted to know
the sine of-- well, I'll just

00:00:53.310 --> 00:00:54.270
write it a little differently.

00:00:54.270 --> 00:01:00.010
What if I wanted to figure out
the sine of a plus-- I'll

00:01:00.010 --> 00:01:02.010
write it this way-- minus c?

00:01:02.010 --> 00:01:04.990
Which is the same thing
as a minus c, right?

00:01:04.990 --> 00:01:07.590
Well, we could just use this
formula up here to say well,

00:01:07.590 --> 00:01:15.730
that's equal to the sine of a
times the cosine of minus c

00:01:15.730 --> 00:01:24.020
plus the sine of minus c
times the cosine of a.

00:01:24.020 --> 00:01:26.480
And we know, and I guess this
is another assumption that

00:01:26.480 --> 00:01:28.580
we're going to have to have
going into this video, that the

00:01:28.580 --> 00:01:35.410
cosine of minus c is equal
to just the cosine of c.

00:01:35.410 --> 00:01:37.690
That the cosine is
an even function.

00:01:37.690 --> 00:01:40.620
And you could look at that by
looking at the graph of the

00:01:40.620 --> 00:01:43.420
cosine function, or even at
the unit circle itself.

00:01:43.420 --> 00:01:45.240
And that the sine is
an odd function.

00:01:45.240 --> 00:01:48.770
That the sine of minus
c is actually equal

00:01:48.770 --> 00:01:53.010
to minus sine of c.

00:01:53.010 --> 00:01:55.860
So we can use both of that
information to rewrite the

00:01:55.860 --> 00:02:01.710
second line up here to say that
the sine of a minus c is equal

00:02:01.710 --> 00:02:05.450
to the sine of a times
the cosine of c.

00:02:05.450 --> 00:02:07.570
Because cosine of minus
c is the same thing

00:02:07.570 --> 00:02:09.320
as the cosine of c.

00:02:09.320 --> 00:02:11.650
Times the cosine of c.

00:02:11.650 --> 00:02:14.510
And then, minus the sine of c.

00:02:14.510 --> 00:02:16.710
Instead of writing this,
I could write this.

00:02:16.710 --> 00:02:23.220
Minus the sine of c
times the cosine of a.

00:02:23.220 --> 00:02:26.080
So that we kind of pseudo
proved this by knowing this

00:02:26.080 --> 00:02:27.590
and this ahead of time.

00:02:27.590 --> 00:02:28.190
Fair enough.

00:02:28.190 --> 00:02:30.920
And I'm going to use all of
these to kind of prove a bunch

00:02:30.920 --> 00:02:34.110
of more trig identities
that I'm going to need.

00:02:34.110 --> 00:02:40.280
So the other trig identity is
that the cosine of a plus b is

00:02:40.280 --> 00:02:43.760
equal to the cosine of a-- you
don't mix up the cosines and

00:02:43.760 --> 00:02:45.520
the sines in this situation.

00:02:45.520 --> 00:02:48.595
Cosine of a times
the sine of b.

00:02:48.595 --> 00:02:51.180
And this is minus--
well, sorry.

00:02:51.180 --> 00:02:53.720
I just said you don't mix it
up and then I mixed them up.

00:02:53.720 --> 00:03:03.010
Times the cosine of b minus
sine of a times the sine of b.

00:03:03.010 --> 00:03:08.140
Now, if you wanted to know what
the cosine of a minus b is,

00:03:08.140 --> 00:03:09.890
well, you use these
same properties.

00:03:09.890 --> 00:03:12.760
Cosine of minus b, that's still
going to be cosine on b.

00:03:12.760 --> 00:03:16.960
So that's going to be the
cosine of a times the cosine--

00:03:16.960 --> 00:03:20.530
cosine of minus b is the
same thing as cosine of b.

00:03:20.530 --> 00:03:23.760
But here you're going to have
sine of minus b, which is the

00:03:23.760 --> 00:03:26.600
same thing as the
minus sine of b.

00:03:26.600 --> 00:03:31.060
And that minus will cancel that
out, so it'll be plus sine

00:03:31.060 --> 00:03:34.070
of a times the sine of b.

00:03:34.070 --> 00:03:34.880
So it's a little tricky.

00:03:34.880 --> 00:03:37.610
When you have a plus sign
here you get a minus there.

00:03:37.610 --> 00:03:39.410
When you don't minus
sign there, you get

00:03:39.410 --> 00:03:40.570
a plus sign there.

00:03:40.570 --> 00:03:41.520
But fair enough.

00:03:41.520 --> 00:03:44.230
I don't want to dwell on that
too much because we have many

00:03:44.230 --> 00:03:46.750
more identities to show.

00:03:46.750 --> 00:03:50.340
So what if I wanted an
identity for let's

00:03:50.340 --> 00:03:53.440
say, the cosine of 2a?

00:03:53.440 --> 00:03:56.760
So the cosine of 2a.

00:03:56.760 --> 00:04:01.500
Well that's just the same thing
as the cosine of a plus a.

00:04:01.500 --> 00:04:03.480
And then we could use this
formula right up here.

00:04:03.480 --> 00:04:07.050
If my second a is just my b,
then this is just equal to

00:04:07.050 --> 00:04:14.030
cosine of a times the cosine
of a minus the sine of

00:04:14.030 --> 00:04:18.320
a times the sine of a.

00:04:18.320 --> 00:04:21.560
My b is also an a in this
situation, which I could

00:04:21.560 --> 00:04:26.570
rewrite as, this is equal to
the cosine squared of a.

00:04:26.570 --> 00:04:30.960
I just wrote cosine of a times
itself twice or times itself.

00:04:30.960 --> 00:04:34.740
Minus sine squared of a.

00:04:34.740 --> 00:04:37.910
This is one I guess
identity already.

00:04:37.910 --> 00:04:40.630
Cosine of 2a is equal to the
cosine squared of a minus

00:04:40.630 --> 00:04:42.240
the sine squared of a.

00:04:42.240 --> 00:04:46.140
Let me box off my identities
that we're showing

00:04:46.140 --> 00:04:47.410
in this video.

00:04:47.410 --> 00:04:49.780
So I just showed you that one.

00:04:49.780 --> 00:04:51.050
What if I'm not satisfied?

00:04:51.050 --> 00:04:53.700
What if I just want it
in terms of cosines?

00:04:53.700 --> 00:04:56.820
Well, we could break out
the unit circle definition

00:04:56.820 --> 00:04:58.450
of our trig functions.

00:04:58.450 --> 00:05:00.740
This is kind of the most
fundamental identity.

00:05:00.740 --> 00:05:05.190
The sine squared of a
plus the cosine squared

00:05:05.190 --> 00:05:06.700
of a is equal to 1.

00:05:06.700 --> 00:05:10.010
Or you could write that--
let me think of the

00:05:10.010 --> 00:05:11.410
best way to do this.

00:05:11.410 --> 00:05:15.590
You could write that the sine
squared of a is equal to

00:05:15.590 --> 00:05:18.670
1 minus the cosine
sign squared of a.

00:05:18.670 --> 00:05:21.330
And then we could take this
and substitute it right here.

00:05:21.330 --> 00:05:24.110
So we could rewrite this
identity as being equal to the

00:05:24.110 --> 00:05:28.610
cosine squared of a minus
the sine squared of a.

00:05:28.610 --> 00:05:32.060
But the sine squared of
a is this right there.

00:05:32.060 --> 00:05:35.690
So minus-- I'll do it
in a different color.

00:05:35.690 --> 00:05:39.310
Minus 1 minus cosine
squared of a.

00:05:39.310 --> 00:05:42.260
That's what I just substituted
for the sine squared of a.

00:05:42.260 --> 00:05:47.050
And so this is equal to the
cosine squared of a minus 1

00:05:47.050 --> 00:05:49.890
plus the cosine squared of a.

00:05:49.890 --> 00:05:52.270
Which is equal to--
we're just adding.

00:05:52.270 --> 00:05:54.030
I'll just continue
on the right.

00:05:54.030 --> 00:05:56.400
We have 1 cosine squared of a
plus another cosine squared

00:05:56.400 --> 00:06:02.780
of a, so it's 2 cosine
squared of a minus 1.

00:06:02.780 --> 00:06:04.610
And all of that is
equal to cosine of 2a.

00:06:08.520 --> 00:06:11.330
Now what if I wanted to get
an identity that gave me

00:06:11.330 --> 00:06:13.710
what cosine squared of
a is in terms of this?

