WEBVTT
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I usually like to start a
lecture with something new,
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but this time I'm going to make
an exception,
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and start with the finishing up
on Friday because it involves a
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little more practice with
complex numbers.
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I think that's what a large
number of you are still fairly
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weak in.
So, to briefly remind you,
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it will be sort of
self-contained,
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but still, it will use complex
numbers.
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And, I think it's a good way to
start today.
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So, remember,
the basic problem was to solve
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something with,
where the input was sinusoidal
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in particular.
The k was on both sides,
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and the input looked like
cosine omega t.
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And, the plan of the solution
consisted of transporting the
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problem to the complex domain.
So, you look for a complex
00:00:57.000 --> 00:01:03.000
solution, and you complexify the
right hand side of the equation,
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as well.
So, cosine omega t
00:01:07.000 --> 00:01:13.000
becomes the real part of
this complex function.
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The reason for doing that,
remember, was because it's
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easier to handle when you solve
linear equations.
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It's much easier to handle
exponentials on the right-hand
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side than it is to handle sines
and cosines because exponentials
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are so easy to integrate when
you multiply them by other
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exponentials.
So, the result was,
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after doing that,
y tilda turned out to be one,
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after I scale the coefficient,
one over one plus omega over k
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And then, the rest was e to the
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i times (omega t minus phi),
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where phi had a certain
meaning.
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It was the arc tangent of a,
it was a phase lag.
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And, this was then,
I had to take the real part of
00:02:09.000 --> 00:02:15.000
this to get the final answer,
which came out to be something
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like one over the square root of
one plus the amplitude one omega
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/ k squared, and then the rest
was cosine omega t plus minus
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phi.
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It's easier to see that that
part is the real part of this;
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the problem is,
of course you have to convert
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this.
Sorry, this should be i omega
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t, in which case you don't need
the parentheses,
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either.
So, the problem was to use the
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polar representation of this
complex number to convert it
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into something whose amplitude
was this, and whose angle was
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minus phi.
Now, that's what we call the
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polar method,
going polar.
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I'd like, now,
for the first few minutes of
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the period, to talk about the
other method,
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the Cartesian method.
I think for a long while,
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many of you will be more
comfortable with that anyway.
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Although, one of the objects of
the course should be to get you
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equally comfortable with the
polar representation of complex
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numbers.
So, if we try to do the same
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thing going Cartesian,
what's going to happen?
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Well, I guess the same point
here.
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So, the starting point is still
y tilde equals one over,
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sorry, this should have an i
here, one plus i times omega
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over k, e to the i omega t.
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But now, what you're going to
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do is turn this into its
Cartesian, turn both of these
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into their Cartesian
representations as a plus ib.
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So, if you do that Cartesianly,
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of course, what you have to do
is the standard thing about
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dividing complex numbers or
taking the reciprocals that I
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told you at the very beginning
of complex numbers.
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You multiply the top and bottom
by the complex conjugate of this
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in order to make the bottom
real.
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So, what does this become?
This becomes one minus i times
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omega over k divided by the
product of this in its complex
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conjugate, which is the real
number, one plus omega over k
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squared
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So, I've now converted this to
the a plus bi form.
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I have also to convert the
right-hand side to the a plus bi
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form.
So, it will look like cosine
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omega t plus i sine omega t.
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Having done that,
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I take the last step,
which is to take the real part
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of that.
Remember, the reason I want the
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real part is because this input
was the real part of the complex
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input.
So, once you've got the complex
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solution, you have to take its
real part to go back into the
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domain you started with,
of real numbers,
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from the domain of complex
numbers.
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So, I want the real part is
going to be, the real part of
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that is, first of all,
there's a factor out in front.
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That's entirely real.
Let's put that out in front,
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so doesn't bother us
particularly.
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And now, I need the product of
this complex number and that
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complex number.
But, I only want the real part
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of it.
So, I'm not going to multiply
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it out and get four terms.
I'm just going to look at the
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two terms that I do want.
I don't want the others.
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All right, the real part is
cosine omega t,
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from the product of this and
that.
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And, the rest of the real part
will be the product of the two i
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terms.
But, it's i times negative i,
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which makes one.
And therefore,
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it's omega over k times sine
omega t.
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Now, that's the answer.
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And that's the answer,
too; they must be equal,
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unless there's a contradiction
in mathematics.
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But, it's extremely important.
And that's the other reason why
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I'm giving you this,
that you learn in this course
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to be able to convert quickly
and automatically things that
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look like this into things that
look like that.
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And, that's done by means of a
basic formula,
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which occurs at the end of the
notes for reference,
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as I optimistically say,
although I think for a lot of
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you will not be referenced,
stuff in the category of,
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yeah, I think I've vaguely seen
that somewhere.
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But, well, we never used it for
anything.
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Okay, you're going to use it
all term.
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So, the formula is,
the famous trigonometric
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identity, which is,
so, the problem is to convert
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this into the other guy.
And, the thing which is going
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to do that, enable one to
combine the sine and the cosine
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terms, is the famous formula
that a times the cosine,
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I'm going to use theta to make
it as neutral as possible,
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--
-- so, theta you can think of
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as being replaced by omega t in
this particular application of
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the formula.
But, I'll just use a general
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angle theta, which doesn't
suggest anything in particular.
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So, the problem is,
you have something which is a
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combination with real
coefficients of cosine and sine,
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and the important thing is that
these numbers be the same.
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In practice,
that means that the omega t,
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you're not allowed to have
omega one t here,
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and some other frequency,
omega two t here.
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That would correspond to using
theta one here,
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and theta two here.
And, though there is a formula
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for combining that,
nobody remembers it,
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and it's, in general,
less universally useful than
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the first.
If you're going to memorize a
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formula, and learn this one,
it's best to start with the
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ones where the two are equal.
That's the basic formula.
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The others are variations of
it, but there is a sizable
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variations.
All right, so the answer is
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that this is equal to some other
constant, real constant,
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times the cosine of theta minus
phi.
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Of course, most people remember
this vaguely.
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What they don't remember is
what the c and the phi are,
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how to calculate them.
I don't suggest you memorize
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the formulas for them.
You can if you wish.
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Instead, memorize the picture,
which is much easier.
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Memorize that a and b are the
two sides of a right triangle.
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Phi is the angle opposite the b
side, and c is the length of the
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hypotenuse.
Okay, that's worth putting up.
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I think that's a pink formula.
It's even worth two of those,
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but I will thrift.
Now, let's apply it to this
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case to see that it gives the
right answer.
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So, to use this formula,
how I use it?
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Well, I should take,
I will reproduce the left-hand
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side.
So that part,
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I just copy.
And, how about the right?
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Well, the amplitude,
it's combined into a single
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cosine term whose amplitude is,
well, the two sides of the
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right triangle are one,
and omega over k.
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The hypotenuse in that case is
going to be, well,
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why don't we write it here?
So, we have one,
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and omega over k.
And, here's phi.
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So, the hypotenuse is going to
be the square root of one plus
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omega over k squared.
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And, that's going to be
multiplied by the cosine of
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omega t minus this phase lag
angle phi.
