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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

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solution, and you complexify the
right hand side of the equation,

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as well.
So, cosine omega t

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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

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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

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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

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trigonometric identity,
there's not a lot of difference

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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

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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

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roots, and whatnot.
The only thing is,

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these two are the same real
mess.

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That amounts to just another
pure oscillation with the same

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frequency as the old guy,
and with the amplitude changed,

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and with a phase shift,
move to the right or left.

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So, this is no more general
than that.

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Notice they both have two
parameters in them,

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these two coefficients.
This one has the two parameters

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in an altered form.
Okay, well, I wanted,

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because of the importance of
this formula,

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I wanted to take a couple of
minutes out for a proof of the

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formula, --

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-- just to give you chance to
stare at it a little more now.

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There are three proofs I know.
I'm sure there are 27.

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The Pythagorean theorem now has
several hundred.

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But, there are three basic
proofs.

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There is the one I will not
give you, I'll call the high

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school proof,
which is the only one one

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normally finds in books,
physics textbooks or other

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textbooks.
The high school proof takes the

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right-hand side,
applies the formula for the

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00:14:14,000 --> 00:14:20,000
cosine of the difference of two
angles, which it assumes you had

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in trigonometry,
and then converts it into this.

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It shows you that once you've
done that, that a turns out to

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be c cosine phi
and b, the number b is c sine

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phi,
and therefore it identifies the

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two sides.
Now, the thing that's of course

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correct and it's the simplest
possible argument,

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the thing that's no good about
it is that the direction at

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which it goes is from here to
here.

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Well, everybody knew that.
If I gave you this and told

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you, write it out in terms of
cosine and sine,

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I would assume it dearly hope
that practically all of you can

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do that.
Unfortunately,

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when you want to use the
formula, it's this way you want

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to use it in the opposite
direction.

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You are starting with this,
and want to convert it to that.

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Now, the proof,
therefore, will not be of much

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help.
It requires you to go in the

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backwards direction,
and match up coefficients.

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It's much better to go
forwards.

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Now, there are two proofs that
go forwards.

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00:15:25,000 --> 00:15:31,000
There's the 18.02 proof.
Since I didn't teach most of

235
00:15:28,000 --> 00:15:34,000
you 18.02, I can't be sure you
had it.

236
00:15:32,000 --> 00:15:38,000
So, I'll spend one minute
giving it to you.

237
00:15:36,000 --> 00:15:42,000
What is the 18.02 proof?
It is the following picture.

238
00:15:42,000 --> 00:15:48,000
I think this requires deep
colored chalk.

239
00:15:46,000 --> 00:15:52,000
This is going to be pretty
heavy.

240
00:15:50,000 --> 00:15:56,000
All right, first of all,
the a and the b are the given.

241
00:15:55,000 --> 00:16:01,000
So, I'm going to put in that
vector.

242
00:16:01,000 --> 00:16:07,000
So, there is the vector whose
sides are, whose components are

243
00:16:05,000 --> 00:16:11,000
a and b.
I'll write it without the i and

244
00:16:08,000 --> 00:16:14,000
j.
I hope you had from Jerison

245
00:16:11,000 --> 00:16:17,000
that form for the vector,
if you don't like that,

246
00:16:14,000 --> 00:16:20,000
write ai plus bj,
okay?

247
00:16:17,000 --> 00:16:23,000
Now, there's another vector
lurking around.

248
00:16:20,000 --> 00:16:26,000
It's the unit vector whose,
I'll write it this way,

249
00:16:24,000 --> 00:16:30,000
u because it's a unit vector,
and theta to indicate that it's

250
00:16:29,000 --> 00:16:35,000
angle is theta.
Now, the reason for doing that

251
00:16:34,000 --> 00:16:40,000
is because you see that the
left-hand side is a dot product

252
00:16:38,000 --> 00:16:44,000
of two vectors.
The left-hand side of the

253
00:16:41,000 --> 00:16:47,000
identity is the dot product of
the vector <a,

254
00:16:45,000 --> 00:16:51,000
b> with the vector whose
components are cosine theta and

255
00:16:49,000 --> 00:16:55,000
sine theta.

256
00:16:52,000 --> 00:16:58,000
That's what I'm calling this
unit vector.

257
00:16:55,000 --> 00:17:01,000
It's a unit vector because
cosine squared plus sine squared

258
00:16:59,000 --> 00:17:05,000
is one.

259
00:17:04,000 --> 00:17:10,000
Now, all this formula is,
is saying that scalar product,

260
00:17:08,000 --> 00:17:14,000
the dot product of those two
vectors, can be evaluated if you

261
00:17:13,000 --> 00:17:19,000
know their components by the
left-hand side of the formula.

262
00:17:18,000 --> 00:17:24,000
And, if you don't know their
components, it can be evaluated

263
00:17:24,000 --> 00:17:30,000
in another way,
the geometric evaluation,

264
00:17:27,000 --> 00:17:33,000
which goes, what is it?
It's a magnitude of one,

265
00:17:31,000 --> 00:17:37,000
times the magnitude of the
other, times the cosine of the

266
00:17:36,000 --> 00:17:42,000
included angle.
Now, what's the included angle?

267
00:17:42,000 --> 00:17:48,000
Well, theta is this angle from
the horizontal to that unit

268
00:17:49,000 --> 00:17:55,000
vector.
The angle phi is this angle,

269
00:17:54,000 --> 00:18:00,000
from this picture here.
And therefore,

270
00:17:58,000 --> 00:18:04,000
the included angle between
(u)theta and my pink vector is

271
00:18:05,000 --> 00:18:11,000
theta minus phi.
That's the formula.

272
00:18:12,000 --> 00:18:18,000
It comes from two ways of
calculating the scalar product

273
00:18:16,000 --> 00:18:22,000
of the vector whose coefficients
are, and the unit vector

274
00:18:21,000 --> 00:18:27,000
whose components are cosine
theta and sine theta.

275
00:18:25,000 --> 00:18:31,000
All right, well,

276
00:18:28,000 --> 00:18:34,000
you should, that was 18.02.

277
00:18:35,000 --> 00:18:41,000
There must be an 18.03 proof
also. Yes.