00:06:13.710 --> 00:06:15.320
Well we could just
solve for that.

00:06:15.320 --> 00:06:17.930
If we add 1 to both sides of
this equation, actually,

00:06:17.930 --> 00:06:18.510
let me write this.

00:06:18.510 --> 00:06:21.460
This is one of our
other identities.

00:06:21.460 --> 00:06:28.430
But if we add 1 to both sides
of that equation we get 2 times

00:06:28.430 --> 00:06:36.750
the cosine squared of a is
equal to cosine of 2a plus 1.

00:06:36.750 --> 00:06:40.620
And if we divide both sides of
this by 2 we get the cosine

00:06:40.620 --> 00:06:47.290
squared of a is equal to 1/2--
now we could rearrange these

00:06:47.290 --> 00:06:54.660
just to do it-- times 1
plus the cosine of 2a.

00:06:54.660 --> 00:06:56.810
And we're done.

00:06:56.810 --> 00:06:59.640
And we have another identity.

00:06:59.640 --> 00:07:04.830
Cosine squared of a, sometimes
it's called the power reduction

00:07:04.830 --> 00:07:07.380
identity right there.

00:07:07.380 --> 00:07:09.440
Now what if we wanted
something in terms of

00:07:09.440 --> 00:07:11.200
the sine squared of a?

00:07:11.200 --> 00:07:14.170
Well then maybe we could go
back up here and we know from

00:07:14.170 --> 00:07:16.300
this identity that the sine
squared of a is equal to 1

00:07:16.300 --> 00:07:17.750
minus cosine squared of a.

00:07:17.750 --> 00:07:18.930
Or we could have
gone the other way.

00:07:18.930 --> 00:07:22.910
We could have subtracted sine
squared of a from both sides

00:07:22.910 --> 00:07:25.000
and we could have gotten--
let me go down there.

00:07:25.000 --> 00:07:27.570
If I subtracted sine squared of
a from both sides you could get

00:07:27.570 --> 00:07:34.050
cosine squared of a is equal
to 1 minus sine squared of a.

00:07:34.050 --> 00:07:36.980
And then we could go back into
this formula right up here and

00:07:36.980 --> 00:07:41.490
we could write down-- and I'll
do it in this blue color.

00:07:41.490 --> 00:07:48.480
We could write down that the
cosine of 2a is equal to--

00:07:48.480 --> 00:07:50.750
instead of writing a cosine
squared of a, I'll write this-

00:07:50.750 --> 00:07:55.780
is equal to 1 minus sine
squared of a minus

00:07:55.780 --> 00:07:56.795
sine squared of a.

00:08:00.660 --> 00:08:05.245
So my cosine of 2a is equal to?

00:08:05.245 --> 00:08:05.910
Let's see.

00:08:05.910 --> 00:08:07.650
I have a minus sine squared
of a minus another

00:08:07.650 --> 00:08:08.420
sine squared of a.

00:08:08.420 --> 00:08:13.600
So I have 1 minus 2
sine squared of a.

00:08:13.600 --> 00:08:15.480
So here's another identity.

00:08:15.480 --> 00:08:19.190
Another way to write
my cosine of 2a.

00:08:19.190 --> 00:08:22.740
We're discovering a lot of ways
to write our cosine of 2a.

00:08:22.740 --> 00:08:25.840
Now if we wanted to solve for
sine squared of 2a we could

00:08:25.840 --> 00:08:27.610
add it to both sides
of the equation.

00:08:27.610 --> 00:08:29.950
So let me do that and I'll
just write it here for

00:08:29.950 --> 00:08:32.554
the sake of saving space.

00:08:32.554 --> 00:08:36.230
Let me scroll down
a little bit.

00:08:36.230 --> 00:08:37.210
So I'm going to go here.

00:08:37.210 --> 00:08:40.520
If I just add 2 sine squared
of a to both sides of this, I

00:08:40.520 --> 00:08:51.480
get 2 sine squared of a plus
cosine of 2a is equal to 1.

00:08:51.480 --> 00:08:53.690
Subtract cosine of
2a from both sides.

00:08:53.690 --> 00:09:01.010
You get 2 sine squared of a is
equal to 1 minus cosine of 2a.

00:09:01.010 --> 00:09:05.420
Then you divide both sides of
this by 2 and you get sine

00:09:05.420 --> 00:09:12.280
squared of a is equal to 1/2
times 1 minus cosine of 2a.

00:09:12.280 --> 00:09:17.250
And we have our other discovery
I guess we could call it.

00:09:17.250 --> 00:09:18.600
Our finding.

00:09:18.600 --> 00:09:19.520
And it's interesting.

00:09:19.520 --> 00:09:21.050
It's always interesting
to look at the symmetry.

00:09:21.050 --> 00:09:24.000
Cosine squared-- they're
identical except for you have a

00:09:24.000 --> 00:09:27.720
plus 2a here for the cosine
squared and you have a minus

00:09:27.720 --> 00:09:30.600
cosine of 2a here for
the sine squared.

00:09:30.600 --> 00:09:33.440
So we've already found a
lot of interesting things.

00:09:33.440 --> 00:09:37.080
Let's see if we can do
anything with the sine of 2a.

00:09:40.120 --> 00:09:42.980
Let me pick a new color
here that I haven't used.

00:09:42.980 --> 00:09:44.880
Well, I've pretty much
used all my colors.

00:09:44.880 --> 00:09:50.120
So if I want to figure out the
sine of 2a, this is equal

00:09:50.120 --> 00:09:54.380
to the sine of a plus a.

00:09:54.380 --> 00:10:03.610
Which is equal to the sine of a
times the co-- well, I don't

00:10:03.610 --> 00:10:04.640
want to make it that thick.

00:10:04.640 --> 00:10:10.150
Times the cosine of a plus--
and this cosine of a,

00:10:10.150 --> 00:10:11.100
that's the second a.

00:10:11.100 --> 00:10:12.020
Actually, you could
view it that way.

00:10:12.020 --> 00:10:15.570
Plus the sine-- I'm just
using the sine of a plus b.

00:10:15.570 --> 00:10:18.000
Plus the sine of the
second a times the

00:10:18.000 --> 00:10:19.870
cosine of the first a.

00:10:19.870 --> 00:10:22.030
I just wrote the same thing
twice, so this is just people

00:10:22.030 --> 00:10:26.370
to 2 sine of a, cosine of a.

00:10:26.370 --> 00:10:27.490
That was a little bit easier.

00:10:27.490 --> 00:10:32.030
So sine of 2a is equal to that.

00:10:32.030 --> 00:10:33.140
So that's another result.

00:10:35.920 --> 00:10:38.160
I know I'm a little bit tired
by playing with all of

00:10:38.160 --> 00:10:39.640
these sine and cosines.

00:10:39.640 --> 00:10:41.990
And I was able to get all the
results that I needed for my

00:10:41.990 --> 00:10:44.580
calculus problem, so hopefully
this was a good review for

00:10:44.580 --> 00:10:48.220
you because it was a
good review for me.

00:10:48.220 --> 00:10:49.360
You can write these
things down.

00:10:49.360 --> 00:10:51.840
You can memorize them if you
want, but the really important

00:10:51.840 --> 00:10:54.350
take away is to realize that
you really can derive all of

00:10:54.350 --> 00:10:59.590
these formulas really from
these initial formulas

00:10:59.590 --> 00:11:00.250
that we just had.

00:11:00.250 --> 00:11:02.910
And even these, I also have
proofs to show you how to get

00:11:02.910 --> 00:11:05.100
these from just the basic
definitions of your

00:11:05.100 --> 00:11:06.700
trig functions.


Laplace transform of the dirac delta function

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:02.440
In the last video, I introduced
you to what is

00:00:02.440 --> 00:00:04.850
probably the most bizarro
function that you've

00:00:04.850 --> 00:00:05.950
encountered so far.

00:00:05.950 --> 00:00:09.235
And that was the Dirac
delta function.

00:00:13.530 --> 00:00:16.190
And I defined it to be--
and I'll do the

00:00:16.190 --> 00:00:17.400
shifted version of it.

00:00:17.400 --> 00:00:21.970
You're already hopefully
reasonably familiar with it.

00:00:21.970 --> 00:00:25.380
So Dirac delta of t minus c.