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You can write,
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if you wish,
phi equals the arc tangent,
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but you are not learning a lot
by that.
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Phi is the arc tangent of omega
over k.
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That's okay,
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but it's true.
But, notice there's
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cancellation now.
This over that is equal to
00:11:23.000 --> 00:11:29.000
what?
Well, it's equal to this.
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And, so when we get in this
way, by combining these two
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factors, one gets exactly the
same formula that we got before.
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So, as you can see,
in some sense,
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there's not,
if you can remember this
00:11:41.000 --> 00:11:47.000
trigonometric identity,
there's not a lot of difference
00:11:45.000 --> 00:11:51.000
between the two methods except
that this one requires this
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extra step.
The answer will come out in
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this form, and you then,
to see what it really looks
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like, really have to convert it
to this form,
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the form in which you can see
what the phase lag and the
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amplitude is.
It's amazing how many people
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who should know,
this includes working
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mathematicians,
theoretical mathematicians,
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includes even possibly the
authors of your textbooks.
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I'm not sure,
but I've caught them in this,
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too, who in this form,
everybody remembers that it's
00:12:24.000 --> 00:12:30.000
something like that.
Unfortunately,
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when it occurs as the answer in
an answer book,
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the numbers are some colossal
mess here plus some colossal
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mess here.
And theta is,
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again, a real mess,
involving roots and some cube
00:12:44.000 --> 00:12:50.000
roots, and whatnot.
The only thing is,
00:12:48.000 --> 00:12:54.000
these two are the same real
mess.
00:12:52.000 --> 00:12:58.000
That amounts to just another
pure oscillation with the same
00:12:58.000 --> 00:13:04.000
frequency as the old guy,
and with the amplitude changed,
00:13:05.000 --> 00:13:11.000
and with a phase shift,
move to the right or left.
00:13:12.000 --> 00:13:18.000
So, this is no more general
than that.
00:13:14.000 --> 00:13:20.000
Notice they both have two
parameters in them,
00:13:18.000 --> 00:13:24.000
these two coefficients.
This one has the two parameters
00:13:22.000 --> 00:13:28.000
in an altered form.
Okay, well, I wanted,
00:13:26.000 --> 00:13:32.000
because of the importance of
this formula,
00:13:29.000 --> 00:13:35.000
I wanted to take a couple of
minutes out for a proof of the
00:13:34.000 --> 00:13:40.000
formula, --
00:13:42.000 --> 00:13:48.000
-- just to give you chance to
stare at it a little more now.
00:13:46.000 --> 00:13:52.000
There are three proofs I know.
I'm sure there are 27.
00:13:50.000 --> 00:13:56.000
The Pythagorean theorem now has
several hundred.
00:13:53.000 --> 00:13:59.000
But, there are three basic
proofs.
00:13:56.000 --> 00:14:02.000
There is the one I will not
give you, I'll call the high
00:14:00.000 --> 00:14:06.000
school proof,
which is the only one one
00:14:03.000 --> 00:14:09.000
normally finds in books,
physics textbooks or other
00:14:07.000 --> 00:14:13.000
textbooks.
The high school proof takes the
00:14:11.000 --> 00:14:17.000
right-hand side,
applies the formula for the
00:14:14.000 --> 00:14:20.000
cosine of the difference of two
angles, which it assumes you had
00:14:19.000 --> 00:14:25.000
in trigonometry,
and then converts it into this.
00:14:22.000 --> 00:14:28.000
It shows you that once you've
done that, that a turns out to
00:14:27.000 --> 00:14:33.000
be c cosine phi
and b, the number b is c sine
00:14:31.000 --> 00:14:37.000
phi,
and therefore it identifies the
00:14:35.000 --> 00:14:41.000
two sides.
Now, the thing that's of course
00:14:39.000 --> 00:14:45.000
correct and it's the simplest
possible argument,
00:14:42.000 --> 00:14:48.000
the thing that's no good about
it is that the direction at
00:14:47.000 --> 00:14:53.000
which it goes is from here to
here.
00:14:49.000 --> 00:14:55.000
Well, everybody knew that.
If I gave you this and told
00:14:53.000 --> 00:14:59.000
you, write it out in terms of
cosine and sine,
00:14:56.000 --> 00:15:02.000
I would assume it dearly hope
that practically all of you can
00:15:01.000 --> 00:15:07.000
do that.
Unfortunately,
00:15:03.000 --> 00:15:09.000
when you want to use the
formula, it's this way you want
00:15:07.000 --> 00:15:13.000
to use it in the opposite
direction.
00:15:09.000 --> 00:15:15.000
You are starting with this,
and want to convert it to that.
00:15:13.000 --> 00:15:19.000
Now, the proof,
therefore, will not be of much
00:15:15.000 --> 00:15:21.000
help.
It requires you to go in the
00:15:17.000 --> 00:15:23.000
backwards direction,
and match up coefficients.
00:15:20.000 --> 00:15:26.000
It's much better to go
forwards.
00:15:22.000 --> 00:15:28.000
Now, there are two proofs that
go forwards.
00:15:25.000 --> 00:15:31.000
There's the 18.02 proof.
Since I didn't teach most of
00:15:28.000 --> 00:15:34.000
you 18.02, I can't be sure you
had it.
00:15:32.000 --> 00:15:38.000
So, I'll spend one minute
giving it to you.
00:15:36.000 --> 00:15:42.000
What is the 18.02 proof?
It is the following picture.
00:15:42.000 --> 00:15:48.000
I think this requires deep
colored chalk.
00:15:46.000 --> 00:15:52.000
This is going to be pretty
heavy.
00:15:50.000 --> 00:15:56.000
All right, first of all,
the a and the b are the given.
00:15:55.000 --> 00:16:01.000
So, I'm going to put in that
vector.
00:16:01.000 --> 00:16:07.000
So, there is the vector whose
sides are, whose components are
00:16:05.000 --> 00:16:11.000
a and b.
I'll write it without the i and
00:16:08.000 --> 00:16:14.000
j.
I hope you had from Jerison
00:16:11.000 --> 00:16:17.000
that form for the vector,
if you don't like that,
00:16:14.000 --> 00:16:20.000
write ai plus bj,
okay?
00:16:17.000 --> 00:16:23.000
Now, there's another vector
lurking around.
00:16:20.000 --> 00:16:26.000
It's the unit vector whose,
I'll write it this way,
00:16:24.000 --> 00:16:30.000
u because it's a unit vector,
and theta to indicate that it's
00:16:29.000 --> 00:16:35.000
angle is theta.
Now, the reason for doing that
00:16:34.000 --> 00:16:40.000
is because you see that the
left-hand side is a dot product
00:16:38.000 --> 00:16:44.000
of two vectors.
The left-hand side of the
00:16:41.000 --> 00:16:47.000
identity is the dot product of
the vector 00:16:51.000
b> with the vector whose
components are cosine theta and
00:16:49.000 --> 00:16:55.000
sine theta.
00:16:52.000 --> 00:16:58.000
That's what I'm calling this
unit vector.
00:16:55.000 --> 00:17:01.000
It's a unit vector because
cosine squared plus sine squared
00:16:59.000 --> 00:17:05.000
is one.