278
00:18:36,000 --> 00:18:42,000
What's the 18.03 proof?
The 18.03 proof uses complex

279
00:18:43,000 --> 00:18:49,000
numbers.
It says, look,

280
00:18:46,000 --> 00:18:52,000
take the left side.
Instead of viewing it as the

281
00:18:53,000 --> 00:18:59,000
dot product of two vectors,
there's another way.

282
00:19:02,000 --> 00:19:08,000
You can think of it as the part
of the products of two complex

283
00:19:06,000 --> 00:19:12,000
numbers.
So, the 18.03 argument,

284
00:19:09,000 --> 00:19:15,000
really, the complex number
argument says,

285
00:19:12,000 --> 00:19:18,000
look, multiply together a minus
bi and the complex

286
00:19:17,000 --> 00:19:23,000
number cosine theta plus i sine
theta.

287
00:19:21,000 --> 00:19:27,000
There are different ways of

288
00:19:24,000 --> 00:19:30,000
explaining why I want to put the
minus i there instead of i.

289
00:19:28,000 --> 00:19:34,000
But, the simplest is because I
want, when I take the real part,

290
00:19:33,000 --> 00:19:39,000
to get the left-hand side.
I will.

291
00:19:37,000 --> 00:19:43,000
If I take the real part of
this, I'm going to get a cosine

292
00:19:42,000 --> 00:19:48,000
theta plus b sine theta

293
00:19:46,000 --> 00:19:52,000
because of negative i and i make
one,

294
00:19:51,000 --> 00:19:57,000
multiplied together.
All right, that's the left-hand

295
00:19:55,000 --> 00:20:01,000
side.
And now, the right-hand side,

296
00:19:58,000 --> 00:20:04,000
I'm going to use polar
representation instead.

297
00:20:03,000 --> 00:20:09,000
What's the polar representation
of this guy?

298
00:20:06,000 --> 00:20:12,000
Well, if <a,
b> has the angle theta,

299
00:20:09,000 --> 00:20:15,000
then a negative b,
a minus bi goes down

300
00:20:13,000 --> 00:20:19,000
below.
It has the angle minus phi.

301
00:20:16,000 --> 00:20:22,000
So, this is,
has magnitude.

302
00:20:18,000 --> 00:20:24,000
It is polar representation.
Its magnitude is a squared plus

303
00:20:23,000 --> 00:20:29,000
b squared,
and its angle is negative phi,

304
00:20:27,000 --> 00:20:33,000
not positive phi because this a
minus bi goes below

305
00:20:32,000 --> 00:20:38,000
the axis if a and b are
positive.

306
00:20:36,000 --> 00:20:42,000
So, it's e to the minus i phi.

307
00:20:39,000 --> 00:20:45,000
That's the first guy.
And, how about the second guy?

308
00:20:43,000 --> 00:20:49,000
Well, the second guy is e to
the i theta.

309
00:20:47,000 --> 00:20:53,000
So, what's the product?
It is a squared plus b squared,

310
00:20:51,000 --> 00:20:57,000
the square root,
times e to the i times (theta

311
00:20:54,000 --> 00:21:00,000
minus phi).

312
00:20:58,000 --> 00:21:04,000
And now, what do I want?
The real part of this,

313
00:21:01,000 --> 00:21:07,000
and I want the real part of
this.

314
00:21:05,000 --> 00:21:11,000
So, let's just say take the
real parts of both sides.

315
00:21:08,000 --> 00:21:14,000
If I take the real part of the
left-hand side,

316
00:21:11,000 --> 00:21:17,000
I get a cosine theta plus b
sine theta.

317
00:21:14,000 --> 00:21:20,000
If I take a real part of this

318
00:21:17,000 --> 00:21:23,000
side, I get square root of a
squared plus b squared times e,

319
00:21:21,000 --> 00:21:27,000
times the cosine,
that's the real part,

320
00:21:23,000 --> 00:21:29,000
right, of theta minus phi,
which is just what it's

321
00:21:26,000 --> 00:21:32,000
supposed to be.

322
00:21:31,000 --> 00:21:37,000
Well, with three different
arguments, I'm really pounding

323
00:21:35,000 --> 00:21:41,000
the table on this formula.
But, I think there's something

324
00:21:40,000 --> 00:21:46,000
to be learned from at least two
of them.

325
00:21:44,000 --> 00:21:50,000
And, you know,
I'm still, for awhile,

326
00:21:47,000 --> 00:21:53,000
I will never miss an
opportunity to bang complex

327
00:21:51,000 --> 00:21:57,000
numbers into your head because,
in some sense,

328
00:21:55,000 --> 00:22:01,000
you have to reproduce the
experience of the race.

329
00:22:01,000 --> 00:22:07,000
As I mentioned in the notes,
it took mathematicians 300 or

330
00:22:04,000 --> 00:22:10,000
400 years to get used to complex
numbers.

331
00:22:07,000 --> 00:22:13,000
So, if it takes you three or
four weeks, that's not too bad.

332
00:22:32,000 --> 00:22:38,000
Now, for the rest of the period
I'd like to go back to the

333
00:22:36,000 --> 00:22:42,000
linear equations,
and try to put into perspective

334
00:22:40,000 --> 00:22:46,000
and summarize,
and tell you a couple of things

335
00:22:43,000 --> 00:22:49,000
which I had to leave out,
but which are,

336
00:22:46,000 --> 00:22:52,000
in my view, extremely
important.

337
00:22:48,000 --> 00:22:54,000
And, up to now,
I don't want to leave you with

338
00:22:52,000 --> 00:22:58,000
any misapprehensions.
So, I'm going to summarize this

339
00:22:56,000 --> 00:23:02,000
way, whereas last lecture I went
from the most general equation

340
00:23:01,000 --> 00:23:07,000
to the most special.
I'd like to just write them

341
00:23:06,000 --> 00:23:12,000
down in the reverse order,
now.

342
00:23:09,000 --> 00:23:15,000
So, we are talking about basic
linear equations.