00:00:25.380 --> 00:00:31.370
We can say that it equals 0,
when t does not equal c, so it

00:00:31.370 --> 00:00:33.775
equals 0 everywhere, but it

00:00:33.775 --> 00:00:35.450
essentially pops up to infinity.

00:00:35.450 --> 00:00:37.470
And we have to be careful
with this infinity.

00:00:37.470 --> 00:00:38.760
I'm going to write
it in quotes.

00:00:38.760 --> 00:00:40.970
It pops up to infinity.

00:00:40.970 --> 00:00:42.700
And we even saw in the previous
video, it's kind of

00:00:42.700 --> 00:00:45.100
different degrees of infinity,
because you can still multiply

00:00:45.100 --> 00:00:48.510
this by other numbers to get
larger Dirac delta functions

00:00:48.510 --> 00:00:50.560
when t is equal to c.

00:00:50.560 --> 00:00:53.170
But more important than this,
and this is kind of a

00:00:53.170 --> 00:00:56.750
pseudodefinition here, is the
idea that when we take the

00:00:56.750 --> 00:01:00.860
integral, when we take the area
under the curve over the

00:01:00.860 --> 00:01:04.839
entire x- or the entire t-axis,
I guess we could say,

00:01:04.839 --> 00:01:08.730
when we take the area under this
curve, and obviously, it

00:01:08.730 --> 00:01:13.890
equals zero everywhere except
at t is equal to c, when we

00:01:13.890 --> 00:01:17.930
take this area, this is the
important point, that the area

00:01:17.930 --> 00:01:19.180
is equal to 1.

00:01:22.630 --> 00:01:25.830
And so this is what I meant by
pseudoinfinity, because if I

00:01:25.830 --> 00:01:32.730
have 2 times the Dirac delta
function, and if I'm taking

00:01:32.730 --> 00:01:37.140
the area under the curve of
that, of 2 times the Dirac

00:01:37.140 --> 00:01:41.370
delta function t minus c dt,
this should be equal to 2

00:01:41.370 --> 00:01:46.410
times-- the area of just under
the Dirac delta function 2

00:01:46.410 --> 00:01:50.570
times from minus infinity to
infinity of the delta function

00:01:50.570 --> 00:01:55.120
shifted by c dt, which is just
2 times-- we already showed

00:01:55.120 --> 00:01:58.090
you, I just said, by definition,
this is 1, so this

00:01:58.090 --> 00:02:00.060
will be equal to 2.

00:02:00.060 --> 00:02:04.620
So if I put a 2 out here, this
infinity will have to be twice

00:02:04.620 --> 00:02:07.370
as high, so that the
area is now 2.

00:02:07.370 --> 00:02:09.870
That's why I put that infinity
in parentheses.

00:02:09.870 --> 00:02:11.180
But it's an interesting
function.

00:02:11.180 --> 00:02:14.100
I talked about it at the end of
the last video that it can

00:02:14.100 --> 00:02:17.720
help model things that kind of
jar things all of a sudden,

00:02:17.720 --> 00:02:21.970
but they impart a fixed amount
of impulse on something and a

00:02:21.970 --> 00:02:24.010
fixed amount of change
in momentum.

00:02:24.010 --> 00:02:26.660
And we'll understand that a
little bit more in the future.

00:02:26.660 --> 00:02:30.300
But let's kind of get the
mathematical tools completely

00:02:30.300 --> 00:02:30.720
understood.

00:02:30.720 --> 00:02:33.590
And let's try to figure out what
the Dirac delta function

00:02:33.590 --> 00:02:36.200
does when we multiply it, what
it does to the Laplace

00:02:36.200 --> 00:02:38.550
transform when we multiply
it times some function.

00:02:38.550 --> 00:02:42.105
So let's say I have my Dirac
delta function and I'm going

00:02:42.105 --> 00:02:43.170
to shift it.

00:02:43.170 --> 00:02:44.560
That's a little bit
more interesting.

00:02:44.560 --> 00:02:46.480
And if you want to unshift
it, you just say, OK,

00:02:46.480 --> 00:02:47.440
well, c equals 0.

00:02:47.440 --> 00:02:49.030
What happens when c equals 0?

00:02:49.030 --> 00:02:52.520
And I'm going to shift it and
multiply it times some

00:02:52.520 --> 00:02:54.720
arbitrary function f of t.

00:02:54.720 --> 00:02:57.590
If I wanted to figure out the
Laplace transform of just the

00:02:57.590 --> 00:02:59.800
delta function by itself,
I could say f of

00:02:59.800 --> 00:03:01.390
t is equal to 1.

00:03:01.390 --> 00:03:08.020
So let's take our Laplace
transform of this.

00:03:08.020 --> 00:03:10.220
And we can just use the
definition of the Laplace

00:03:10.220 --> 00:03:16.740
transform, so this is equal to
the area from 0 to infinity,

00:03:16.740 --> 00:03:21.510
or we could call it the integral
from 0 to infinity of

00:03:21.510 --> 00:03:24.340
e to the minus -- that's just
part of the Laplace transform

00:03:24.340 --> 00:03:27.095
definition-- times this thing--
and I'll just write it

00:03:27.095 --> 00:03:31.080
in this order-- times
f of t times

00:03:31.080 --> 00:03:33.960
our Dirac delta function.

00:03:33.960 --> 00:03:38.600
Delta t minus c and times dt.

00:03:41.360 --> 00:03:43.090
Now, here I'm going to
make a little bit of

00:03:43.090 --> 00:03:44.670
an intuitive argument.

00:03:44.670 --> 00:03:47.760
A lot of the math we do is kind
of-- especially if you

00:03:47.760 --> 00:03:50.410
want to be very rigorous and
formal, the Dirac delta

00:03:50.410 --> 00:03:53.660
function starts to break down a
lot of tools that you might

00:03:53.660 --> 00:03:55.170
have not realized it would
break down, but I think

00:03:55.170 --> 00:03:56.920
intuitively, we can still
work with it.

00:03:56.920 --> 00:04:00.425
So I'm going to solve this
integral for you intuitively,

00:04:00.425 --> 00:04:02.990
and I think it'll
make some sense.

00:04:02.990 --> 00:04:05.380
So let's draw this.

00:04:05.380 --> 00:04:08.790
Let me draw this, what
we're trying to do.

00:04:08.790 --> 00:04:10.310
So let me draw what we're
trying to take

00:04:10.310 --> 00:04:11.375
the integral of.

00:04:11.375 --> 00:04:13.600
And we only care from zero to
infinity, so I'll only do it

00:04:13.600 --> 00:04:14.610
from zero to infinity.

00:04:14.610 --> 00:04:17.820
And I'll assume that c is
greater than zero, that the

00:04:17.820 --> 00:04:19.560
delta function pops
up someplace in

00:04:19.560 --> 00:04:21.870
the positive t-axis.

00:04:21.870 --> 00:04:25.620
So what is this first part
going to look like?

00:04:25.620 --> 00:04:27.610
What is that going to look
like? e to the minus

00:04:27.610 --> 00:04:28.780
st times f of t?

00:04:28.780 --> 00:04:29.220
I don't know.

00:04:29.220 --> 00:04:31.460
It's going to be some function.
e to the minus st

00:04:31.460 --> 00:04:34.320
starts at 1 and drops down, but
we're multiplying it times

00:04:34.320 --> 00:04:38.270
some arbitrary function, so I'll
just draw it like this.

00:04:38.270 --> 00:04:40.510
Maybe it looks something
like this.

00:04:40.510 --> 00:04:45.570
This right here is e to the
minus st times f of t.

00:04:45.570 --> 00:04:48.860
And the f of t is what kind of
gives it its arbitrary shape.

00:04:48.860 --> 00:04:50.200
Fair enough.

00:04:50.200 --> 00:04:53.110
Now, let's graph our Dirac
delta function.

00:04:53.110 --> 00:04:57.980
With zero everywhere except
right at c, right at c right

00:04:57.980 --> 00:05:01.840
there, it pops up infinitely
high, but we only draw an

00:05:01.840 --> 00:05:05.540
arrow that is of height 1 to
show that its area is 1.

00:05:05.540 --> 00:05:07.260
I mean, normally when you graph
things you don't draw

00:05:07.260 --> 00:05:10.110
arrows, but this arrow shows
that the area under this

00:05:10.110 --> 00:05:12.290
infinitely high thing is 1.

00:05:12.290 --> 00:05:14.260
So we do a 1 there.

