Uploaded by UCBerkeley on 17.11.2009

Transcript:

Yeah, we'll just, settle down.

OK.

It's good to see you all, and welcome back to

parametric equations.

I want to begin by revisiting what I talked about at

the end of last lecture.

Because, I didn't have enough time to explain it in detail.

I talked about arc lengths of curves, and I wrote a

formula for the arc length.

And, I wanted to tell you a little more about how

this formula is derived.

So, we look at a piece of a curve on a plane, which

is given by, it which is given in parametric form.

And, we'll look at the segment of this curve for

t between alpha and beta.

So, this point would correspond to t equals alpha, and this

point will correspond to t equals beta.

Remember, we discussed last time that in the case of areas,

it was important to keep track of which endpoint was which.

And, so we have to go, when we compute the area, we have

to go from left to right.

It just might happen that the value of t on the left would

be larger than the value of t on the right.

So, then we end up with an integral where the limits

are kind of unusual.

The lower limit is larger than the upper limit.

But, it's no problem, you know, that if you have such an

integral, it's just minus of the integral in which the

limits are switched, they're reversed.

So, there is this subtle point when you do areas.

When you do arc length, there is no such subtly.

It actually doesn't matter.

And you could here t equals theta or t equals alpha.

Either way, you're going to get a good formula for it.

OK.

So, how do we find the arc length of this curve?

The idea, which in fact you will find in many places in

this course, and in all of calculus, is to approximate

the curve by a union of small segments.

Like this.

So, you break it into to many, many pieces.

And, then each of those segments, you want to

approximate by an interval, by a segment of a straight line.

So, that is a crucial idea of all of calculus.

Because you remember, we discussed last time that lines

have the simplest curves.

So, whenever you can approach, approximate well a curve, a

general curve, by segment of a line, you're doing well.

You are able to obtain good results.

It's not a good idea to approximate the entire curve

by a straight line segment.

Because, you see that it looks very different from this curve.

And, the length of this, would certainly not be the same as

the length of this curve.

And, in fact, the curve could be much worse, and sometimes

it could be much more wiggly like this.

So, certainly the length does depend on the

shape of the curve.

However, when we break into small pieces, each piece can

be successfully approximated by a straight line segment.

[LAUGHTER]

You know how these days, you know, you have this

advertisements and sublime advertisements so.

I almost feel like, yeah, we should get product placements,

and make some money on this.

Nice watch, you know.

Especially this time, you know, time of financial crisis.

I think university can do really well.

Given that we have, hopefully, a very large worldwide

audience for this.

Because, what else people want to watch, you know, at

home, but multi-variable calculus, [LAUGHTER]

come to think about it.

And, not just before they go to sleep.

Anyway, going back to this.

Let's just blow up one small segment of this curve, so it

would look something like this.

That's just one of those little guys.

And, now we approximate it by a straight line, OK.

And, so then we approximate the length of this segment by the

length of the straight line segment, and that we can

easily compute from the Pythagoras Theorem.

This would be the displacement delta x, displacement in x,

this would be displacement in y, and this would be delta l.

Now, I want to say that this is for particular segment, and

let's say the segments will be numbered.

There will be segments numbered from 1 to N, where N is some

large number; say, a thousand.

On this picture it's about 10 or so.

But, you want to make many, you want to break it

into many pieces.

So, this will be really delta xi and yi, delta li, And, delta

li you find by Pythagoras Theorem to be delta xi squared

plus delta yi squared, square root.

And now, so this is nice, but I would like to rewrite this in a

slightly more convenient form.

You will see why it's more convenient.

I want to divide and multiply by delta ti.

Where, delta ti is the range of our parameter t from

this point to this point.

So, what I do is I divide by this, and I also

multiply by this.

So, you know, here I'm dividing -- I have a squared root but

-- in each square, in each square I divide by delta ti.

So, the net result is that I'm dividing by delta ti.

It's like I'm dividing by delta ti.

But, then at the time I'm multiplying by delta ti, so

the result is the same.

But, this one will be more convenient for us.

And now, the arc length of our curve, which is the sum of the

arc length of this little segments of the curve, can be

approximated by the sum of the arc length of the

straight line segments.

So, we are going to get the sum of this delta li with

I going from 1 to N.

This is to say that we have delta l1, the lengths of the

first segment, delta l2, the lengths of the second segment,

and so on, up to the lengths of the N segment, delta lN.