343
00:23:13,000 --> 00:23:19,000
First order,
of course, we haven't moved as

344
00:23:16,000 --> 00:23:22,000
a second order yet.
So, the most special one,

345
00:23:20,000 --> 00:23:26,000
and the one we talked about
essentially all of the previous

346
00:23:25,000 --> 00:23:31,000
two times, or last Friday,
anyway, was the equation where

347
00:23:30,000 --> 00:23:36,000
the k, the coefficient of y,
is constant,

348
00:23:34,000 --> 00:23:40,000
and where you also get it on
the right-hand side quite

349
00:23:39,000 --> 00:23:45,000
providentially.
So, this is the most special

350
00:23:44,000 --> 00:23:50,000
form, and it's the one which
governed what I will call the

351
00:23:48,000 --> 00:23:54,000
temperature-concentration model,
or if you want to be grown up,

352
00:23:53,000 --> 00:23:59,000
the conduction-diffusion model,
conduction-diffusion which

353
00:23:57,000 --> 00:24:03,000
describes the processes,
which the equation is modeling,

354
00:24:01,000 --> 00:24:07,000
whereas these simply described
the variables of things,

355
00:24:05,000 --> 00:24:11,000
which you usually are trying to
calculate when you use the

356
00:24:10,000 --> 00:24:16,000
equation.
Now, there are a class of

357
00:24:13,000 --> 00:24:19,000
things where the thing is
constant, but where the k does

358
00:24:17,000 --> 00:24:23,000
not appear naturally on the
right hand side.

359
00:24:20,000 --> 00:24:26,000
And, you're going to encounter
them pretty quickly in physics,

360
00:24:24,000 --> 00:24:30,000
for one place.
So, I better not try to sweep

361
00:24:27,000 --> 00:24:33,000
those under the rug.
Let's just call that q of t.

362
00:24:32,000 --> 00:24:38,000
And finally,
there is the most general case,

363
00:24:36,000 --> 00:24:42,000
where you allow k to be
non-constant.

364
00:24:40,000 --> 00:24:46,000
That's the one we began,
when we talked about the linear

365
00:24:45,000 --> 00:24:51,000
equation.
And you know how to solve it in

366
00:24:49,000 --> 00:24:55,000
general by a definite or an
indefinite integral.

367
00:24:54,000 --> 00:25:00,000
Now, there's one other thing,
which I want to talk about.

368
00:25:01,000 --> 00:25:07,000
I will do all these in a
certain order.

369
00:25:03,000 --> 00:25:09,000
But, from the beginning,
you should keep in mind that

370
00:25:07,000 --> 00:25:13,000
there's another between the
first two cases.

371
00:25:10,000 --> 00:25:16,000
Between the first two cases,
there's another extremely

372
00:25:14,000 --> 00:25:20,000
important distinction,
and that is as to whether k is

373
00:25:18,000 --> 00:25:24,000
positive or not.
Up to now, we've always had k

374
00:25:21,000 --> 00:25:27,000
positive.
So, I'm going to put that here.

375
00:25:24,000 --> 00:25:30,000
So, it's understood when I
write these, that k is positive.

376
00:25:30,000 --> 00:25:36,000
I want to talk about that,
too.

377
00:25:32,000 --> 00:25:38,000
But, first things first.
The first thing I wanted to do

378
00:25:36,000 --> 00:25:42,000
was to show you that this,
the first case,

379
00:25:39,000 --> 00:25:45,000
the most special case,
does not just apply to this.

380
00:25:43,000 --> 00:25:49,000
It applies to other things,
too.

381
00:25:45,000 --> 00:25:51,000
Let me give you a mixing
problem.

382
00:25:48,000 --> 00:25:54,000
The typical mixing problem
gives another example.

383
00:25:51,000 --> 00:25:57,000
You've already done in
recitation, and you did one for

384
00:25:55,000 --> 00:26:01,000
the problem set,
the problem of the two rooms

385
00:25:59,000 --> 00:26:05,000
filled with smoke.
But, let me do it just using

386
00:26:04,000 --> 00:26:10,000
letters, so that the ideas stand
out a little more clearly,

387
00:26:08,000 --> 00:26:14,000
and you are not preoccupied
with the numbers,

388
00:26:11,000 --> 00:26:17,000
and calculating with the
numbers, and trying to get

389
00:26:15,000 --> 00:26:21,000
numerical examples.
So, it's as simple as k sub

390
00:26:19,000 --> 00:26:25,000
mixing.
It looks like this.

391
00:26:21,000 --> 00:26:27,000
You have a tank,
a room, I don't know,

392
00:26:23,000 --> 00:26:29,000
where everything's getting
mixed in.

393
00:26:26,000 --> 00:26:32,000
It has a certain volume,
which I will call v.

394
00:26:31,000 --> 00:26:37,000
Something is flowing in,
a gas or a liquid.

395
00:26:34,000 --> 00:26:40,000
And, r will be the flow rate,
in some units.

396
00:26:38,000 --> 00:26:44,000
Now, since it can't pile up
inside this sealed container,

397
00:26:44,000 --> 00:26:50,000
which I'm sure is full,
the flow rate out must also be

398
00:26:49,000 --> 00:26:55,000
r.
And, what we're interested in

399
00:26:51,000 --> 00:26:57,000
is the amount of salt.
So, x, let's suppose these are

400
00:26:56,000 --> 00:27:02,000
fluid flows, and the dissolved
substance that I'm talking about

401
00:27:02,000 --> 00:27:08,000
is not carbon monoxide,
it's salt, any dissolved

402
00:27:06,000 --> 00:27:12,000
substance, some pollutant or
whatever the problem calls for.

403
00:27:14,000 --> 00:27:20,000
Let's use salt.
So, it's the amount of salt in

404
00:27:20,000 --> 00:27:26,000
the tank at time t.
I'm interested in knowing how

405
00:27:28,000 --> 00:27:34,000
that varies with time.
Now, there's nothing to be said

406
00:27:34,000 --> 00:27:40,000
about how it flows out.
What flows out,

407
00:27:36,000 --> 00:27:42,000
of course, is what happens to
be in the tank.

408
00:27:39,000 --> 00:27:45,000
But, I do have to say what
flows in.