00:05:14.260 --> 00:05:16.630
So if we multiply this, we care
about the area under this

00:05:16.630 --> 00:05:17.410
whole thing.

00:05:17.410 --> 00:05:21.120
When we multiply these two
functions, when we multiply

00:05:21.120 --> 00:05:24.050
this times this times the delta
function, this is-- let

00:05:24.050 --> 00:05:24.570
me write this.

00:05:24.570 --> 00:05:27.800
This is the delta function
shifted to c.

00:05:27.800 --> 00:05:30.950
If I multiply that times
that, what do I get?

00:05:30.950 --> 00:05:34.060
This is kind of the key
intuition here.

00:05:34.060 --> 00:05:37.410
Let me redraw my axes.

00:05:37.410 --> 00:05:41.500
Let me see if I can do it
a little bit straighter.

00:05:41.500 --> 00:05:44.930
Don't judge me by the
straightness of my axes.

00:05:44.930 --> 00:05:46.260
So that's t.

00:05:46.260 --> 00:05:48.480
So what happens when I
multiply these two?

00:05:48.480 --> 00:05:52.480
Everywhere, when t equals
anything other than c, the

00:05:52.480 --> 00:05:55.570
Dirac delta function is zero.

00:05:55.570 --> 00:05:57.290
So it's zero times anything.

00:05:57.290 --> 00:05:58.560
I don't care what this function
is going to do, it's

00:05:58.560 --> 00:05:59.920
going to be zero.

00:05:59.920 --> 00:06:02.760
So it's going to be zero
everywhere, except something

00:06:02.760 --> 00:06:07.050
interesting happens at
t is equal to c.

00:06:07.050 --> 00:06:10.110
At t equals c, what's the
value of the function?

00:06:10.110 --> 00:06:12.000
Well, it's going to
be the value of

00:06:12.000 --> 00:06:14.070
the Dirac delta function.

00:06:14.070 --> 00:06:17.760
It's going to be the Dirac delta
function times whatever

00:06:17.760 --> 00:06:20.030
height this is.

00:06:20.030 --> 00:06:23.410
This is going to be this point
right here or this right

00:06:23.410 --> 00:06:26.530
there, that point.

00:06:26.530 --> 00:06:30.030
This is going to be this
function evaluated at c.

00:06:30.030 --> 00:06:33.830
I'll mark it right here on the
y-axis, or on the f of t,

00:06:33.830 --> 00:06:34.950
whatever you want to call it.

00:06:34.950 --> 00:06:39.765
This is going to be e to the
minus sc times f of c.

00:06:39.765 --> 00:06:42.970
All I'm doing is I'm just
evaluating this function at c,

00:06:42.970 --> 00:06:45.150
so that's the point
right there.

00:06:45.150 --> 00:06:48.180
So if you take this point, which
is just some number, it

00:06:48.180 --> 00:06:51.840
could be 5, 5 times this, you're
just getting 5 times

00:06:51.840 --> 00:06:53.140
the Dirac delta function.

00:06:53.140 --> 00:06:54.540
Or in this case, it's not 5.

00:06:54.540 --> 00:06:55.930
It's this little more
abstract thing.

00:06:55.930 --> 00:06:58.660
I could just draw
it like this.

00:06:58.660 --> 00:07:02.080
When I multiply this thing times
my little delta function

00:07:02.080 --> 00:07:04.260
there, I get this.

00:07:04.260 --> 00:07:08.560
The height, it's a delta
function, but it's scaled now.

00:07:08.560 --> 00:07:11.420
It's scaled, so now my
new thing is going

00:07:11.420 --> 00:07:12.950
to look like this.

00:07:12.950 --> 00:07:17.080
If I just multiply that times
that, I essentially get e to

00:07:17.080 --> 00:07:20.820
the minus sc times f of c.

00:07:20.820 --> 00:07:23.240
This might look like some fancy
function, but it's just

00:07:23.240 --> 00:07:25.600
a number when we consider
it in terms of t.

00:07:25.600 --> 00:07:29.240
s, it becomes something when we
go into the Laplace world,

00:07:29.240 --> 00:07:32.250
but from t's point of view,
it's just a constant.

00:07:32.250 --> 00:07:34.100
All of these are just constants,
so this might as

00:07:34.100 --> 00:07:35.430
well just be 5.

00:07:35.430 --> 00:07:41.060
So it's this constant times my
Dirac delta function, times

00:07:41.060 --> 00:07:43.360
delta of t minus c.

00:07:43.360 --> 00:07:46.600
When I multiply that thing times
that thing, all I'm left

00:07:46.600 --> 00:07:48.680
with is this thing.

00:07:48.680 --> 00:07:51.315
And this height is still going
to be infinitely high, but

00:07:51.315 --> 00:07:57.070
it's infinitely high scaled in
such a way that its area is

00:07:57.070 --> 00:07:59.130
going to be not 1.

00:07:59.130 --> 00:08:00.500
And I'll show it to you.

00:08:00.500 --> 00:08:02.850
So what's the integral
of this thing?

00:08:02.850 --> 00:08:04.980
Taking the integral of this
thing from minus infinity to

00:08:04.980 --> 00:08:07.870
infinity, since this thing is
this thing, it should be the

00:08:07.870 --> 00:08:09.380
same thing as taking the
integral of this thing from

00:08:09.380 --> 00:08:10.710
minus infinity to infinity.

00:08:10.710 --> 00:08:12.220
So let's do that.

00:08:12.220 --> 00:08:13.500
Actually, we don't have to do
it from minus infinity.

00:08:13.500 --> 00:08:15.220
I said from zero to infinity.

00:08:15.220 --> 00:08:19.850
So if we take from zero to
infinity, what I'm saying is

00:08:19.850 --> 00:08:21.390
taking this integral
is equivalent

00:08:21.390 --> 00:08:23.160
to taking this integral.

00:08:23.160 --> 00:08:31.260
So e to the minus sc f of c
times my delta function t

00:08:31.260 --> 00:08:34.270
minus c dt.

00:08:34.270 --> 00:08:37.179
Now, this thing right here, let
me make this very clear,

00:08:37.179 --> 00:08:41.280
I'm claiming that this is
equivalent to this.

00:08:41.280 --> 00:08:44.530
Because everywhere else, the
delta function zeroes out this

00:08:44.530 --> 00:08:47.300
function, so we only care about
this function, or e to

00:08:47.300 --> 00:08:50.430
the minus st f of t when
t is equal to c.

00:08:50.430 --> 00:08:52.810
And so that's why we were able
to turn it into a constant.

00:08:52.810 --> 00:08:54.920
But since this is a constant,
we can bring it out of the

00:08:54.920 --> 00:08:59.380
integral, and so this is equal
to-- I'm going to go backwards

00:08:59.380 --> 00:09:01.860
here just to kind of save space
and still give you these

00:09:01.860 --> 00:09:03.120
things to look at.

00:09:03.120 --> 00:09:07.690
If we take out the constants
from inside of the integral,

00:09:07.690 --> 00:09:14.540
we get e to the minus sc times f
of c times the integral from

00:09:14.540 --> 00:09:21.650
0 to infinity of f
of t minus c dt.

00:09:21.650 --> 00:09:23.830
Oh sorry, not f of t minus c.

00:09:27.520 --> 00:09:28.370
This is not an f.

00:09:28.370 --> 00:09:29.380
I have to be very careful.

00:09:29.380 --> 00:09:31.000
This is a delta.

00:09:31.000 --> 00:09:33.160
Let me do that in a
different color.

00:09:33.160 --> 00:09:35.960
I took out the constant terms
there, and it's going to be a

00:09:35.960 --> 00:09:39.710
delta of t minus c dt.

00:09:42.410 --> 00:09:43.760
Let me get the right color.

00:09:46.350 --> 00:09:49.300
Now, what is this thing
by definition?

00:09:49.300 --> 00:09:50.370
This thing is 1.

00:09:50.370 --> 00:09:52.420
I mean, we could put it from
minus infinity to infinity, it

00:09:52.420 --> 00:09:53.150
doesn't matter.

00:09:53.150 --> 00:09:55.170
The only time where it has any
area is right under c.

00:09:55.170 --> 00:09:57.110
So this thing is equal to 1.

00:09:57.110 --> 00:10:01.390
So this whole integral right
there has been reduced to this

00:10:01.390 --> 00:10:04.040
right there, because this
is just equal to 1.