I hope you remember this notation from the previous

calculus courses, which means the summation

of all those pieces.

OK.

So, I continue the formula, and I substitute this

expression here.

So, what I get is the sum from 1 to N, and here I put the

squared root of delta xi divided by delta ti squared

plus delta yi, delta ti squared delta ti.

And, now the point is, so you this expression, which it

starts approximating the length of the curve, the finer

the partition becomes.

The more number N, capital N becomes.

So, for a thousand pieces, you would get a good approximation,

for a million pieces you'll get an even better approximation.

So, in the limit, when N goes to infinity, this will

actually give us the arc length of the curve.

And, this sum -- this is called, the sum is called the

partial sum, and the integral is defined as the limit of

this kind of partial sum.

And, so if you remember how the integral in one variable is

defined, and this is really going to be an integral in one

variable, because it's just a summation in one variable.

So, this is going to be precisely what we call the

integral from t equal alpha, and t equal beta, which are the

endpoints of this expression, in which you substitute

instead of delta x, delta t, you substitute dx, dt.

This becomes well approximated in the limit, becomes equal to,

it is well approximated for finite end, but becomes equal

to dx, dt, the derivative of x with respect to t.

And, likewise this one becomes, perhaps I should draw it like

this, and this one becomes dy, dt.

So, this whole thing becomes dx, dt squared plus

dy, dt squared, dt.

And, that's the formula which I wrote down at

the end of last lecture.

I explained it very quickly, so I didn't, I skipped this

intermediate steps about summation and taking a limit,

but that's the formula we get by doing this calculation.

If you will, we can write it down even more precisely of

remembering that x is equal to a function f of t, and y is

equal to a function g of t.

So, we can write it as an integral from alpha to beta,

square root of f prime of t squared plus g prime

of t squared, dt.

And, this is a very representative example for many

other things, which we will study in this course.

Because, oftentimes we will try to approximate things, kind of,

we will try to approximate various quantities for curvy

objects like curves or surfaces by sums of the same kind of

quantities for straight objects like straight line intervals or

squares or parallelograms, and things like that.

And, inevitably we end up with an expression like this.

Where you have a summation over all pieces in the partition.

Where you have some expression involving your parameter, in

the case t times delta ti.

And, under good circumstances.

And, in this course, we don't really discuss the sort of the

subtle points here, about what are the conditions

on the functions f and g.

But, let's just say, good circumstances which are

known and well understood.

All the functions, which we will study in this course, will

satisfy those conditions.

Under those conditions, this sum in limit when N goes to

infinity, when partition becomes more and more fine,

becomes an integral like this.

And, it's very easy to read off the expression of the integral

from the expression of this partial sum as you can

see from this example.

So this is a good guiding principal for many other

things that come later.

For example even in this chapter, we discuss also the

area, surface of, a revolution, and also in, Cartesian

coordinates and polar coordinates.

And, the idea is always the same.

Some I'm not going to explain it each time.

I'm going to sort of stress it here in this particular

example, and I want to say that, in those other cases it's

going to work in a very similar way.

But, you know, as always in math or in other subjects,

there are somehow two part of the story.

The first part of the story is to get the formula, and here

I give you an intuitive derivation of the formula.

I give you a rough explanation of why this formula is true.

But, once you derive this formula, you can sort of,

you can forget about this derivation, and you

can just use it.

So, when you go do homework exercises, from a practical

point of view, you don't need to remember every steop

of this derivation.

I think it is worth wild to really understand, to think

about it and to understand how these things work.

Not only for this particular example, but for other problems

or questions which will come next, which will

come in the future.

But, at the end of the day when you do homework, all you

need to know is the formula.

And, you need to know how to work with these formulas.

So, in this case it is very easy to work with the formula,

because normally you are going to be given the

parameterization, or maybe you have to find it yourself.

But, once you have the parmaterization, you have

formula, you just plug things in, and you get an integral.

What kind of integral.

Well, you get a single integral, an integral with

one variable, right.

And, so you get something which was the subject matter of a

single variable calculus.

So, at this point, you have to remember all the tricks and

rules that you learned in Math 1A-1B on how to compute

integrals in one variable.

This is not the subject of this course.

In other words, we already, we used that information

in this course.

So, what we've done here is we have reduced the problem which

arises in multi-variable calculus.