409
00:27:41,000 --> 00:27:47,000
Now, the only convenient way to
describe the in-flow is in terms

410
00:27:46,000 --> 00:27:52,000
of its concentration.
The salt will be dissolved in

411
00:27:49,000 --> 00:27:55,000
the in-flowing water,
and so there will be a certain

412
00:27:52,000 --> 00:27:58,000
concentration.
And, as you will see,

413
00:27:55,000 --> 00:28:01,000
for a secret reason,
I'm going to give that the

414
00:27:58,000 --> 00:28:04,000
subscript e.
So, e is the concentration of

415
00:28:02,000 --> 00:28:08,000
the incoming salt,
in other words,

416
00:28:05,000 --> 00:28:11,000
in the fluid,
how many grams are there per

417
00:28:09,000 --> 00:28:15,000
liter in the incoming fluid.
That's the data.

418
00:28:13,000 --> 00:28:19,000
So, this is the data.
r is part of the data.

419
00:28:17,000 --> 00:28:23,000
r is the flow rate.
v is the volume.

420
00:28:20,000 --> 00:28:26,000
I think I won't bother writing
that down, and the problem is to

421
00:28:25,000 --> 00:28:31,000
determine what happens to x of
t.

422
00:28:30,000 --> 00:28:36,000
Now, I strongly recommend you
not attempt to work directly

423
00:28:34,000 --> 00:28:40,000
with the concentrations unless
you feel you have a really good

424
00:28:39,000 --> 00:28:45,000
physical feeling for
concentrations.

425
00:28:42,000 --> 00:28:48,000
I strongly recommend you work
with a variable that you are

426
00:28:46,000 --> 00:28:52,000
given, namely,
the dependent variable,

427
00:28:49,000 --> 00:28:55,000
which is the amount of salt,
grams.

428
00:28:52,000 --> 00:28:58,000
Well, because it's something
you can physically think about.

429
00:28:57,000 --> 00:29:03,000
It's coming in,
it's getting mixed up,

430
00:29:00,000 --> 00:29:06,000
and some of it is going out.
So, the basic equation is going

431
00:29:08,000 --> 00:29:14,000
to be that the rate of change of
salt in the tank is the rate of

432
00:29:18,000 --> 00:29:24,000
salt inflow, let me write salt
inflow, minus the rate of salt

433
00:29:27,000 --> 00:29:33,000
outflow.
Okay, at what rate is salt

434
00:29:33,000 --> 00:29:39,000
flowing in?
Well, the flow rate is the flow

435
00:29:39,000 --> 00:29:45,000
rate of the liquid.
I multiply the flow rate,

436
00:29:43,000 --> 00:29:49,000
1 L per minute times the
concentration,

437
00:29:47,000 --> 00:29:53,000
3 g per liter.
That means 3 g per minute.

438
00:29:51,000 --> 00:29:57,000
It's going to be,
therefore, the product of the

439
00:29:56,000 --> 00:30:02,000
flow rate and the concentration,
incoming concentration.

440
00:30:03,000 --> 00:30:09,000
How about the rate of the salt
outflow?

441
00:30:05,000 --> 00:30:11,000
Well, again,
the rate of the liquid outflow

442
00:30:08,000 --> 00:30:14,000
is r.
And, what is the concentration

443
00:30:11,000 --> 00:30:17,000
of salt in the outflow?
I must use, when you talk flow

444
00:30:15,000 --> 00:30:21,000
rates, the other factor must be
the concentration,

445
00:30:18,000 --> 00:30:24,000
not the amount.
So, what is the concentration

446
00:30:22,000 --> 00:30:28,000
in the outflow?
Well, it's the amount of salt

447
00:30:25,000 --> 00:30:31,000
in the tank divided by its
volume.

448
00:30:29,000 --> 00:30:35,000
So, the analog,
the concentration here is x

449
00:30:32,000 --> 00:30:38,000
divided by v.
Now, here's a typical messy

450
00:30:36,000 --> 00:30:42,000
equation, dx / dt,
let's write it in the standard,

451
00:30:40,000 --> 00:30:46,000
linear form,
plus r times x over v equals r

452
00:30:44,000 --> 00:30:50,000
times the given concentration,
which is a function of time.

453
00:30:50,000 --> 00:30:56,000
Now, this is going to be some
given function,

454
00:30:54,000 --> 00:31:00,000
and there will be no reason
whatsoever why you can't solve

455
00:30:59,000 --> 00:31:05,000
it in that form.
And, that's normally what you

456
00:31:04,000 --> 00:31:10,000
will do.
Nonetheless,

457
00:31:05,000 --> 00:31:11,000
in trying to understand how it
fits into this paradigm,

458
00:31:09,000 --> 00:31:15,000
which kind of equation is it?
Well, clearly there's an

459
00:31:12,000 --> 00:31:18,000
awkwardness in that on the
right-hand side,

460
00:31:15,000 --> 00:31:21,000
we have concentration,
and on the left-hand side,

461
00:31:19,000 --> 00:31:25,000
we seem to have amounts.
Now, the way to understand the

462
00:31:22,000 --> 00:31:28,000
equation as opposed to the way
to solve it, well,

463
00:31:26,000 --> 00:31:32,000
it's a step on the way to
solving it.

464
00:31:28,000 --> 00:31:34,000
But, I emphasize,
you can and normally will solve

465
00:31:32,000 --> 00:31:38,000
it in exactly that form.
But, to understand what's

466
00:31:36,000 --> 00:31:42,000
happening, it's better to
express it in terms of

467
00:31:40,000 --> 00:31:46,000
concentration entirely,
and that's why it's called the

468
00:31:43,000 --> 00:31:49,000
concentration,
or the diffusion,

469
00:31:45,000 --> 00:31:51,000
concentration-diffusion
equation.

470
00:31:48,000 --> 00:31:54,000
So, I'm going to convert this
to concentrations.

471
00:31:51,000 --> 00:31:57,000
Now, there's no problem here.
x over v is the concentration

472
00:31:55,000 --> 00:32:01,000
in the tank.
And now, immediately,

473
00:31:57,000 --> 00:32:03,000
you see, hey,
it looks like it's going to

474
00:32:00,000 --> 00:32:06,000
come out just in the first form.
But, wait a minute.

475
00:32:06,000 --> 00:32:12,000
How about the x?
How do I convert that?

476
00:32:10,000 --> 00:32:16,000
Well, what's the relation
between x?