00:17:04.000 --> 00:17:10.000
Now, all this formula is,
is saying that scalar product,
00:17:08.000 --> 00:17:14.000
the dot product of those two
vectors, can be evaluated if you
00:17:13.000 --> 00:17:19.000
know their components by the
left-hand side of the formula.
00:17:18.000 --> 00:17:24.000
And, if you don't know their
components, it can be evaluated
00:17:24.000 --> 00:17:30.000
in another way,
the geometric evaluation,
00:17:27.000 --> 00:17:33.000
which goes, what is it?
It's a magnitude of one,
00:17:31.000 --> 00:17:37.000
times the magnitude of the
other, times the cosine of the
00:17:36.000 --> 00:17:42.000
included angle.
Now, what's the included angle?
00:17:42.000 --> 00:17:48.000
Well, theta is this angle from
the horizontal to that unit
00:17:49.000 --> 00:17:55.000
vector.
The angle phi is this angle,
00:17:54.000 --> 00:18:00.000
from this picture here.
And therefore,
00:17:58.000 --> 00:18:04.000
the included angle between
(u)theta and my pink vector is
00:18:05.000 --> 00:18:11.000
theta minus phi.
That's the formula.
00:18:12.000 --> 00:18:18.000
It comes from two ways of
calculating the scalar product
00:18:16.000 --> 00:18:22.000
of the vector whose coefficients
are, and the unit vector
00:18:21.000 --> 00:18:27.000
whose components are cosine
theta and sine theta.
00:18:25.000 --> 00:18:31.000
All right, well,
00:18:28.000 --> 00:18:34.000
you should, that was 18.02.
00:18:35.000 --> 00:18:41.000
There must be an 18.03 proof
also. Yes.
00:18:36.000 --> 00:18:42.000
What's the 18.03 proof?
The 18.03 proof uses complex
00:18:43.000 --> 00:18:49.000
numbers.
It says, look,
00:18:46.000 --> 00:18:52.000
take the left side.
Instead of viewing it as the
00:18:53.000 --> 00:18:59.000
dot product of two vectors,
there's another way.
00:19:02.000 --> 00:19:08.000
You can think of it as the part
of the products of two complex
00:19:06.000 --> 00:19:12.000
numbers.
So, the 18.03 argument,
00:19:09.000 --> 00:19:15.000
really, the complex number
argument says,
00:19:12.000 --> 00:19:18.000
look, multiply together a minus
bi and the complex
00:19:17.000 --> 00:19:23.000
number cosine theta plus i sine
theta.
00:19:21.000 --> 00:19:27.000
There are different ways of
00:19:24.000 --> 00:19:30.000
explaining why I want to put the
minus i there instead of i.
00:19:28.000 --> 00:19:34.000
But, the simplest is because I
want, when I take the real part,
00:19:33.000 --> 00:19:39.000
to get the left-hand side.
I will.
00:19:37.000 --> 00:19:43.000
If I take the real part of
this, I'm going to get a cosine
00:19:42.000 --> 00:19:48.000
theta plus b sine theta
00:19:46.000 --> 00:19:52.000
because of negative i and i make
one,
00:19:51.000 --> 00:19:57.000
multiplied together.
All right, that's the left-hand
00:19:55.000 --> 00:20:01.000
side.
And now, the right-hand side,
00:19:58.000 --> 00:20:04.000
I'm going to use polar
representation instead.
00:20:03.000 --> 00:20:09.000
What's the polar representation
of this guy?
00:20:06.000 --> 00:20:12.000
Well, if has the angle theta,
00:20:09.000 --> 00:20:15.000
then a negative b,
a minus bi goes down
00:20:13.000 --> 00:20:19.000
below.
It has the angle minus phi.
00:20:16.000 --> 00:20:22.000
So, this is,
has magnitude.
00:20:18.000 --> 00:20:24.000
It is polar representation.
Its magnitude is a squared plus
00:20:23.000 --> 00:20:29.000
b squared,
and its angle is negative phi,
00:20:27.000 --> 00:20:33.000
not positive phi because this a
minus bi goes below
00:20:32.000 --> 00:20:38.000
the axis if a and b are
positive.
00:20:36.000 --> 00:20:42.000
So, it's e to the minus i phi.
00:20:39.000 --> 00:20:45.000
That's the first guy.
And, how about the second guy?
00:20:43.000 --> 00:20:49.000
Well, the second guy is e to
the i theta.
00:20:47.000 --> 00:20:53.000
So, what's the product?
It is a squared plus b squared,
00:20:51.000 --> 00:20:57.000
the square root,
times e to the i times (theta
00:20:54.000 --> 00:21:00.000
minus phi).
00:20:58.000 --> 00:21:04.000
And now, what do I want?
The real part of this,
00:21:01.000 --> 00:21:07.000
and I want the real part of
this.
00:21:05.000 --> 00:21:11.000
So, let's just say take the
real parts of both sides.
00:21:08.000 --> 00:21:14.000
If I take the real part of the
left-hand side,
00:21:11.000 --> 00:21:17.000
I get a cosine theta plus b
sine theta.
00:21:14.000 --> 00:21:20.000
If I take a real part of this
00:21:17.000 --> 00:21:23.000
side, I get square root of a
squared plus b squared times e,
00:21:21.000 --> 00:21:27.000
times the cosine,
that's the real part,
00:21:23.000 --> 00:21:29.000
right, of theta minus phi,
which is just what it's
00:21:26.000 --> 00:21:32.000
supposed to be.
00:21:31.000 --> 00:21:37.000
Well, with three different
arguments, I'm really pounding
00:21:35.000 --> 00:21:41.000
the table on this formula.
But, I think there's something
00:21:40.000 --> 00:21:46.000
to be learned from at least two
of them.
00:21:44.000 --> 00:21:50.000
And, you know,
I'm still, for awhile,
00:21:47.000 --> 00:21:53.000
I will never miss an
opportunity to bang complex
00:21:51.000 --> 00:21:57.000
numbers into your head because,
in some sense,
00:21:55.000 --> 00:22:01.000
you have to reproduce the
experience of the race.
00:22:01.000 --> 00:22:07.000
As I mentioned in the notes,
it took mathematicians 300 or
00:22:04.000 --> 00:22:10.000
400 years to get used to complex
numbers.
00:22:07.000 --> 00:22:13.000
So, if it takes you three or
four weeks, that's not too bad.
00:22:32.000 --> 00:22:38.000
Now, for the rest of the period
I'd like to go back to the
00:22:36.000 --> 00:22:42.000
linear equations,
and try to put into perspective
00:22:40.000 --> 00:22:46.000
and summarize,
and tell you a couple of things
00:22:43.000 --> 00:22:49.000
which I had to leave out,
but which are,
00:22:46.000 --> 00:22:52.000
in my view, extremely
important.
00:22:48.000 --> 00:22:54.000
And, up to now,
I don't want to leave you with
00:22:52.000 --> 00:22:58.000
any misapprehensions.
So, I'm going to summarize this
00:22:56.000 --> 00:23:02.000
way, whereas last lecture I went
from the most general equation
00:23:01.000 --> 00:23:07.000
to the most special.