The problem which involves two variables, right, x and y.

It is a curve on the plane.

And, we want to calculate its arc length.

But, what we have done is we have reduced the problem to a

question in single variable calculus.

And, the question really is to calculate an integral

in one variable.

How do you calculate our integral.

Well, at this point you don't need to know anything about

multi-variable calculus.

You switch back to the single variable calculus, and you have

at your disposal all kinds of methods, and tools, and tricks

that you've learned before.

And, so this is a good time to refresh your memory

on this, right.

To look back, in your notes, or in the book, and to see how

to compute those integrals.

But, this is, this will be assumed.

We're not going to dwell on this too much.

There are some standard methods like change of variables, you

know, integration by parts, and things like that.

So, you have to also remember the anti-derivatives of various

trigonometric functions, for example, there is special

functions, like exponential functions, logarithm,

and things like that.

And, so you just take that toolbox of single variable

calculus, and you apply it here, OK.

So, in a way, in this course, and in the homework, and in

the tests, it is not our goals to your knowledge

necessarily on 1B.

In other words, I'm not going to give you very hard integrals

to work on, but still the basic methods you have to know.

And, even if it is a simple integral, you still need to

know some basic methods to be able to compute it.

Alright.

So, that's about arc length.

And next, we have one more calculation, one more integral

expression, which is not for the length, but it's for

surface area, namely surface area, area of the surface

of a revolution.

Which means, that we look at the surface obtained by

rotating a curve about x-axis or y-axis.

Let's start, let's look at the first of the case of x-axis.

So, here, again, we have, let's say we have some curve, a

segment of a curve between some points here, a and b, say.

And, we rotate it about the x-axis.

So, what we get is, let me actually make a slightly

better picture for you.

[UNINTELLIGIBLE]

And, so when rotate each point on the curve makes a circle.

But, for example this point makes a circle.

And, then this point makes a smaller circle,

because it's closer.

And, the curve itself appears many times.

So, it look something like this.

And, then here also you have a circle, here you have

a circle, and so on.

So, you get kind of a cylindrical looking picture.

In fact, you will get a cylinder if your curve were a

straight line, or segment of a straight line, parallel

to the x-axis.

So, the simplest, let me erase it again.

The simplest example would be if you take this, and then

the surface is just like this, right.

So, this is a kind of wiggly kind of a, like a vase, if you

want to think about it this way, which is turned

on its side.

So, you want to compute the area of this, of this object,

and a cylinder is one example.

Another example, is actually a sphere.

Because, you can think of a sphere as a surface of

revolution over 1/2 a circle.

Because, if you start with 1/2 circle, which has the endpoints

precisely on the x-axis when you rotate it, you're

going to get a sphere.

So, this is kind of a generalization of a sphere and

a cylinder at the same time.

So, it's kind of useful to know what the surface area is.

For instance, this way, you can derive a formula for the area

of a sphere, which is very useful.

So here, the idea, again there are two parts.

One is derivation of the formula, which kind of proceeds

in a very similar way, as here.

And, the second is using the formula.

Once, you get the formula, all you need to do is to substitute

the information you are given, and then use tools and methods

from single variable calculus.

So, what is the formula.

The formula looks like this.

It is the integral from alpha to beta, 2pi, y, and then the

familiar expression, which we have here, for the

arc length, OK.

And, where does this formula come from.

Well, it comes in exactly the same way.

Just like here, I have to break everything into small pieces.

And, so a small piece would be say, I break on the

curve into small pieces.

And, then for each, for each small piece, I'm going to end

up with a little cylinder on this surface, which

looks like this.

It almost looks like a cylinder.

And, so I will approximate the area of the cylinder by taking

the product of the arc length of this curve.

In other words, sort of the height of this cylinder.

And, 2 pi times the radius of this circle, which is the

circumference of the circle as we discussed last time, right.

So, this part comes from the circumference of the circle.

And, this part comes from the arc length of the

segment of the curve.

So, the formula is not at all surprising.

It's just that the formula realizes a very simple fact,

that if you have a cylinder, then the area of the cylinder

is just going to be equal to 2 pi times the radius

of this cylinder.

Which in this case, the radius is y.

That's why you get

2pi, y times the lengths of the side.

The lengths of the side.

And, the lengths of the side is the arc length

that we talked about.

So, it's given by this formula.

That's how you get this.