477
00:32:13,000 --> 00:32:19,000
So, if the concentration in the
tank is equal to x over v,

478
00:32:19,000 --> 00:32:25,000
so the tank concentration,
then x is equal to C times the

479
00:32:25,000 --> 00:32:31,000
constant, V, and dx / dt,
therefore, will be c times dC /

480
00:32:31,000 --> 00:32:37,000
dt.
You see that?

481
00:32:34,000 --> 00:32:40,000
Now, that's not in standard
form.

482
00:32:38,000 --> 00:32:44,000
Let's put it in standard form.
To put it in standard form,

483
00:32:44,000 --> 00:32:50,000
I see, now, that it's not r
that's the critical quantity.

484
00:32:51,000 --> 00:32:57,000
It's r divided by v.
So, it's dC / dt,

485
00:32:55,000 --> 00:33:01,000
C prime, plus r divided by v,
--

486
00:33:00,000 --> 00:33:06,000
-- I'm going to call that k,
k C, no let's not,

487
00:33:04,000 --> 00:33:10,000
r divided by v is equal to r
divided by v times Ce.

488
00:33:09,000 --> 00:33:15,000
That's the equation expressed

489
00:33:14,000 --> 00:33:20,000
in a form where the
concentration is the dependent

490
00:33:18,000 --> 00:33:24,000
variable, rather than the amount
of salt itself.

491
00:33:23,000 --> 00:33:29,000
And, you can see it falls
exactly in this category.

492
00:33:27,000 --> 00:33:33,000
That means that I can talk
about it.

493
00:33:31,000 --> 00:33:37,000
The natural way to talk about
this equation is in terms of,

494
00:33:36,000 --> 00:33:42,000
the same way we talked about
the temperature equation.

495
00:33:43,000 --> 00:33:49,000
I said concentration.
I mean, that concentration has

496
00:33:47,000 --> 00:33:53,000
nothing to do with this
concentration.

497
00:33:50,000 --> 00:33:56,000
This is the diffusion model,
where salt solution outside,

498
00:33:54,000 --> 00:34:00,000
cell in the middle,
salt diffusing through a

499
00:33:58,000 --> 00:34:04,000
semi-permeable membrane into
that, uses Newton's law of

500
00:34:03,000 --> 00:34:09,000
diffusion, except he didn't do a
law of diffusion.

501
00:34:08,000 --> 00:34:14,000
But, he is sticky.
His name is attached to

502
00:34:10,000 --> 00:34:16,000
everything.
So, that's this concentration

503
00:34:13,000 --> 00:34:19,000
model.
It's the one entirely analogous

504
00:34:15,000 --> 00:34:21,000
to the temperature.
And the physical setup is the

505
00:34:18,000 --> 00:34:24,000
same.
This one is entirely different.

506
00:34:21,000 --> 00:34:27,000
Mixing in this form of this
problem has really nothing to do

507
00:34:24,000 --> 00:34:30,000
with this model whatsoever.
But, nor does that

508
00:34:27,000 --> 00:34:33,000
concentration had anything to do
with this concentration,

509
00:34:31,000 --> 00:34:37,000
which refers to the result of
the mixing in the tank.

510
00:34:36,000 --> 00:34:42,000
But, what happens is the
differential equation is the

511
00:34:40,000 --> 00:34:46,000
same.
The language of input and

512
00:34:43,000 --> 00:34:49,000
response that we talked about is
also available here.

513
00:34:47,000 --> 00:34:53,000
So, everything is the same.
And, the most interesting thing

514
00:34:52,000 --> 00:34:58,000
is that it shows that the analog
of the conductivity,

515
00:34:57,000 --> 00:35:03,000
the k, the analog of
conductivity and diffusivity is

516
00:35:01,000 --> 00:35:07,000
this quantity.
I should not be considering r

517
00:35:06,000 --> 00:35:12,000
and v by themselves.
I should be considering as the

518
00:35:11,000 --> 00:35:17,000
basic quantity,
the ratio of those two.

519
00:35:15,000 --> 00:35:21,000
Now, why is that,
is the basic parameter.

520
00:35:19,000 --> 00:35:25,000
What is this?
Well, r is the rate of outflow,

521
00:35:23,000 --> 00:35:29,000
and the rate of inflow,
what's r over v?

522
00:35:27,000 --> 00:35:33,000
r over v is the fractional rate
of outflow.

523
00:35:31,000 --> 00:35:37,000
In other words,
if r over v is one tenth,

524
00:35:35,000 --> 00:35:41,000
it means that 1/10 of the tank
will be emptied in a minute,

525
00:35:40,000 --> 00:35:46,000
say.
In other words,

526
00:35:44,000 --> 00:35:50,000
we lumped these two constants
into a single k,

527
00:35:49,000 --> 00:35:55,000
and at the same time have
simplified the units.

528
00:35:53,000 --> 00:35:59,000
What are the units?
This is volume per minute.

529
00:35:58,000 --> 00:36:04,000
This is volume.
So, it's simply reciprocal

530
00:36:02,000 --> 00:36:08,000
minutes, reciprocal time,
which was the same units of

531
00:36:07,000 --> 00:36:13,000
that diffusivity and
conductivity had,

532
00:36:11,000 --> 00:36:17,000
reciprocal time.
The space variables have

533
00:36:16,000 --> 00:36:22,000
entirely disappeared.
So, it that way,

534
00:36:19,000 --> 00:36:25,000
it's simplified.
It's simplified conceptually,

535
00:36:22,000 --> 00:36:28,000
and now, you can answer the
same type of questions we asked

536
00:36:27,000 --> 00:36:33,000
before about this.
I think it would be better for

537
00:36:32,000 --> 00:36:38,000
us to move on,
though.

538
00:36:33,000 --> 00:36:39,000
Well, just an example,
one really simple thing,

539
00:36:37,000 --> 00:36:43,000
so, suppose since we spent so
much time worrying about what

540
00:36:41,000 --> 00:36:47,000
was happening with sinusoid
inputs, I mean,

541
00:36:45,000 --> 00:36:51,000
when could Ce be sinusoidal,
for example?