I'd like to just write them
00:23:06.000 --> 00:23:12.000
down in the reverse order,
now.
00:23:09.000 --> 00:23:15.000
So, we are talking about basic
linear equations.
00:23:13.000 --> 00:23:19.000
First order,
of course, we haven't moved as
00:23:16.000 --> 00:23:22.000
a second order yet.
So, the most special one,
00:23:20.000 --> 00:23:26.000
and the one we talked about
essentially all of the previous
00:23:25.000 --> 00:23:31.000
two times, or last Friday,
anyway, was the equation where
00:23:30.000 --> 00:23:36.000
the k, the coefficient of y,
is constant,
00:23:34.000 --> 00:23:40.000
and where you also get it on
the right-hand side quite
00:23:39.000 --> 00:23:45.000
providentially.
So, this is the most special
00:23:44.000 --> 00:23:50.000
form, and it's the one which
governed what I will call the
00:23:48.000 --> 00:23:54.000
temperature-concentration model,
or if you want to be grown up,
00:23:53.000 --> 00:23:59.000
the conduction-diffusion model,
conduction-diffusion which
00:23:57.000 --> 00:24:03.000
describes the processes,
which the equation is modeling,
00:24:01.000 --> 00:24:07.000
whereas these simply described
the variables of things,
00:24:05.000 --> 00:24:11.000
which you usually are trying to
calculate when you use the
00:24:10.000 --> 00:24:16.000
equation.
Now, there are a class of
00:24:13.000 --> 00:24:19.000
things where the thing is
constant, but where the k does
00:24:17.000 --> 00:24:23.000
not appear naturally on the
right hand side.
00:24:20.000 --> 00:24:26.000
And, you're going to encounter
them pretty quickly in physics,
00:24:24.000 --> 00:24:30.000
for one place.
So, I better not try to sweep
00:24:27.000 --> 00:24:33.000
those under the rug.
Let's just call that q of t.
00:24:32.000 --> 00:24:38.000
And finally,
there is the most general case,
00:24:36.000 --> 00:24:42.000
where you allow k to be
non-constant.
00:24:40.000 --> 00:24:46.000
That's the one we began,
when we talked about the linear
00:24:45.000 --> 00:24:51.000
equation.
And you know how to solve it in
00:24:49.000 --> 00:24:55.000
general by a definite or an
indefinite integral.
00:24:54.000 --> 00:25:00.000
Now, there's one other thing,
which I want to talk about.
00:25:01.000 --> 00:25:07.000
I will do all these in a
certain order.
00:25:03.000 --> 00:25:09.000
But, from the beginning,
you should keep in mind that
00:25:07.000 --> 00:25:13.000
there's another between the
first two cases.
00:25:10.000 --> 00:25:16.000
Between the first two cases,
there's another extremely
00:25:14.000 --> 00:25:20.000
important distinction,
and that is as to whether k is
00:25:18.000 --> 00:25:24.000
positive or not.
Up to now, we've always had k
00:25:21.000 --> 00:25:27.000
positive.
So, I'm going to put that here.
00:25:24.000 --> 00:25:30.000
So, it's understood when I
write these, that k is positive.
00:25:30.000 --> 00:25:36.000
I want to talk about that,
too.
00:25:32.000 --> 00:25:38.000
But, first things first.
The first thing I wanted to do
00:25:36.000 --> 00:25:42.000
was to show you that this,
the first case,
00:25:39.000 --> 00:25:45.000
the most special case,
does not just apply to this.
00:25:43.000 --> 00:25:49.000
It applies to other things,
too.
00:25:45.000 --> 00:25:51.000
Let me give you a mixing
problem.
00:25:48.000 --> 00:25:54.000
The typical mixing problem
gives another example.
00:25:51.000 --> 00:25:57.000
You've already done in
recitation, and you did one for
00:25:55.000 --> 00:26:01.000
the problem set,
the problem of the two rooms
00:25:59.000 --> 00:26:05.000
filled with smoke.
But, let me do it just using
00:26:04.000 --> 00:26:10.000
letters, so that the ideas stand
out a little more clearly,
00:26:08.000 --> 00:26:14.000
and you are not preoccupied
with the numbers,
00:26:11.000 --> 00:26:17.000
and calculating with the
numbers, and trying to get
00:26:15.000 --> 00:26:21.000
numerical examples.
So, it's as simple as k sub
00:26:19.000 --> 00:26:25.000
mixing.
It looks like this.
00:26:21.000 --> 00:26:27.000
You have a tank,
a room, I don't know,
00:26:23.000 --> 00:26:29.000
where everything's getting
mixed in.
00:26:26.000 --> 00:26:32.000
It has a certain volume,
which I will call v.
00:26:31.000 --> 00:26:37.000
Something is flowing in,
a gas or a liquid.
00:26:34.000 --> 00:26:40.000
And, r will be the flow rate,
in some units.
00:26:38.000 --> 00:26:44.000
Now, since it can't pile up
inside this sealed container,
00:26:44.000 --> 00:26:50.000
which I'm sure is full,
the flow rate out must also be
00:26:49.000 --> 00:26:55.000
r.
And, what we're interested in
00:26:51.000 --> 00:26:57.000
is the amount of salt.
So, x, let's suppose these are
00:26:56.000 --> 00:27:02.000
fluid flows, and the dissolved
substance that I'm talking about
00:27:02.000 --> 00:27:08.000
is not carbon monoxide,
it's salt, any dissolved
00:27:06.000 --> 00:27:12.000
substance, some pollutant or
whatever the problem calls for.
00:27:14.000 --> 00:27:20.000
Let's use salt.
So, it's the amount of salt in
00:27:20.000 --> 00:27:26.000
the tank at time t.
I'm interested in knowing how
00:27:28.000 --> 00:27:34.000
that varies with time.
Now, there's nothing to be said
00:27:34.000 --> 00:27:40.000
about how it flows out.
What flows out,
00:27:36.000 --> 00:27:42.000
of course, is what happens to
be in the tank.
00:27:39.000 --> 00:27:45.000
But, I do have to say what
flows in.
00:27:41.000 --> 00:27:47.000
Now, the only convenient way to
describe the in-flow is in terms
00:27:46.000 --> 00:27:52.000
of its concentration.
The salt will be dissolved in
00:27:49.000 --> 00:27:55.000
the in-flowing water,
and so there will be a certain
00:27:52.000 --> 00:27:58.000
concentration.
And, as you will see,
00:27:55.000 --> 00:28:01.000
for a secret reason,
I'm going to give that the
00:27:58.000 --> 00:28:04.000
subscript e.
So, e is the concentration of
00:28:02.000 --> 00:28:08.000
the incoming salt,
in other words,
00:28:05.000 --> 00:28:11.000
in the fluid,
how many grams are there per
00:28:09.000 --> 00:28:15.000
liter in the incoming fluid.
That's the data.
00:28:13.000 --> 00:28:19.000
So, this is the data.
r is part of the data.
00:28:17.000 --> 00:28:23.000
r is the flow rate.
v is the volume.