00:10:04.040 --> 00:10:07.460
So the Laplace transform of
our shifted delta function

00:10:07.460 --> 00:10:13.060
times some other function is
equal to e to the minus sc

00:10:13.060 --> 00:10:15.200
times f of c.

00:10:15.200 --> 00:10:18.160
Let me write that
again down here.

00:10:18.160 --> 00:10:20.650
Let me write it all at once.

00:10:20.650 --> 00:10:25.590
So the Laplace transform of our
shifted delta function t

00:10:25.590 --> 00:10:32.900
minus c times some function
f of t, it equals e

00:10:32.900 --> 00:10:35.570
to the minus c.

00:10:35.570 --> 00:10:38.420
Essentially, we're just
evaluating e to the minus st

00:10:38.420 --> 00:10:39.240
evaluated at c.

00:10:39.240 --> 00:10:44.110
So e to the minus
cs times f of c.

00:10:44.110 --> 00:10:47.420
We're essentially just
evaluating these things at c.

00:10:47.420 --> 00:10:48.680
This is what it equals.

00:10:48.680 --> 00:10:51.160
So from this we can get a lot
of interesting things.

00:10:51.160 --> 00:10:55.080
What is the Laplace transform--
actually, what is

00:10:55.080 --> 00:10:57.490
the Laplace transform
of just the plain

00:10:57.490 --> 00:11:00.250
vanilla delta function?

00:11:00.250 --> 00:11:05.880
Well, in this case, we have c
is equal to 0, and f of t is

00:11:05.880 --> 00:11:06.880
equal to 1.

00:11:06.880 --> 00:11:08.340
It's just a constant term.

00:11:08.340 --> 00:11:11.090
So if we do that, then the
Laplace transform of this

00:11:11.090 --> 00:11:16.095
thing is just going to be e to
the minus 0 times s times 1,

00:11:16.095 --> 00:11:18.040
which is just equal to 1.

00:11:18.040 --> 00:11:21.740
So the Laplace transform of our
delta function is 1, which

00:11:21.740 --> 00:11:23.850
is a nice clean thing
to find out.

00:11:23.850 --> 00:11:26.270
And then if we wanted to just
figure out the Laplace

00:11:26.270 --> 00:11:28.530
transform of our shifted
function, the Laplace

00:11:28.530 --> 00:11:35.870
transform of our shifted delta
function, this is just a

00:11:35.870 --> 00:11:37.813
special case where f
of t is equal to 1.

00:11:40.440 --> 00:11:42.870
We could write it times 1, where
f of t is equal to 1.

00:11:42.870 --> 00:11:47.810
So this is going to be equal to
e to the minus cs times f

00:11:47.810 --> 00:11:51.840
of c, but f is just a constant,
f is just 1 here.

00:11:51.840 --> 00:11:55.670
So it's times 1, or it's
just e to the mine cs.

00:11:55.670 --> 00:12:01.550
So just like that, using a kind
of visual evaluation of

00:12:01.550 --> 00:12:03.820
the integral, we were able
to figure out the Laplace

00:12:03.820 --> 00:12:07.850
transforms for a bunch of
different situations involving

00:12:07.850 --> 00:12:09.510
the Dirac delta function.

00:12:09.510 --> 00:12:12.400
Anyway, hopefully, you found
that reasonably useful.


Dirac delta function

WEBVTT
Kind: captions
Language: en

00:00:00.390 --> 00:00:03.050
When I introduced you to the
unit step function, I said,

00:00:03.050 --> 00:00:06.440
you know, this type of function,
it's more exotic and

00:00:06.440 --> 00:00:10.240
a little unusual relative to
what you've seen in just a

00:00:10.240 --> 00:00:12.900
traditional Calculus course,
what you've seen in maybe your

00:00:12.900 --> 00:00:13.930
Algebra courses.

00:00:13.930 --> 00:00:15.990
But the reason why this was
introduced is because a lot of

00:00:15.990 --> 00:00:18.590
physical systems kind
of behave this way.

00:00:18.590 --> 00:00:21.700
That all of a sudden nothing
happens for a long period of

00:00:21.700 --> 00:00:22.530
time and then bam!

00:00:22.530 --> 00:00:23.630
Something happens.

00:00:23.630 --> 00:00:24.450
And you go like that.

00:00:24.450 --> 00:00:26.440
And it doesn't happen exactly
like this, but it can be

00:00:26.440 --> 00:00:29.330
approximated by the unit
step function.

00:00:29.330 --> 00:00:34.390
Similarly, sometimes you have
nothing happening for a long

00:00:34.390 --> 00:00:35.590
period of time.

00:00:35.590 --> 00:00:36.870
Nothing happens for
a long period of

00:00:36.870 --> 00:00:39.350
time, and then whack!

00:00:39.350 --> 00:00:42.760
Something hits you really hard
and then goes away, and then

00:00:42.760 --> 00:00:45.300
nothing happens for a very
long period of time.

00:00:45.300 --> 00:00:46.920
And you'll learn this in the
future, you can kind of view

00:00:46.920 --> 00:00:48.200
this is an impulse.

00:00:48.200 --> 00:00:49.680
And we'll talk about
unit impulse

00:00:49.680 --> 00:00:50.600
functions and all of that.

00:00:50.600 --> 00:00:53.760
So wouldn't it be neat if we had
some type of function that

00:00:53.760 --> 00:00:57.130
could model this type
of behavior?

00:00:57.130 --> 00:01:00.360
And in our ideal function,
what would happen is that

00:01:00.360 --> 00:01:04.910
nothing happens until we get
to some point and then bam!

00:01:04.910 --> 00:01:07.150
It would get infinitely strong,
but maybe it has a

00:01:07.150 --> 00:01:08.630
finite area.

00:01:08.630 --> 00:01:11.970
And then it would go back to
zero and then go like that.

00:01:11.970 --> 00:01:16.660
So it'd be infinitely high right
at 0 right there, and

00:01:16.660 --> 00:01:17.630
then it continues there.

00:01:17.630 --> 00:01:21.530
And let's say that the area
under this, it becomes very--

00:01:21.530 --> 00:01:23.880
to call this a function is
actually kind of pushing it,

00:01:23.880 --> 00:01:26.650
and this is beyond the math of
this video, but we'll call it

00:01:26.650 --> 00:01:27.990
a function in this video.

00:01:27.990 --> 00:01:32.300
But you say, well, what good
is this function for?

00:01:32.300 --> 00:01:33.610
How can you even
manipulate it?

00:01:33.610 --> 00:01:37.800
And I'm going to make one more
definition of this function.

00:01:37.800 --> 00:01:40.070
Let's say we call this function
represented by the

00:01:40.070 --> 00:01:43.090
delta, and that's what we do
represent this function by.

00:01:43.090 --> 00:01:45.420
It's called the Dirac
delta function.

00:01:45.420 --> 00:01:48.510
And we'll just informally say,
look, when it's in infinity,

00:01:48.510 --> 00:01:52.480
it pops up to infinity
when x equal to 0.

00:01:52.480 --> 00:01:56.525
And it's zero everywhere else
when x is not equal to 0.

00:01:56.525 --> 00:01:58.040
And you say, how do
I deal with that?

00:01:58.040 --> 00:01:59.840
How do I take the integral
of that?

00:01:59.840 --> 00:02:02.220
And to help you with that, I'm
going to make a definition.

00:02:02.220 --> 00:02:04.620
I'm going to tell you what
the integral of this is.

00:02:04.620 --> 00:02:06.690
This is part of the definition
of the function.

00:02:06.690 --> 00:02:09.470
I'm going to tell you that if I
were to take the integral of

00:02:09.470 --> 00:02:12.510
this function from minus
infinity to infinity, so

00:02:12.510 --> 00:02:16.750
essentially over the entire real
number line, if I take

00:02:16.750 --> 00:02:22.010
the integral of this function,
I'm defining it

00:02:22.010 --> 00:02:24.490
to be equal to 1.

00:02:24.490 --> 00:02:25.590
I'm defining this.

00:02:25.590 --> 00:02:27.600
Now, you might say, Sal, you
didn't prove it to me.

00:02:27.600 --> 00:02:28.670
No, I'm defining it.

00:02:28.670 --> 00:02:31.960
I'm telling you that this delta
of x is a function such

00:02:31.960 --> 00:02:34.640
that its integral is 1.