542
00:36:48,000 --> 00:36:54,000
Well, roughly sinusoidal if,
for example,

543
00:36:51,000 --> 00:36:57,000
some factory were polluting.
If this were a lake,

544
00:36:55,000 --> 00:37:01,000
and some factory were polluting
it, but in the beginning,

545
00:36:59,000 --> 00:37:05,000
at the beginning of the day,
they produced a lot of the

546
00:37:03,000 --> 00:37:09,000
pollutant, and by the end of the
day when it wound down,

547
00:37:07,000 --> 00:37:13,000
it might well happen that the
concentration of pollutants in

548
00:37:12,000 --> 00:37:18,000
the incoming stream would vary
sinusoidally with a 24 hour

549
00:37:16,000 --> 00:37:22,000
cycle.
And then, we would be asking,

550
00:37:22,000 --> 00:37:28,000
so, suppose this varies
sinusoidally.

551
00:37:26,000 --> 00:37:32,000
In other words,
it's like cosine omega t.

552
00:37:31,000 --> 00:37:37,000
I'm asking, how closely does

553
00:37:36,000 --> 00:37:42,000
the concentration in the tank
follow C sub e?

554
00:37:44,000 --> 00:37:50,000
Now, what would that depend
upon?

555
00:37:48,000 --> 00:37:54,000
Think about it.
Well, the answer,

556
00:37:52,000 --> 00:37:58,000
suppose k is large.
Closely, let's just analyze one

557
00:37:58,000 --> 00:38:04,000
case, if k is big.
Now, what would make k big?

558
00:38:03,000 --> 00:38:09,000
We know that from the
temperature thing.

559
00:38:06,000 --> 00:38:12,000
If the conductivity is high,
then the inner temperature will

560
00:38:10,000 --> 00:38:16,000
follow the outer temperature
closely, and the same thing with

561
00:38:15,000 --> 00:38:21,000
the diffusion model.
But we, of course,

562
00:38:17,000 --> 00:38:23,000
therefore, since the equation
is the same, we must get the

563
00:38:21,000 --> 00:38:27,000
same result here.
Now, what would make k big?

564
00:38:25,000 --> 00:38:31,000
If r is big,
if the flow rate is very fast,

565
00:38:28,000 --> 00:38:34,000
we will expect the
concentration of the inside of

566
00:38:31,000 --> 00:38:37,000
that tank to match fairly
closely the concentration of the

567
00:38:35,000 --> 00:38:41,000
pollutant, of the incoming salt
solution, or,

568
00:38:38,000 --> 00:38:44,000
if the tank is very small.
For fixed flow rates,

569
00:38:43,000 --> 00:38:49,000
if the tank is very small,
well, then it gets emptied

570
00:38:47,000 --> 00:38:53,000
quickly.
So, both of these are,

571
00:38:49,000 --> 00:38:55,000
I think, intuitive results.
And, of course,

572
00:38:52,000 --> 00:38:58,000
as before, we got them from
that, by trying to analyze the

573
00:38:56,000 --> 00:39:02,000
final form of the solution.
In other words,

574
00:38:59,000 --> 00:39:05,000
we got them by looking at that
form of the solution up there,

575
00:39:03,000 --> 00:39:09,000
and seeing if k is big.
As k increases,

576
00:39:07,000 --> 00:39:13,000
what happens to the amplitude,
and what happens to the phase

577
00:39:12,000 --> 00:39:18,000
lag?
But, that summarizes the two.

578
00:39:14,000 --> 00:39:20,000
So, this means,
closely means,

579
00:39:17,000 --> 00:39:23,000
that the phase lag is,
big or small?

580
00:39:20,000 --> 00:39:26,000
The lag is small.
And, the amplitude is,

581
00:39:23,000 --> 00:39:29,000
well, the amplitude,
the biggest the amplitude could

582
00:39:27,000 --> 00:39:33,000
ever be is one because that's
the amplitude of this.

583
00:39:33,000 --> 00:39:39,000
So, the amplitude is near one,
one because that's the

584
00:39:38,000 --> 00:39:44,000
amplitude of the incoming
signal, input,

585
00:39:41,000 --> 00:39:47,000
whatever you want to call it.
Okay, now, I'd like to spend

586
00:39:47,000 --> 00:39:53,000
the rest of the time talking
about the failures of number

587
00:39:52,000 --> 00:39:58,000
one, and when you have to use
number two, and when even number

588
00:39:58,000 --> 00:40:04,000
two is no good.
So, let me end first-order

589
00:40:03,000 --> 00:40:09,000
equations by putting my worst
foot forward.

590
00:40:08,000 --> 00:40:14,000
Well, I'm just trying to avoid
disappointment at

591
00:40:13,000 --> 00:40:19,000
misapprehensions from you.
I'll watch you leave this room

592
00:40:19,000 --> 00:40:25,000
and say, well,
he said that,

593
00:40:22,000 --> 00:40:28,000
okay.
So, the first one you're going

594
00:40:26,000 --> 00:40:32,000
to encounter very shortly where
one is not satisfied,

595
00:40:32,000 --> 00:40:38,000
but two is, so on some examples
where you need two,

596
00:40:38,000 --> 00:40:44,000
well, it's going to happen
right here.

597
00:40:44,000 --> 00:40:50,000
Somebody, sooner or later,
it's going to draw on that

598
00:40:47,000 --> 00:40:53,000
loathsome orange chalk,
which is unerasable,

599
00:40:50,000 --> 00:40:56,000
something which looks like
that.

600
00:40:52,000 --> 00:40:58,000
Remember, you saw it here
first.

601
00:40:54,000 --> 00:41:00,000
r, yeah, we had that.
Okay, see, I had it in high

602
00:40:58,000 --> 00:41:04,000
school too.
That's the capacitance.

603
00:41:00,000 --> 00:41:06,000
This is the resistance.
That's the electromotive force:

604
00:41:04,000 --> 00:41:10,000
battery, or a source of
alternating current,

605
00:41:07,000 --> 00:41:13,000
something like that.
Now, of course,

606
00:41:11,000 --> 00:41:17,000
what you're interested in is
how the current flows in the

607
00:41:15,000 --> 00:41:21,000
circuit.
Since current across the

608
00:41:18,000 --> 00:41:24,000
capacitance doesn't make sense,
you have to talk about the

609
00:41:23,000 --> 00:41:29,000
charge on the capacitance.
So, q, it's customary in a

610
00:41:27,000 --> 00:41:33,000
circle this simple to use as the
variable not current,

611
00:41:32,000 --> 00:41:38,000
but the charge on the
capacitance.