00:28:20.000 --> 00:28:26.000
I think I won't bother writing
that down, and the problem is to
00:28:25.000 --> 00:28:31.000
determine what happens to x of
t.
00:28:30.000 --> 00:28:36.000
Now, I strongly recommend you
not attempt to work directly
00:28:34.000 --> 00:28:40.000
with the concentrations unless
you feel you have a really good
00:28:39.000 --> 00:28:45.000
physical feeling for
concentrations.
00:28:42.000 --> 00:28:48.000
I strongly recommend you work
with a variable that you are
00:28:46.000 --> 00:28:52.000
given, namely,
the dependent variable,
00:28:49.000 --> 00:28:55.000
which is the amount of salt,
grams.
00:28:52.000 --> 00:28:58.000
Well, because it's something
you can physically think about.
00:28:57.000 --> 00:29:03.000
It's coming in,
it's getting mixed up,
00:29:00.000 --> 00:29:06.000
and some of it is going out.
So, the basic equation is going
00:29:08.000 --> 00:29:14.000
to be that the rate of change of
salt in the tank is the rate of
00:29:18.000 --> 00:29:24.000
salt inflow, let me write salt
inflow, minus the rate of salt
00:29:27.000 --> 00:29:33.000
outflow.
Okay, at what rate is salt
00:29:33.000 --> 00:29:39.000
flowing in?
Well, the flow rate is the flow
00:29:39.000 --> 00:29:45.000
rate of the liquid.
I multiply the flow rate,
00:29:43.000 --> 00:29:49.000
1 L per minute times the
concentration,
00:29:47.000 --> 00:29:53.000
3 g per liter.
That means 3 g per minute.
00:29:51.000 --> 00:29:57.000
It's going to be,
therefore, the product of the
00:29:56.000 --> 00:30:02.000
flow rate and the concentration,
incoming concentration.
00:30:03.000 --> 00:30:09.000
How about the rate of the salt
outflow?
00:30:05.000 --> 00:30:11.000
Well, again,
the rate of the liquid outflow
00:30:08.000 --> 00:30:14.000
is r.
And, what is the concentration
00:30:11.000 --> 00:30:17.000
of salt in the outflow?
I must use, when you talk flow
00:30:15.000 --> 00:30:21.000
rates, the other factor must be
the concentration,
00:30:18.000 --> 00:30:24.000
not the amount.
So, what is the concentration
00:30:22.000 --> 00:30:28.000
in the outflow?
Well, it's the amount of salt
00:30:25.000 --> 00:30:31.000
in the tank divided by its
volume.
00:30:29.000 --> 00:30:35.000
So, the analog,
the concentration here is x
00:30:32.000 --> 00:30:38.000
divided by v.
Now, here's a typical messy
00:30:36.000 --> 00:30:42.000
equation, dx / dt,
let's write it in the standard,
00:30:40.000 --> 00:30:46.000
linear form,
plus r times x over v equals r
00:30:44.000 --> 00:30:50.000
times the given concentration,
which is a function of time.
00:30:50.000 --> 00:30:56.000
Now, this is going to be some
given function,
00:30:54.000 --> 00:31:00.000
and there will be no reason
whatsoever why you can't solve
00:30:59.000 --> 00:31:05.000
it in that form.
And, that's normally what you
00:31:04.000 --> 00:31:10.000
will do.
Nonetheless,
00:31:05.000 --> 00:31:11.000
in trying to understand how it
fits into this paradigm,
00:31:09.000 --> 00:31:15.000
which kind of equation is it?
Well, clearly there's an
00:31:12.000 --> 00:31:18.000
awkwardness in that on the
right-hand side,
00:31:15.000 --> 00:31:21.000
we have concentration,
and on the left-hand side,
00:31:19.000 --> 00:31:25.000
we seem to have amounts.
Now, the way to understand the
00:31:22.000 --> 00:31:28.000
equation as opposed to the way
to solve it, well,
00:31:26.000 --> 00:31:32.000
it's a step on the way to
solving it.
00:31:28.000 --> 00:31:34.000
But, I emphasize,
you can and normally will solve
00:31:32.000 --> 00:31:38.000
it in exactly that form.
But, to understand what's
00:31:36.000 --> 00:31:42.000
happening, it's better to
express it in terms of
00:31:40.000 --> 00:31:46.000
concentration entirely,
and that's why it's called the
00:31:43.000 --> 00:31:49.000
concentration,
or the diffusion,
00:31:45.000 --> 00:31:51.000
concentration-diffusion
equation.
00:31:48.000 --> 00:31:54.000
So, I'm going to convert this
to concentrations.
00:31:51.000 --> 00:31:57.000
Now, there's no problem here.
x over v is the concentration
00:31:55.000 --> 00:32:01.000
in the tank.
And now, immediately,
00:31:57.000 --> 00:32:03.000
you see, hey,
it looks like it's going to
00:32:00.000 --> 00:32:06.000
come out just in the first form.
But, wait a minute.
00:32:06.000 --> 00:32:12.000
How about the x?
How do I convert that?
00:32:10.000 --> 00:32:16.000
Well, what's the relation
between x?
00:32:13.000 --> 00:32:19.000
So, if the concentration in the
tank is equal to x over v,
00:32:19.000 --> 00:32:25.000
so the tank concentration,
then x is equal to C times the
00:32:25.000 --> 00:32:31.000
constant, V, and dx / dt,
therefore, will be c times dC /
00:32:31.000 --> 00:32:37.000
dt.
You see that?
00:32:34.000 --> 00:32:40.000
Now, that's not in standard
form.
00:32:38.000 --> 00:32:44.000
Let's put it in standard form.
To put it in standard form,
00:32:44.000 --> 00:32:50.000
I see, now, that it's not r
that's the critical quantity.
00:32:51.000 --> 00:32:57.000
It's r divided by v.
So, it's dC / dt,
00:32:55.000 --> 00:33:01.000
C prime, plus r divided by v,
--
00:33:00.000 --> 00:33:06.000
-- I'm going to call that k,
k C, no let's not,
00:33:04.000 --> 00:33:10.000
r divided by v is equal to r
divided by v times Ce.
00:33:09.000 --> 00:33:15.000
That's the equation expressed
00:33:14.000 --> 00:33:20.000
in a form where the
concentration is the dependent
00:33:18.000 --> 00:33:24.000
variable, rather than the amount
of salt itself.
00:33:23.000 --> 00:33:29.000
And, you can see it falls
exactly in this category.
00:33:27.000 --> 00:33:33.000
That means that I can talk
about it.
00:33:31.000 --> 00:33:37.000
The natural way to talk about
this equation is in terms of,
00:33:36.000 --> 00:33:42.000
the same way we talked about
the temperature equation.
00:33:43.000 --> 00:33:49.000
I said concentration.
I mean, that concentration has
00:33:47.000 --> 00:33:53.000
nothing to do with this
concentration.
00:33:50.000 --> 00:33:56.000
This is the diffusion model,
where salt solution outside,
00:33:54.000 --> 00:34:00.000
cell in the middle,
salt diffusing through a
00:33:58.000 --> 00:34:04.000
semi-permeable membrane into
that, uses Newton's law of
00:34:03.000 --> 00:34:09.000
diffusion, except he didn't do a
law of diffusion.