Once you get it, and you substitute f and g for x and

y, and you get a single variable integral.

Any questions about this.

Yes. [INAUDIBLE]

Do you have to add the sides?

OK.

Good question.

It depends on what is being asked.

First of all, let me repeat the question.

The question is whether in calculating this we have to

add the areas of the top and the bottom of this, right.

That's the question.

So, it depends on what is asked.

If you're asked to calculate.

If you are told that you have to look at the figure which is

which includes both the surface of revolution, and the

top and the bottom.

And, you are asked to calculate the area of the whole thing,

then you have to take the sum of three terms.

One is given by this integral.

And, that is, strictly speaking, is the

area of revolution.

The area of revolution by itself does not include

top and bottom.

But, if you're asked, you can add those two pieces as well.

Any other questions.

[INAUDIBLE]

What if you rotate it with something besides

the axes, right.

That's a good question.

So, here I had talked about rotating about the x-axis.

But, in principle, we could rotate -- well the next level

would be to look at, rotate around the y-axis, right.

But, a formula would be very similar.

It will be instead of 2 pi, y, we will get 2 pi x.

But, now the question is suppose that it's rotated about

a different axis which is neither of these two, but

some other axis like this.

Well, in this case, the formula can also be adopted.

And, but to really do justice to this, you have know the

general rules for changing variables under linear

transformations.

That's the subject of Math 54.

So, in this course we're not going to focus on this kind of

questions -- rotation around lines other than their

coordinate axes.

But, in principle, you could.

And, the way, roughly speaking you do it is by making

a transformation of the whole picture.

By rotating the picture, so that that line becomes one

of the coordinate axis.

And, then doing the calculation using the formula that we got.

There was more question.

You're.

Did you still have a question?

Same question.

OK.

Good.

OK, one more.

[INAUDIBLE]

Why does this formula use the, most of them, this formula in a

special case when x is t and y is equal to f of t, which is as

we discussed, the case of graph of a function, y equals f of x.

This formula becomes the old formula, which we had in

single variable calculus.

So, it is really a generalization.

OK.

So, let's move on.

So, to the next subject, and the next subject

is polar coordinates.

Up to now, we have studied various questions about curves.

And, in all of this discussion, our initial point was, a

certain parameterization of this curve.

In other words, an expression for both x and y, coordinates

of points, which are on this curve, as functions of an

auxiliary parameter, t.

So, what are x and y here? x and y refer to the

coordinates of the point.

So, we are using here a way to parameterize points on the

plane by pairs of numbers, x and y, by their coordinates.

And, as I said already at the first lecture, you should

think of this as a way of addressing those points.

In other words, you can think of this as a unique address

of this point amongst all the points on the plane.

This system of coordinates, x and y is called a Cartesian

coordinate system.

In honor of a French Mathematician,

Philosopher, Descartes.

But, in fact, there are other systems of coordinates, which

in many situations are more convenient and more useful

than the Cartesian system of coordinates.

And, the typical example of a different coordinate system is

polar coordinate system, which we are going to talk about now.

OK.

So, what is the polar coordinate system.

A polar coordinate system is a different way to assign

an address to a given point on the plane.

And, it is defined by a different rule.

So, what is this rule.

Let me do it here.

So, the rule is instead of projecting a point onto the x

and y coordinates -- I still draw those coordinates because,

just because kind of a tribute to the fact how deeply

entrenched this Cartesian coordinate system is in our

minds, because I kind of said, it's like, formatting, think of

it as formatting a disc.

The plane doesn't have any coordinate system, but I kind

of like to draw it, to just indicate that we are viewing it

as a plane on which we are going to draw curves and do

various mathematical calculations.

However, given a point now, we are not going to assign to

it an address by dropping perpendicular lines onto x and

y axes, the way we did before.

But, instead we will measure different characteristics.

Namely, we'll measure the distance to the origin, OK.

And, so we will call this r.

Now, If we just measure this, that's not going to give this

point a unique address.

Because, there are many points on the circle, for which the

distance to origin is equal to r.

In fact, we know precisely what this set is, this

is a circle of radius r.

So, there are way too many.

What we are striving to do on the other hand is to find

a way to assign to each point a unique address.

So, just measuring this by itself is not going to help us.

We need additional information.

And, what gives us additional information, which already

uniquely determines the point, is the angle, which the segment

connecting the origin and our point makes with x-axis.