612
00:41:36,000 --> 00:41:42,000
And then Kirchhoff's,
you are also supposed to know

613
00:41:40,000 --> 00:41:46,000
that the derivative,
that the time derivative of q

614
00:41:45,000 --> 00:41:51,000
is what's called the current in
the circuit.

615
00:41:49,000 --> 00:41:55,000
That sort of intuitive.
But, i in a physics class,

616
00:41:53,000 --> 00:41:59,000
j in an electrical engineering
class, and why,

617
00:41:57,000 --> 00:42:03,000
not the letter Y,
but why is that?

618
00:42:02,000 --> 00:42:08,000
That's because of electrical
engineers use lots of lots of

619
00:42:06,000 --> 00:42:12,000
complex numbers and therefore,
you have to call current j,

620
00:42:11,000 --> 00:42:17,000
I guess.
I think they do j in physics,

621
00:42:15,000 --> 00:42:21,000
too, now.
No, no they don't.

622
00:42:17,000 --> 00:42:23,000
I don't know.
So, i is ambiguous if you are

623
00:42:21,000 --> 00:42:27,000
in that particular subject.
And it's customary to use,

624
00:42:25,000 --> 00:42:31,000
I don't know.
Now it's completely safe.

625
00:42:29,000 --> 00:42:35,000
Okay, where are we?
Well, the law is,

626
00:42:32,000 --> 00:42:38,000
the basic differential equation
is Kirchhoff's voltage law,

627
00:42:36,000 --> 00:42:42,000
but the sum of the voltage
drops across these three has to

628
00:42:40,000 --> 00:42:46,000
be zero.
So, it's R times i,

629
00:42:42,000 --> 00:42:48,000
which is dq / dt.
That's Ohm's law.

630
00:42:44,000 --> 00:42:50,000
That's the voltage drop across
resistance.

631
00:42:47,000 --> 00:42:53,000
The voltage drop across the
capacitance is Coulomb's law,

632
00:42:51,000 --> 00:42:57,000
one form of Coulomb's law.
It's q divided by C.

633
00:42:55,000 --> 00:43:01,000
And, that has to be the voltage
drop.

634
00:42:57,000 --> 00:43:03,000
And then, there is some sign
convention.

635
00:43:02,000 --> 00:43:08,000
So, this is either plus or
minus, depending on your sign

636
00:43:06,000 --> 00:43:12,000
conventions, but it's E of t.
Now, if I put that in standard

637
00:43:11,000 --> 00:43:17,000
form, in standard form I
probably should say q prime plus

638
00:43:15,000 --> 00:43:21,000
q over RC equals,
well, I suppose,

639
00:43:18,000 --> 00:43:24,000
E over R.
And, this is what would appear

640
00:43:23,000 --> 00:43:29,000
in the equation.
But, it's not the natural

641
00:43:26,000 --> 00:43:32,000
thing.
The k is one over RC.

642
00:43:30,000 --> 00:43:36,000
And, that's the reciprocal.

643
00:43:32,000 --> 00:43:38,000
The RC constant is what
everybody knows is important

644
00:43:35,000 --> 00:43:41,000
when you talk about a little
circuit of that form.

645
00:43:39,000 --> 00:43:45,000
On the other hand,
the right-hand side,

646
00:43:41,000 --> 00:43:47,000
it's quite unnatural to try to
stick in the right-hand side

647
00:43:46,000 --> 00:43:52,000
that same RC.
Call this EC over RC.

648
00:43:48,000 --> 00:43:54,000
People don't do that,
and therefore,

649
00:43:50,000 --> 00:43:56,000
it doesn't really fall into the
paradigm of that first equation.

650
00:43:55,000 --> 00:44:01,000
It's the second equation that
really falls into the category.

651
00:43:59,000 --> 00:44:05,000
Another simple example of this
is chained to k,

652
00:44:02,000 --> 00:44:08,000
radioactively changed to k.
Well, let's say the radioactive

653
00:44:08,000 --> 00:44:14,000
substance, A,
decays into,

654
00:44:10,000 --> 00:44:16,000
let's say, one atom of this
produces one atom of that for

655
00:44:14,000 --> 00:44:20,000
simplicity.
So, it decays into B,

656
00:44:17,000 --> 00:44:23,000
which then still is radioactive
and decays further.

657
00:44:21,000 --> 00:44:27,000
Okay, what's the differential
equation, which is going to be,

658
00:44:26,000 --> 00:44:32,000
it's going to govern this
situation?

659
00:44:29,000 --> 00:44:35,000
What I want to know is how much
B there is at any given time.

660
00:44:35,000 --> 00:44:41,000
So, I want a differential
equation for the quantity of the

661
00:44:39,000 --> 00:44:45,000
radioactive product at any given
time.

662
00:44:41,000 --> 00:44:47,000
Well, what's it going to look
like?

663
00:44:44,000 --> 00:44:50,000
Well, it's the amount coming in
minus the amount going out,

664
00:44:48,000 --> 00:44:54,000
so to speak.
The rate of inflow minus the

665
00:44:51,000 --> 00:44:57,000
rate of outflow,
except it's not the same type

666
00:44:55,000 --> 00:45:01,000
of physical flow we had before.
How fast is it coming in?

667
00:44:59,000 --> 00:45:05,000
Well, A is decaying at a
certain rate,

668
00:45:01,000 --> 00:45:07,000
and so the rate at which A
decays is by the basic

669
00:45:05,000 --> 00:45:11,000
radioactive law.
It's k1, it's constant,

670
00:45:09,000 --> 00:45:15,000
decay constant,
times the amount of A present.

671
00:45:12,000 --> 00:45:18,000
If I used the differential

672
00:45:15,000 --> 00:45:21,000
equation with A here,
I'd have to put a negative sign

673
00:45:18,000 --> 00:45:24,000
because it's the rate at which
that stuff is leaving A.