00:34:08.000 --> 00:34:14.000
But, he is sticky.
His name is attached to
00:34:10.000 --> 00:34:16.000
everything.
So, that's this concentration
00:34:13.000 --> 00:34:19.000
model.
It's the one entirely analogous
00:34:15.000 --> 00:34:21.000
to the temperature.
And the physical setup is the
00:34:18.000 --> 00:34:24.000
same.
This one is entirely different.
00:34:21.000 --> 00:34:27.000
Mixing in this form of this
problem has really nothing to do
00:34:24.000 --> 00:34:30.000
with this model whatsoever.
But, nor does that
00:34:27.000 --> 00:34:33.000
concentration had anything to do
with this concentration,
00:34:31.000 --> 00:34:37.000
which refers to the result of
the mixing in the tank.
00:34:36.000 --> 00:34:42.000
But, what happens is the
differential equation is the
00:34:40.000 --> 00:34:46.000
same.
The language of input and
00:34:43.000 --> 00:34:49.000
response that we talked about is
also available here.
00:34:47.000 --> 00:34:53.000
So, everything is the same.
And, the most interesting thing
00:34:52.000 --> 00:34:58.000
is that it shows that the analog
of the conductivity,
00:34:57.000 --> 00:35:03.000
the k, the analog of
conductivity and diffusivity is
00:35:01.000 --> 00:35:07.000
this quantity.
I should not be considering r
00:35:06.000 --> 00:35:12.000
and v by themselves.
I should be considering as the
00:35:11.000 --> 00:35:17.000
basic quantity,
the ratio of those two.
00:35:15.000 --> 00:35:21.000
Now, why is that,
is the basic parameter.
00:35:19.000 --> 00:35:25.000
What is this?
Well, r is the rate of outflow,
00:35:23.000 --> 00:35:29.000
and the rate of inflow,
what's r over v?
00:35:27.000 --> 00:35:33.000
r over v is the fractional rate
of outflow.
00:35:31.000 --> 00:35:37.000
In other words,
if r over v is one tenth,
00:35:35.000 --> 00:35:41.000
it means that 1/10 of the tank
will be emptied in a minute,
00:35:40.000 --> 00:35:46.000
say.
In other words,
00:35:44.000 --> 00:35:50.000
we lumped these two constants
into a single k,
00:35:49.000 --> 00:35:55.000
and at the same time have
simplified the units.
00:35:53.000 --> 00:35:59.000
What are the units?
This is volume per minute.
00:35:58.000 --> 00:36:04.000
This is volume.
So, it's simply reciprocal
00:36:02.000 --> 00:36:08.000
minutes, reciprocal time,
which was the same units of
00:36:07.000 --> 00:36:13.000
that diffusivity and
conductivity had,
00:36:11.000 --> 00:36:17.000
reciprocal time.
The space variables have
00:36:16.000 --> 00:36:22.000
entirely disappeared.
So, it that way,
00:36:19.000 --> 00:36:25.000
it's simplified.
It's simplified conceptually,
00:36:22.000 --> 00:36:28.000
and now, you can answer the
same type of questions we asked
00:36:27.000 --> 00:36:33.000
before about this.
I think it would be better for
00:36:32.000 --> 00:36:38.000
us to move on,
though.
00:36:33.000 --> 00:36:39.000
Well, just an example,
one really simple thing,
00:36:37.000 --> 00:36:43.000
so, suppose since we spent so
much time worrying about what
00:36:41.000 --> 00:36:47.000
was happening with sinusoid
inputs, I mean,
00:36:45.000 --> 00:36:51.000
when could Ce be sinusoidal,
for example?
00:36:48.000 --> 00:36:54.000
Well, roughly sinusoidal if,
for example,
00:36:51.000 --> 00:36:57.000
some factory were polluting.
If this were a lake,
00:36:55.000 --> 00:37:01.000
and some factory were polluting
it, but in the beginning,
00:36:59.000 --> 00:37:05.000
at the beginning of the day,
they produced a lot of the
00:37:03.000 --> 00:37:09.000
pollutant, and by the end of the
day when it wound down,
00:37:07.000 --> 00:37:13.000
it might well happen that the
concentration of pollutants in
00:37:12.000 --> 00:37:18.000
the incoming stream would vary
sinusoidally with a 24 hour
00:37:16.000 --> 00:37:22.000
cycle.
And then, we would be asking,
00:37:22.000 --> 00:37:28.000
so, suppose this varies
sinusoidally.
00:37:26.000 --> 00:37:32.000
In other words,
it's like cosine omega t.
00:37:31.000 --> 00:37:37.000
I'm asking, how closely does
00:37:36.000 --> 00:37:42.000
the concentration in the tank
follow C sub e?
00:37:44.000 --> 00:37:50.000
Now, what would that depend
upon?
00:37:48.000 --> 00:37:54.000
Think about it.
Well, the answer,
00:37:52.000 --> 00:37:58.000
suppose k is large.
Closely, let's just analyze one
00:37:58.000 --> 00:38:04.000
case, if k is big.
Now, what would make k big?
00:38:03.000 --> 00:38:09.000
We know that from the
temperature thing.
00:38:06.000 --> 00:38:12.000
If the conductivity is high,
then the inner temperature will
00:38:10.000 --> 00:38:16.000
follow the outer temperature
closely, and the same thing with
00:38:15.000 --> 00:38:21.000
the diffusion model.
But we, of course,
00:38:17.000 --> 00:38:23.000
therefore, since the equation
is the same, we must get the
00:38:21.000 --> 00:38:27.000
same result here.
Now, what would make k big?
00:38:25.000 --> 00:38:31.000
If r is big,
if the flow rate is very fast,
00:38:28.000 --> 00:38:34.000
we will expect the
concentration of the inside of
00:38:31.000 --> 00:38:37.000
that tank to match fairly
closely the concentration of the
00:38:35.000 --> 00:38:41.000
pollutant, of the incoming salt
solution, or,
00:38:38.000 --> 00:38:44.000
if the tank is very small.
For fixed flow rates,
00:38:43.000 --> 00:38:49.000
if the tank is very small,
well, then it gets emptied
00:38:47.000 --> 00:38:53.000
quickly.
So, both of these are,
00:38:49.000 --> 00:38:55.000
I think, intuitive results.
And, of course,
00:38:52.000 --> 00:38:58.000
as before, we got them from
that, by trying to analyze the
00:38:56.000 --> 00:39:02.000
final form of the solution.
In other words,
00:38:59.000 --> 00:39:05.000
we got them by looking at that
form of the solution up there,
00:39:03.000 --> 00:39:09.000
and seeing if k is big.
As k increases,
00:39:07.000 --> 00:39:13.000
what happens to the amplitude,
and what happens to the phase
00:39:12.000 --> 00:39:18.000
lag?
But, that summarizes the two.
00:39:14.000 --> 00:39:20.000
So, this means,
closely means,
00:39:17.000 --> 00:39:23.000
that the phase lag is,
big or small?