00:02:34.640 --> 00:02:38.170
So it has this infinitely
narrow base that goes

00:02:38.170 --> 00:02:42.810
infinitely high, and the area
under this-- I'm telling you--

00:02:42.810 --> 00:02:44.090
is of area 1.

00:02:44.090 --> 00:02:47.380
And you're like, hey, Sal,
that's a crazy function.

00:02:47.380 --> 00:02:50.490
I want a little bit better
understanding of how someone

00:02:50.490 --> 00:02:52.110
can construct a function
like this.

00:02:52.110 --> 00:02:56.520
So let's see if we can satisfy
that a little bit more.

00:02:56.520 --> 00:02:57.880
But then once that's satisfied,
then we're going to

00:02:57.880 --> 00:03:00.540
start taking the Laplace
transform of this, and then

00:03:00.540 --> 00:03:03.240
we'll start manipulating
it and whatnot.

00:03:03.240 --> 00:03:06.240
Let's see, let me complete
this delta right here.

00:03:06.240 --> 00:03:08.420
Let's say that I constructed
another function.

00:03:08.420 --> 00:03:14.110
Let's call it d sub tau And this
is all just to satisfy

00:03:14.110 --> 00:03:17.640
this craving for maybe a better
intuition for how this

00:03:17.640 --> 00:03:20.270
Dirac delta function
can be constructed.

00:03:20.270 --> 00:03:24.930
And let's say my d sub tau of--
well, let me put it as a

00:03:24.930 --> 00:03:27.690
function of t because everything
we're doing in the

00:03:27.690 --> 00:03:29.280
Laplace transform world,
everything's been

00:03:29.280 --> 00:03:30.600
a function of t.

00:03:30.600 --> 00:03:40.000
So let's say that it equals 1
over 2 tau, and you'll see why

00:03:40.000 --> 00:03:42.290
I'm picking these numbers
the way I am.

00:03:42.290 --> 00:03:48.805
1 over 2 tau when t is
less then tau and

00:03:48.805 --> 00:03:51.260
greater than minus tau.

00:03:51.260 --> 00:03:53.640
And let's say it's 0
everywhere else.

00:03:58.470 --> 00:04:01.890
So this type of equation,
this is more reasonable.

00:04:01.890 --> 00:04:04.130
This will actually look like
a combination of unit step

00:04:04.130 --> 00:04:06.120
functions, and we can actually
define it as a combination of

00:04:06.120 --> 00:04:07.460
unit step functions.

00:04:07.460 --> 00:04:10.220
So if I draw, that's
my x-axis.

00:04:14.130 --> 00:04:19.680
And then if I put my
y-axis right here.

00:04:19.680 --> 00:04:20.930
That's my y-axis.

00:04:23.520 --> 00:04:25.710
Sorry, this is a t-axis.

00:04:25.710 --> 00:04:26.990
I have to get out
of that habit.

00:04:26.990 --> 00:04:29.980
This is the t-axis, and, I mean,
we could call it the

00:04:29.980 --> 00:04:33.430
y-axis or the f of t-axis, or
whatever we want to call it.

00:04:33.430 --> 00:04:35.160
That's the dependent variable.

00:04:35.160 --> 00:04:36.910
So what's going to
happen here?

00:04:36.910 --> 00:04:38.820
It's going to be zero everywhere
until we get to

00:04:38.820 --> 00:04:40.870
minus t, and then at minus
t, we're going to

00:04:40.870 --> 00:04:43.290
jump up to some level.

00:04:43.290 --> 00:04:44.820
Just let me put that
point here.

00:04:44.820 --> 00:04:48.825
So this is minus tau, and
this is plus tau.

00:04:52.010 --> 00:04:53.700
So it's going to be zero
everywhere, and then at minus

00:04:53.700 --> 00:04:58.040
tau, we jump to this level, and
then we stay constant at

00:04:58.040 --> 00:05:01.670
that level until we
get to plus tau.

00:05:01.670 --> 00:05:05.430
And that level, I'm saying
is 1 over 2 tau.

00:05:05.430 --> 00:05:11.080
So this point right here on the
dependent axis, this is 1

00:05:11.080 --> 00:05:13.080
over 2 tau.

00:05:13.080 --> 00:05:16.630
So why did I construct this
function this way?

00:05:16.630 --> 00:05:17.710
Well, let's think about it.

00:05:17.710 --> 00:05:21.410
What happens if I take
the integral?

00:05:21.410 --> 00:05:23.640
Let me write a nicer
integral sign.

00:05:23.640 --> 00:05:29.790
If I took the integral from
minus infinity to infinity of

00:05:29.790 --> 00:05:37.700
d sub tau of t dt, what is this
going to be equal to?

00:05:37.700 --> 00:05:41.100
Well, if the integral is just
the area under this curve,

00:05:41.100 --> 00:05:41.700
this is a pretty

00:05:41.700 --> 00:05:43.900
straightforward thing to calculate.

00:05:43.900 --> 00:05:46.440
You just look at this, and you
say, well, first of all, it's

00:05:46.440 --> 00:05:48.480
zero everywhere else.

00:05:48.480 --> 00:05:52.380
It's zero everywhere else, and
it's only the area right here.

00:05:52.380 --> 00:05:56.730
I mean, I could rewrite this
integral as the integral from

00:05:56.730 --> 00:06:00.560
minus tau to tau-- and we don't
care if infinity and

00:06:00.560 --> 00:06:02.870
minus infinity or positive
infinity, because there's no

00:06:02.870 --> 00:06:06.280
area under any of those
points-- of 1

00:06:06.280 --> 00:06:11.230
over 2 tau d tau.

00:06:11.230 --> 00:06:12.900
Sorry, 1 over 2 tau dt.

00:06:12.900 --> 00:06:14.460
So we could write it this
way too, right?

00:06:14.460 --> 00:06:17.910
Because we can just take the
boundaries from here to here,

00:06:17.910 --> 00:06:23.130
because we get nothing whether t
goes to positive infinity or

00:06:23.130 --> 00:06:24.130
minus infinity.

00:06:24.130 --> 00:06:27.380
And then over that boundary, the
function is a constant, 1

00:06:27.380 --> 00:06:29.840
over 2 tau, so we could just
take this integral.

00:06:29.840 --> 00:06:31.310
And either way we evaluate it.

00:06:31.310 --> 00:06:32.990
We don't even have to know
calculus to know what this

00:06:32.990 --> 00:06:34.680
integral's going
to evaluate to.

00:06:34.680 --> 00:06:41.340
This is just the area under
this, which is just the base.

00:06:41.340 --> 00:06:42.130
What's the base?

00:06:42.130 --> 00:06:44.275
The base is 2 tau.

00:06:47.940 --> 00:06:50.550
You have one tau here and
then another tau there.

00:06:50.550 --> 00:06:54.240
So it's equal to 2 tau
times your height.

00:06:54.240 --> 00:06:58.460
And your height, I just
said, is 1 over 2 tau.

00:06:58.460 --> 00:07:01.880
So your area for this function,
or for this

00:07:01.880 --> 00:07:03.560
integral, is going to be 1.

00:07:03.560 --> 00:07:04.680
You could evaluate this.

00:07:04.680 --> 00:07:07.160
You could get this is going to
be equal to-- you take the

00:07:07.160 --> 00:07:10.630
antiderivative of 1 over 2 tau,
you get-- I'll do this

00:07:10.630 --> 00:07:14.230
just to satiate your curiosity--
t over 2 tau, and

00:07:14.230 --> 00:07:18.750
you have to evaluate this
from minus tau to tau.

00:07:18.750 --> 00:07:23.310
And when you would put tau in
there, you get tau over 2 tau,

00:07:23.310 --> 00:07:29.550
and then minus minus tau over
2 tau, and then you get tau

00:07:29.550 --> 00:07:33.850
plus tau over 2 tau, that's
2 tau over 2 tau,

00:07:33.850 --> 00:07:35.110
which is equal to 1.

00:07:35.110 --> 00:07:37.120
Maybe I'm beating
a dead horse.

00:07:37.120 --> 00:07:40.490
I think you're satisfied that
the area under this is going

00:07:40.490 --> 00:07:43.300
to be 1, regardless
of what tau was.

00:07:43.300 --> 00:07:45.650
I kept this abstract.

00:07:45.650 --> 00:07:48.510
Now, if I take smaller and
smaller values of tau, what's

00:07:48.510 --> 00:07:49.450
going to happen?