674
00:45:22,000 --> 00:45:28,000
But, I'm interested in the rate
at which it's coming in to B.

675
00:45:26,000 --> 00:45:32,000
So, it has a positive sign.
And then, the rate at which B

676
00:45:30,000 --> 00:45:36,000
is decaying, and therefore the
quantity of good B is gone,

677
00:45:34,000 --> 00:45:40,000
--
-- that will have some other

678
00:45:38,000 --> 00:45:44,000
constant, B.
So, that will be the equation,

679
00:45:41,000 --> 00:45:47,000
and to avoid having two
dependent variables in there,

680
00:45:46,000 --> 00:45:52,000
we know how A is decaying.
So, it's k1,

681
00:45:49,000 --> 00:45:55,000
some constant times A,
sorry, A will be e to the

682
00:45:53,000 --> 00:45:59,000
negative, you know,
the decay law,

683
00:45:56,000 --> 00:46:02,000
so, times the initial amount
that was there times e to the

684
00:46:01,000 --> 00:46:07,000
negative k1 t.
That's how much A there is at

685
00:46:07,000 --> 00:46:13,000
any given time.
It's decaying by the

686
00:46:10,000 --> 00:46:16,000
radioactive decay law,
minus k2 B.

687
00:46:14,000 --> 00:46:20,000
Okay, so how does the
differential equation look like?

688
00:46:18,000 --> 00:46:24,000
It looks like B prime plus k2 B
equals an exponential,

689
00:46:23,000 --> 00:46:29,000
k1 A zero e to the negative k1
t.

690
00:46:28,000 --> 00:46:34,000
But, there's no reason to

691
00:46:31,000 --> 00:46:37,000
expect that that constant really
has anything to do with k2.

692
00:46:36,000 --> 00:46:42,000
It's unnatural to put it in
that form, which is the correct

693
00:46:41,000 --> 00:46:47,000
one.
Now, in the last two minutes,

694
00:46:49,000 --> 00:46:55,000
I wish to alienate half the
class by pointing out that if k

695
00:47:02,000 --> 00:47:08,000
is less than zero,
none of the terminology of

696
00:47:13,000 --> 00:47:19,000
transient, steady-state input
response applies.

697
00:47:25,000 --> 00:47:31,000
The technique of solving the
equation is identical.

698
00:47:28,000 --> 00:47:34,000
But, you cannot interpret.
So, the technique is the same,

699
00:47:33,000 --> 00:47:39,000
and therefore it's worth
learning.

700
00:47:37,000 --> 00:47:43,000
The technique is the same.
In other words,

701
00:47:41,000 --> 00:47:47,000
the solution will be still e to
the negative kt integral q of t

702
00:47:48,000 --> 00:47:54,000
e to the kt dt plus

703
00:47:54,000 --> 00:48:00,000
a constant times e to the k,
oh, this is terrible,

704
00:48:00,000 --> 00:48:06,000
no.
dy / dt, let's give an example.

705
00:48:04,000 --> 00:48:10,000
The equation I'm going to look
at is something that looks like

706
00:48:10,000 --> 00:48:16,000
this: y equals q of t,
let's say, okay,

707
00:48:13,000 --> 00:48:19,000
a constant, but the constant a
is positive.

708
00:48:17,000 --> 00:48:23,000
So, the constant here is
negative.

709
00:48:19,000 --> 00:48:25,000
Then, when I solve,
my k, in other words,

710
00:48:23,000 --> 00:48:29,000
is now properly written as
negative a.

711
00:48:26,000 --> 00:48:32,000
And therefore,
this formula should now become

712
00:48:30,000 --> 00:48:36,000
not this, but the negative k is
a t.

713
00:48:36,000 --> 00:48:42,000
And, here it's negative a t.
And, here it is positive a t.

714
00:48:41,000 --> 00:48:47,000
Now, why is it,
if this is going to be the

715
00:48:44,000 --> 00:48:50,000
solution, why are all those
things totally irrelevant?

716
00:48:49,000 --> 00:48:55,000
This is not a transient any
longer because if a is positive,

717
00:48:54,000 --> 00:49:00,000
this goes to infinity.
Or, if I go to minus infinity,

718
00:48:59,000 --> 00:49:05,000
then C is negative.
So, it's not transient.

719
00:49:03,000 --> 00:49:09,000
It's not going to zero,
and it depends heavily on the

720
00:49:07,000 --> 00:49:13,000
initial conditions.
That means that of these two

721
00:49:10,000 --> 00:49:16,000
functions, this is the important
guy.

722
00:49:12,000 --> 00:49:18,000
This is just fixed,
some fixed function.

723
00:49:15,000 --> 00:49:21,000
Everything, in other words,
is going to depend upon the

724
00:49:18,000 --> 00:49:24,000
initial conditions,
whereas in the other cases we

725
00:49:21,000 --> 00:49:27,000
have been studying,
the initial conditions after a

726
00:49:25,000 --> 00:49:31,000
while don't matter anymore.
Now, why did I say I would

727
00:49:30,000 --> 00:49:36,000
alienate half of you?
Well, because in what subjects

728
00:49:35,000 --> 00:49:41,000
will a be positive?
In what subjects will k be

729
00:49:39,000 --> 00:49:45,000
negative?
k is typically negative in

730
00:49:43,000 --> 00:49:49,000
biology, economics,
Sloan.

731
00:49:45,000 --> 00:49:51,000
In other words,
the simple thing is think of it

732
00:49:50,000 --> 00:49:56,000
in biology.
What's the simplest equation

733
00:49:53,000 --> 00:49:59,000
for population growth?
Well, it is dP / dt equals

734
00:49:58,000 --> 00:50:04,000
some, if the population is
growing, a times P,

735
00:50:02,000 --> 00:50:08,000
and a is a positive number.
That means P prime minus a P is

736
00:50:09,000 --> 00:50:15,000
zero.
So, the thing I want to leave

737
00:50:13,000 --> 00:50:19,000
you with is this.
If life is involved,

738
00:50:16,000 --> 00:50:22,000
k is likely to be negative.
k is positive when inanimate

739
00:50:22,000 --> 00:50:28,000
things are involved;
I won't say dead,

740
00:50:25,000 --> 00:50:31,000
inanimate.