00:39:20.000 --> 00:39:26.000
The lag is small.
And, the amplitude is,
00:39:23.000 --> 00:39:29.000
well, the amplitude,
the biggest the amplitude could
00:39:27.000 --> 00:39:33.000
ever be is one because that's
the amplitude of this.
00:39:33.000 --> 00:39:39.000
So, the amplitude is near one,
one because that's the
00:39:38.000 --> 00:39:44.000
amplitude of the incoming
signal, input,
00:39:41.000 --> 00:39:47.000
whatever you want to call it.
Okay, now, I'd like to spend
00:39:47.000 --> 00:39:53.000
the rest of the time talking
about the failures of number
00:39:52.000 --> 00:39:58.000
one, and when you have to use
number two, and when even number
00:39:58.000 --> 00:40:04.000
two is no good.
So, let me end first-order
00:40:03.000 --> 00:40:09.000
equations by putting my worst
foot forward.
00:40:08.000 --> 00:40:14.000
Well, I'm just trying to avoid
disappointment at
00:40:13.000 --> 00:40:19.000
misapprehensions from you.
I'll watch you leave this room
00:40:19.000 --> 00:40:25.000
and say, well,
he said that,
00:40:22.000 --> 00:40:28.000
okay.
So, the first one you're going
00:40:26.000 --> 00:40:32.000
to encounter very shortly where
one is not satisfied,
00:40:32.000 --> 00:40:38.000
but two is, so on some examples
where you need two,
00:40:38.000 --> 00:40:44.000
well, it's going to happen
right here.
00:40:44.000 --> 00:40:50.000
Somebody, sooner or later,
it's going to draw on that
00:40:47.000 --> 00:40:53.000
loathsome orange chalk,
which is unerasable,
00:40:50.000 --> 00:40:56.000
something which looks like
that.
00:40:52.000 --> 00:40:58.000
Remember, you saw it here
first.
00:40:54.000 --> 00:41:00.000
r, yeah, we had that.
Okay, see, I had it in high
00:40:58.000 --> 00:41:04.000
school too.
That's the capacitance.
00:41:00.000 --> 00:41:06.000
This is the resistance.
That's the electromotive force:
00:41:04.000 --> 00:41:10.000
battery, or a source of
alternating current,
00:41:07.000 --> 00:41:13.000
something like that.
Now, of course,
00:41:11.000 --> 00:41:17.000
what you're interested in is
how the current flows in the
00:41:15.000 --> 00:41:21.000
circuit.
Since current across the
00:41:18.000 --> 00:41:24.000
capacitance doesn't make sense,
you have to talk about the
00:41:23.000 --> 00:41:29.000
charge on the capacitance.
So, q, it's customary in a
00:41:27.000 --> 00:41:33.000
circle this simple to use as the
variable not current,
00:41:32.000 --> 00:41:38.000
but the charge on the
capacitance.
00:41:36.000 --> 00:41:42.000
And then Kirchhoff's,
you are also supposed to know
00:41:40.000 --> 00:41:46.000
that the derivative,
that the time derivative of q
00:41:45.000 --> 00:41:51.000
is what's called the current in
the circuit.
00:41:49.000 --> 00:41:55.000
That sort of intuitive.
But, i in a physics class,
00:41:53.000 --> 00:41:59.000
j in an electrical engineering
class, and why,
00:41:57.000 --> 00:42:03.000
not the letter Y,
but why is that?
00:42:02.000 --> 00:42:08.000
That's because of electrical
engineers use lots of lots of
00:42:06.000 --> 00:42:12.000
complex numbers and therefore,
you have to call current j,
00:42:11.000 --> 00:42:17.000
I guess.
I think they do j in physics,
00:42:15.000 --> 00:42:21.000
too, now.
No, no they don't.
00:42:17.000 --> 00:42:23.000
I don't know.
So, i is ambiguous if you are
00:42:21.000 --> 00:42:27.000
in that particular subject.
And it's customary to use,
00:42:25.000 --> 00:42:31.000
I don't know.
Now it's completely safe.
00:42:29.000 --> 00:42:35.000
Okay, where are we?
Well, the law is,
00:42:32.000 --> 00:42:38.000
the basic differential equation
is Kirchhoff's voltage law,
00:42:36.000 --> 00:42:42.000
but the sum of the voltage
drops across these three has to
00:42:40.000 --> 00:42:46.000
be zero.
So, it's R times i,
00:42:42.000 --> 00:42:48.000
which is dq / dt.
That's Ohm's law.
00:42:44.000 --> 00:42:50.000
That's the voltage drop across
resistance.
00:42:47.000 --> 00:42:53.000
The voltage drop across the
capacitance is Coulomb's law,
00:42:51.000 --> 00:42:57.000
one form of Coulomb's law.
It's q divided by C.
00:42:55.000 --> 00:43:01.000
And, that has to be the voltage
drop.
00:42:57.000 --> 00:43:03.000
And then, there is some sign
convention.
00:43:02.000 --> 00:43:08.000
So, this is either plus or
minus, depending on your sign
00:43:06.000 --> 00:43:12.000
conventions, but it's E of t.
Now, if I put that in standard
00:43:11.000 --> 00:43:17.000
form, in standard form I
probably should say q prime plus
00:43:15.000 --> 00:43:21.000
q over RC equals,
well, I suppose,
00:43:18.000 --> 00:43:24.000
E over R.
And, this is what would appear
00:43:23.000 --> 00:43:29.000
in the equation.
But, it's not the natural
00:43:26.000 --> 00:43:32.000
thing.
The k is one over RC.
00:43:30.000 --> 00:43:36.000
And, that's the reciprocal.
00:43:32.000 --> 00:43:38.000
The RC constant is what
everybody knows is important
00:43:35.000 --> 00:43:41.000
when you talk about a little
circuit of that form.
00:43:39.000 --> 00:43:45.000
On the other hand,
the right-hand side,
00:43:41.000 --> 00:43:47.000
it's quite unnatural to try to
stick in the right-hand side
00:43:46.000 --> 00:43:52.000
that same RC.
Call this EC over RC.
00:43:48.000 --> 00:43:54.000
People don't do that,
and therefore,
00:43:50.000 --> 00:43:56.000
it doesn't really fall into the
paradigm of that first equation.
00:43:55.000 --> 00:44:01.000
It's the second equation that
really falls into the category.
00:43:59.000 --> 00:44:05.000
Another simple example of this
is chained to k,
00:44:02.000 --> 00:44:08.000
radioactively changed to k.
Well, let's say the radioactive
00:44:08.000 --> 00:44:14.000
substance, A,
decays into,
00:44:10.000 --> 00:44:16.000
let's say, one atom of this
produces one atom of that for
00:44:14.000 --> 00:44:20.000
simplicity.
So, it decays into B,
00:44:17.000 --> 00:44:23.000
which then still is radioactive
and decays further.
00:44:21.000 --> 00:44:27.000
Okay, what's the differential
equation, which is going to be,
00:44:26.000 --> 00:44:32.000
it's going to govern this
situation?
00:44:29.000 --> 00:44:35.000
What I want to know is how much
B there is at any given time.