00:07:49.450 --> 00:07:56.730
If my new tau is going to be
here, let's say my new tau is

00:07:56.730 --> 00:08:00.357
going to be there, I'm just
going to pick up my new tau

00:08:00.357 --> 00:08:03.320
there, then my 1 over
2 tau, the tau is

00:08:03.320 --> 00:08:04.560
now a smaller number.

00:08:04.560 --> 00:08:06.250
So when it's in the denominator,
my 1 over 2 tau

00:08:06.250 --> 00:08:12.210
is going to be something
like this, right?

00:08:12.210 --> 00:08:14.670
I mean, I'm just saying, if I
pick smaller and smaller taus.

00:08:14.670 --> 00:08:18.380
So then if I pick an even
smaller tau than that, then my

00:08:18.380 --> 00:08:20.900
height is going to be
have to be higher.

00:08:20.900 --> 00:08:22.750
My 1 over 2 tau is going
to have to even

00:08:22.750 --> 00:08:25.010
be higher than that.

00:08:25.010 --> 00:08:26.860
And so I think you see where
I'm going with this.

00:08:26.860 --> 00:08:31.080
What happens as the limit
as tau approaches zero?

00:08:31.080 --> 00:08:41.169
So what is the limit as tau
approaches zero of my little d

00:08:41.169 --> 00:08:44.380
sub tau function?

00:08:44.380 --> 00:08:46.070
What's the limit of this?

00:08:46.070 --> 00:08:49.370
Well, these things are going
to go infinitely close to

00:08:49.370 --> 00:08:50.530
zero, but this is the limit.

00:08:50.530 --> 00:08:52.460
They're never going to
be quite at zero.

00:08:52.460 --> 00:08:56.050
And your height here is going to
go infinitely high, but the

00:08:56.050 --> 00:08:59.400
whole time, I said no matter
what my tau is, because it was

00:08:59.400 --> 00:09:01.340
defined very arbitrarily,
was my area is

00:09:01.340 --> 00:09:03.130
always going to be 1.

00:09:03.130 --> 00:09:08.860
So you're going to end up with
your Dirac delta function.

00:09:08.860 --> 00:09:09.590
Let me write it now.

00:09:09.590 --> 00:09:10.430
I was going to write
an x again.

00:09:10.430 --> 00:09:14.630
Your Dirac delta function is a
function of t, and because of

00:09:14.630 --> 00:09:22.130
this, if you ask what's the
limit as tau approaches zero

00:09:22.130 --> 00:09:26.220
of the integral from minus
infinity to infinity of d sub

00:09:26.220 --> 00:09:30.900
tau of t dt, well, this should
still be 1, right?

00:09:30.900 --> 00:09:33.230
Because this thing right here,
this evaluates to 1.

00:09:33.230 --> 00:09:36.630
So as you take the limit as tau
approaches zero-- and I'm

00:09:36.630 --> 00:09:38.940
being very generous with
my definitions

00:09:38.940 --> 00:09:40.020
of limits and whatnot.

00:09:40.020 --> 00:09:41.330
I'm not being very rigorous.

00:09:41.330 --> 00:09:43.665
But I think you can kind of
understand the intuition of

00:09:43.665 --> 00:09:44.470
where I'm going.

00:09:44.470 --> 00:09:46.630
This is going to
be equal to 1.

00:09:46.630 --> 00:09:51.050
And so by the same intuitive
argument, you could say that

00:09:51.050 --> 00:09:55.690
the limit from minus infinity to
infinity of our Dirac delta

00:09:55.690 --> 00:10:01.600
function of t dt is also
going to be 1.

00:10:01.600 --> 00:10:05.000
And likewise, the Dirac delta
function-- I mean, this thing

00:10:05.000 --> 00:10:09.370
pops up to infinity at
t is equal to 0.

00:10:09.370 --> 00:10:14.830
This thing, if I were to draw my
x-axis like that, and then

00:10:14.830 --> 00:10:17.980
right at t equals 0, my
Dirac delta function

00:10:17.980 --> 00:10:19.180
pops up like that.

00:10:19.180 --> 00:10:20.970
And you normally draw
it like that.

00:10:20.970 --> 00:10:23.730
And you normally draw it so
it goes up to 1 to kind

00:10:23.730 --> 00:10:24.670
of depict its area.

00:10:24.670 --> 00:10:26.630
But you actually put an arrow
there, and so this is your

00:10:26.630 --> 00:10:28.270
Dirac delta function.

00:10:28.270 --> 00:10:29.820
But what happens if you
want to shift it?

00:10:32.800 --> 00:10:36.860
How would I represent my--
let's say I want

00:10:36.860 --> 00:10:38.660
to do t minus 3?

00:10:38.660 --> 00:10:40.700
What would the graph
of this be?

00:10:40.700 --> 00:10:43.450
Well, this would just be
shifting it to the right by 3.

00:10:43.450 --> 00:10:46.100
For example, when t equals 3,
this will become the Dirac

00:10:46.100 --> 00:10:47.160
delta of 0.

00:10:47.160 --> 00:10:51.630
So this graph will just
look like this.

00:10:51.630 --> 00:10:54.550
This will be my x-axis.

00:10:54.550 --> 00:10:58.610
And let's say that this
is my y-axis.

00:10:58.610 --> 00:11:00.330
Let me just make that 1.

00:11:00.330 --> 00:11:04.810
And let me just draw some points
here, so it's 1, 2, 3

00:11:04.810 --> 00:11:06.360
That's t is equal to 3.

00:11:06.360 --> 00:11:07.390
Did I say that was the x-axis?

00:11:07.390 --> 00:11:08.930
That's my t-axis.

00:11:08.930 --> 00:11:09.935
This is t equal to 3.

00:11:09.935 --> 00:11:12.760
And what I'm going to do here is
the Dirac delta function is

00:11:12.760 --> 00:11:14.125
going to be zero everywhere.

00:11:18.020 --> 00:11:19.870
But then right at 3, it
goes infinitely high.

00:11:19.870 --> 00:11:22.240
And obviously, we don't have
enough paper to draw an

00:11:22.240 --> 00:11:24.230
infinitely high spike
right there.

00:11:24.230 --> 00:11:26.440
So what we do is we
draw an arrow.

00:11:26.440 --> 00:11:27.660
We draw an arrow there.

00:11:27.660 --> 00:11:30.890
And the arrow, we usually draw
the magnitude of the area

00:11:30.890 --> 00:11:32.480
under that spike.

00:11:32.480 --> 00:11:34.110
So we do it like this.

00:11:34.110 --> 00:11:35.280
And let me be clear.

00:11:35.280 --> 00:11:37.600
This is not saying that the
function just goes to 1 and

00:11:37.600 --> 00:11:38.720
then spikes back down.

00:11:38.720 --> 00:11:41.080
This tells me that the
area under the

00:11:41.080 --> 00:11:42.400
function is equal to 1.

00:11:42.400 --> 00:11:44.810
This spike would have to be
infinitely high to have any

00:11:44.810 --> 00:11:50.250
area, considering it has an
infinitely small base, so the

00:11:50.250 --> 00:11:55.470
area under this impulse function
or under this Dirac

00:11:55.470 --> 00:11:57.250
delta function.

00:11:57.250 --> 00:12:01.940
Now, this one right here is t
minus 3, but your area under

00:12:01.940 --> 00:12:03.740
this is still going to be 1.

00:12:03.740 --> 00:12:05.415
And that's why I made
the arrow go to 1.

00:12:07.960 --> 00:12:10.050
Let's say I wanted to graph--
let me do it in another color.

00:12:10.050 --> 00:12:14.750
Let's say I wanted to graph
2 times the Dirac

00:12:14.750 --> 00:12:17.520
delta of t minus 2.

00:12:17.520 --> 00:12:19.480
How would I graph this?

00:12:19.480 --> 00:12:21.830
Well, I would go to t minus 2.

00:12:21.830 --> 00:12:26.010
When t is equal to 2, you get
the Dirac delta of zero, so

00:12:26.010 --> 00:12:27.490
that's where you would
have your spike.

00:12:27.490 --> 00:12:31.550
And we're multiplying it by 2,
so you would do a spike twice

00:12:31.550 --> 00:12:32.820
as high like this.

00:12:32.820 --> 00:12:35.250
Now, both of these go to
infinity, but this goes twice

00:12:35.250 --> 00:12:36.460
as high to infinity.

00:12:36.460 --> 00:12:38.890
And I know this is all being
a little ridiculous now.

00:12:38.890 --> 00:12:41.940
But the idea here is that the
area under this curve should

00:12:41.940 --> 00:12:43.590
be twice the area under
this curve.

00:12:43.590 --> 00:12:46.320
And that's why we make the arrow
go to 2 to say that the

00:12:46.320 --> 00:12:49.020
area under this arrow is 2.

00:12:49.020 --> 00:12:51.200
The spike would have to
go infinitely high.

00:12:51.200 --> 00:12:54.060
So this is all a little
abstract, but this is a useful

00:12:54.060 --> 00:12:58.770
way to model things that are
kind of very jarring.

00:12:58.770 --> 00:13:00.785
Obviously, nothing actually
behaves like this, but there

00:13:00.785 --> 00:13:06.160
are a lot of phenomena in
physics or the real world that

00:13:06.160 --> 00:13:08.820
have this spiky behavior.

00:13:08.820 --> 00:13:10.460
Instead of trying to say,
oh, what does that spike

00:13:10.460 --> 00:13:11.180
exactly look like?

00:13:11.180 --> 00:13:13.360
We say, hey, that's a Dirac
delta function.

00:13:13.360 --> 00:13:17.420
And we'll dictate its impulse
by something like this.

00:13:17.420 --> 00:13:20.010
And just to give you a little
bit of motivation behind this,

00:13:20.010 --> 00:13:23.200
and I was going to go here in
the last video, but then I

00:13:23.200 --> 00:13:25.360
kind of decided not to.

00:13:25.360 --> 00:13:26.870
But I'm just going to show it,
because I've been doing a lot

00:13:26.870 --> 00:13:28.370
of differential equations and
I've been giving you no

00:13:28.370 --> 00:13:31.066
motivation for how this applies
in the real world.

00:13:31.066 --> 00:13:35.720
But you can imagine, if I have
just a wall, and then I have a

00:13:35.720 --> 00:13:42.740
spring attached to some mass
right there, and let's say

00:13:42.740 --> 00:13:45.550
that this is a natural state
of the spring, so that the

00:13:45.550 --> 00:13:48.790
spring would want to be here,
so it's been stretched a

00:13:48.790 --> 00:13:53.340
distance y from its kind of
natural where it wants to go.

00:13:53.340 --> 00:13:59.160
And let's say I have some
external force right here.

00:13:59.160 --> 00:14:01.330
Let's say I have some external
force right here on the

00:14:01.330 --> 00:14:03.600
spring, and, of course, let's
say it's ice on ice.

00:14:03.600 --> 00:14:05.510
There's no friction
in all of this.

00:14:05.510 --> 00:14:07.940
And I just want to show you
that I can represent the

00:14:07.940 --> 00:14:10.860
behavior of this system with
the differential equation.

00:14:10.860 --> 00:14:13.680
And actually things like the
unit step functions, the Dirac

00:14:13.680 --> 00:14:17.520
delta function, actually start
to become useful in this type

00:14:17.520 --> 00:14:18.660
of environment.

00:14:18.660 --> 00:14:24.610
So we know that F is equal to
mass times acceleration.

00:14:24.610 --> 00:14:26.240
That's basic physics
right there.

00:14:26.240 --> 00:14:28.940
Now, what are all
of the forces on

00:14:28.940 --> 00:14:33.510
this mass right here?

00:14:33.510 --> 00:14:37.490
Well, you have this
force right here.

00:14:37.490 --> 00:14:39.620
And I'll say this is a positive
rightward direction,

00:14:39.620 --> 00:14:42.780
so it's that force, and then
you have a minus force from

00:14:42.780 --> 00:14:44.650
the spring.

00:14:44.650 --> 00:14:47.100
The force from the spring
is Hooke's Law.

00:14:47.100 --> 00:14:51.910
It's proportional to how far
it's been stretched from its

00:14:51.910 --> 00:14:54.620
kind of natural point, so its
force in that direction is

00:14:54.620 --> 00:14:57.880
going to be ky, or you could
call it minus ky, because it's

00:14:57.880 --> 00:14:59.850
going in the opposite direction
of what we've

00:14:59.850 --> 00:15:01.250
already said is a positive
direction.

00:15:01.250 --> 00:15:06.970
So the net forces on this is F
minus ky, and that's equal to

00:15:06.970 --> 00:15:13.470
the mass of our object times
its acceleration.

00:15:13.470 --> 00:15:14.520
Now, what's its acceleration?

00:15:14.520 --> 00:15:20.510
If its position is y, so if y
is equal to position, if we

00:15:20.510 --> 00:15:24.070
take the derivative of y with
respect to t, y prime, which

00:15:24.070 --> 00:15:29.570
we could also say dy dt, this
is going to be its velocity.

00:15:29.570 --> 00:15:32.210
And then if we take the
derivative of that, y prime

00:15:32.210 --> 00:15:36.200
prime, which is equal to d
squared y with respect to dt

00:15:36.200 --> 00:15:37.940
squared, this is equal
to acceleration.

00:15:41.080 --> 00:15:48.340
So instead of writing a, we
could right y prime prime.

00:15:48.340 --> 00:15:51.110
And so, if we just put this
on the other side of the

00:15:51.110 --> 00:15:54.010
equation, what do we get?

00:15:54.010 --> 00:15:58.270
We get the force-- this force,
not just this force; this was

00:15:58.270 --> 00:16:02.540
just F equals ma-- but this
force is equal to the mass of

00:16:02.540 --> 00:16:05.610
our object, times the
acceleration of the object

00:16:05.610 --> 00:16:09.400
plus whatever the spring
constant is for the spring

00:16:09.400 --> 00:16:14.140
plus k times our position,
times y.

00:16:14.140 --> 00:16:17.580
So if you had no outside force,
if this was zero you'd

00:16:17.580 --> 00:16:20.370
have a homogeneous differential
equation.

00:16:20.370 --> 00:16:23.150
And in that case, the spring
would just start

00:16:23.150 --> 00:16:24.560
moving on its own.

00:16:24.560 --> 00:16:27.590
But now this F, all of a
sudden, it's kind of a

00:16:27.590 --> 00:16:31.610
non-homogeneous term, it's what
the outside force you're

00:16:31.610 --> 00:16:33.900
applying to this mass.

00:16:33.900 --> 00:16:36.780
So if this outside force was
some type of Dirac delta

00:16:36.780 --> 00:16:42.310
function-- so let's say it's t
minus 2 is equal to our mass

00:16:42.310 --> 00:16:50.190
times y prime prime plus our
spring constant times y, this

00:16:50.190 --> 00:16:54.180
is saying that at time is equal
to 2 seconds, we're just

00:16:54.180 --> 00:16:56.770
going to jar this thing
to the right.

00:16:56.770 --> 00:16:58.960
And it's going to have an-- and
I'll talk more about it--

00:16:58.960 --> 00:17:00.360
it's going to have
an impulse of 2.

00:17:00.360 --> 00:17:04.280
It's force times time is going
to be-- or its impulse is

00:17:04.280 --> 00:17:05.910
going to have 1.

00:17:05.910 --> 00:17:07.560
And I don't want to get too much
into the physics here,

00:17:07.560 --> 00:17:11.630
but its impulse or its change in
momentum, is going to be of

00:17:11.630 --> 00:17:14.430
magnitude 1, depending on
what our units are.

00:17:14.430 --> 00:17:16.890
But anyway, I just wanted to
take a slight diversion,

00:17:16.890 --> 00:17:20.329
because you might think Sal is
introducing me to these weird,

00:17:20.329 --> 00:17:21.349
exotic functions.

00:17:21.349 --> 00:17:23.569
What are they ever going
to be good for?

00:17:23.569 --> 00:17:27.369
But this is good for the idea
that sometimes you just jar

00:17:27.369 --> 00:17:29.750
this thing by some magnitude
and then let go.

00:17:29.750 --> 00:17:33.790
And you do it kind of infinitely
fast, but you do it

00:17:33.790 --> 00:17:38.540
enough to change the momentum of
this in a well-defined way.

00:17:38.540 --> 00:17:41.140
Anyway, in the next video, we'll
continue with the Dirac

00:17:41.140 --> 00:17:41.870
delta function.

00:17:41.870 --> 00:17:44.830
We'll figure out its Laplace
transform and see what it does

00:17:44.830 --> 00:17:47.650
to the Laplace transforms
of other functions.