00:44:35.000 --> 00:44:41.000
So, I want a differential
equation for the quantity of the
00:44:39.000 --> 00:44:45.000
radioactive product at any given
time.
00:44:41.000 --> 00:44:47.000
Well, what's it going to look
like?
00:44:44.000 --> 00:44:50.000
Well, it's the amount coming in
minus the amount going out,
00:44:48.000 --> 00:44:54.000
so to speak.
The rate of inflow minus the
00:44:51.000 --> 00:44:57.000
rate of outflow,
except it's not the same type
00:44:55.000 --> 00:45:01.000
of physical flow we had before.
How fast is it coming in?
00:44:59.000 --> 00:45:05.000
Well, A is decaying at a
certain rate,
00:45:01.000 --> 00:45:07.000
and so the rate at which A
decays is by the basic
00:45:05.000 --> 00:45:11.000
radioactive law.
It's k1, it's constant,
00:45:09.000 --> 00:45:15.000
decay constant,
times the amount of A present.
00:45:12.000 --> 00:45:18.000
If I used the differential
00:45:15.000 --> 00:45:21.000
equation with A here,
I'd have to put a negative sign
00:45:18.000 --> 00:45:24.000
because it's the rate at which
that stuff is leaving A.
00:45:22.000 --> 00:45:28.000
But, I'm interested in the rate
at which it's coming in to B.
00:45:26.000 --> 00:45:32.000
So, it has a positive sign.
And then, the rate at which B
00:45:30.000 --> 00:45:36.000
is decaying, and therefore the
quantity of good B is gone,
00:45:34.000 --> 00:45:40.000
--
-- that will have some other
00:45:38.000 --> 00:45:44.000
constant, B.
So, that will be the equation,
00:45:41.000 --> 00:45:47.000
and to avoid having two
dependent variables in there,
00:45:46.000 --> 00:45:52.000
we know how A is decaying.
So, it's k1,
00:45:49.000 --> 00:45:55.000
some constant times A,
sorry, A will be e to the
00:45:53.000 --> 00:45:59.000
negative, you know,
the decay law,
00:45:56.000 --> 00:46:02.000
so, times the initial amount
that was there times e to the
00:46:01.000 --> 00:46:07.000
negative k1 t.
That's how much A there is at
00:46:07.000 --> 00:46:13.000
any given time.
It's decaying by the
00:46:10.000 --> 00:46:16.000
radioactive decay law,
minus k2 B.
00:46:14.000 --> 00:46:20.000
Okay, so how does the
differential equation look like?
00:46:18.000 --> 00:46:24.000
It looks like B prime plus k2 B
equals an exponential,
00:46:23.000 --> 00:46:29.000
k1 A zero e to the negative k1
t.
00:46:28.000 --> 00:46:34.000
But, there's no reason to
00:46:31.000 --> 00:46:37.000
expect that that constant really
has anything to do with k2.
00:46:36.000 --> 00:46:42.000
It's unnatural to put it in
that form, which is the correct
00:46:41.000 --> 00:46:47.000
one.
Now, in the last two minutes,
00:46:49.000 --> 00:46:55.000
I wish to alienate half the
class by pointing out that if k
00:47:02.000 --> 00:47:08.000
is less than zero,
none of the terminology of
00:47:13.000 --> 00:47:19.000
transient, steady-state input
response applies.
00:47:25.000 --> 00:47:31.000
The technique of solving the
equation is identical.
00:47:28.000 --> 00:47:34.000
But, you cannot interpret.
So, the technique is the same,
00:47:33.000 --> 00:47:39.000
and therefore it's worth
learning.
00:47:37.000 --> 00:47:43.000
The technique is the same.
In other words,
00:47:41.000 --> 00:47:47.000
the solution will be still e to
the negative kt integral q of t
00:47:48.000 --> 00:47:54.000
e to the kt dt plus
00:47:54.000 --> 00:48:00.000
a constant times e to the k,
oh, this is terrible,
00:48:00.000 --> 00:48:06.000
no.
dy / dt, let's give an example.
00:48:04.000 --> 00:48:10.000
The equation I'm going to look
at is something that looks like
00:48:10.000 --> 00:48:16.000
this: y equals q of t,
let's say, okay,
00:48:13.000 --> 00:48:19.000
a constant, but the constant a
is positive.
00:48:17.000 --> 00:48:23.000
So, the constant here is
negative.
00:48:19.000 --> 00:48:25.000
Then, when I solve,
my k, in other words,
00:48:23.000 --> 00:48:29.000
is now properly written as
negative a.
00:48:26.000 --> 00:48:32.000
And therefore,
this formula should now become
00:48:30.000 --> 00:48:36.000
not this, but the negative k is
a t.
00:48:36.000 --> 00:48:42.000
And, here it's negative a t.
And, here it is positive a t.
00:48:41.000 --> 00:48:47.000
Now, why is it,
if this is going to be the
00:48:44.000 --> 00:48:50.000
solution, why are all those
things totally irrelevant?
00:48:49.000 --> 00:48:55.000
This is not a transient any
longer because if a is positive,
00:48:54.000 --> 00:49:00.000
this goes to infinity.
Or, if I go to minus infinity,
00:48:59.000 --> 00:49:05.000
then C is negative.
So, it's not transient.
00:49:03.000 --> 00:49:09.000
It's not going to zero,
and it depends heavily on the
00:49:07.000 --> 00:49:13.000
initial conditions.
That means that of these two
00:49:10.000 --> 00:49:16.000
functions, this is the important
guy.
00:49:12.000 --> 00:49:18.000
This is just fixed,
some fixed function.
00:49:15.000 --> 00:49:21.000
Everything, in other words,
is going to depend upon the
00:49:18.000 --> 00:49:24.000
initial conditions,
whereas in the other cases we
00:49:21.000 --> 00:49:27.000
have been studying,
the initial conditions after a
00:49:25.000 --> 00:49:31.000
while don't matter anymore.
Now, why did I say I would
00:49:30.000 --> 00:49:36.000
alienate half of you?
Well, because in what subjects
00:49:35.000 --> 00:49:41.000
will a be positive?
In what subjects will k be
00:49:39.000 --> 00:49:45.000
negative?
k is typically negative in
00:49:43.000 --> 00:49:49.000
biology, economics,
Sloan.
00:49:45.000 --> 00:49:51.000
In other words,
the simple thing is think of it
00:49:50.000 --> 00:49:56.000
in biology.
What's the simplest equation
00:49:53.000 --> 00:49:59.000
for population growth?
Well, it is dP / dt equals
00:49:58.000 --> 00:50:04.000
some, if the population is
growing, a times P,
00:50:02.000 --> 00:50:08.000
and a is a positive number.
That means P prime minus a P is
00:50:09.000 --> 00:50:15.000
zero.
So, the thing I want to leave
00:50:13.000 --> 00:50:19.000
you with is this.
If life is involved,
00:50:16.000 --> 00:50:22.000
k is likely to be negative.
k is positive when inanimate
00:50:22.000 --> 00:50:28.000
things are involved;
I won't say dead,
00:50:25.000 --> 00:50:31.000
inanimate.