Let's call this angle theta.

So now you see, as I said, the set of all points, which have

distance are to the origin is the entire circle.

But, within that circle, there's only one point for

which the angle is going to be a particular angle theta.

So, now we've pinned down this point in a unique way,

once we know these two numbers, r and theta.

And, this I call the polar coordinates.

So there are several questions here.

First of all, why do we need another coordinate

system to begin with.

Why can't we be satisfied with the original coordinate

system, the Cartesian coordinate system.

And, the answer to this is that oftentimes if you use the

traditional, the Cartesian coordinate system, you end up,

for example, with various types of integrals.

And, these integrals are going to be single variable

integrals like this, for arc length or surface area.

And, sometimes they're just too hard.

And, even if you are, you had an A plus on 1B, you

wouldn't be able to get a number out of it.

So, even if you apply all the tricks, you're still

unable to solve it.

And, so oftentimes there is a, you should try

a different approach.

And, so oftentimes, the same quantity can be expressed a

different type of integral, right.

And, a way to get to a different type of integral

is to use a different coordinate system.

And, polar coordinate turns out to be very convenient

in many cases.

In many cases it simplifies the answer.

It simplifies the kind of integral that we get.

In fact, just a couple minutes ago, we discussed the

question of rotating around the different lines.

If we were to try to tackle this question by using just the

x and y coordinates, we would not, we will get nowhere.

It would be very difficult.

What I mean to say is that suppose that we were asked to

rotate a curve not around this axis, but around

this axis, you know.

It's a legitimate question.

And, so the answer to this question, the correct answer,

is to realize that in fact, in addition to this coordinate

system, xy, that I kind of draw without thinking on the board,

has as much right to exist as a different coordinate system,

which is obtained by rotating this one by a small angle.

In which, we'll have two different axis, which I

draw with pink chalk.

So, this one I call x prime, and this one

I will call y prime.

And, once you translate it to this coordinate system, the

question becomes exactly identical to the question

we've discussed now, right.

So, this already makes you appreciate the fact that, first

of all there is not a unique coordinate system on the plane,

so it's an illusion that there is a unique coordinate system.

Because, the way I draw it, I draw the horizontal

line, sort of parallel to the floor, right.

But, if I kind of tilt my body a little bit, then it will be

like this, and also if you tilt the floor, right.

So, it's not a good reason.

And, so this tells you already, there is a whole variety of

coordinates you can get by rotating.

All of those coordinate systems though have the same flavor.

They are all Cartesian coordinate systems.

They just rotate one with respect to another

by a certain angle.

This one is sort of radically different.

But, this is already a good illustration that you should

not be stuck with a particular coordinate system.

That oftentimes, to get an answer or to get a good

solution or to get a better approach your problem, it

is advantageous to try a different coordinate system.

So, here we try something which is different, and the advantage

of this is that equations of curves simplify, of certain

curves simplify when you use this coordinate system.

And, of course the curve for which equation simplifies

is the circle.

Circle, I recall can be parameterize using the

traditional Cartesian coordinate system in

the following way.

We write x is cosine t, and y is sine t.

That's not too bad.

But, see we are using two trigonometric functions.

By the way, this is for a circle of radius 1, but if you

want a circle of radius R, let's call it R capital

to distinguish it.

So the circle of radius R.

For example, a circle of radius 5 would have x, 5 cosine t and

y, 5 sine t.

That's not too bad, but this has some telemetric functions,

which are not, not kind of not elementary functions.

On the other hand, in polar coordinate system, the equation

we can try to write equations for this curve as well, for

this circle as well, using this new coordinate system.

And, the equation will simply be r equals R.

So, this is in polar coordinate system.

So, it's the same curve, the circle, but represented in two

different coordinate systems.

Here, we use cosine and sine.

Here we use nothing.

It's just r equals to R capital.

Keep in mind that hear, the small r and the big R play

completely different roles.

This r is one of the two polar coordinates.

This is coordinate like x or y.

And, this R is a number.

Because, I'm asking you to write an equation for

the circle of radius R.

So, for example it's a number which could be equally to any

number you want, like 5, for example.

So, in this case the equation would read just R equals 5.

So, it's this type of equations that we get when looking at

a circle from the point of view of polar coordinates.

And, surely this is a much simpler equation than this one.

It doesn't involve sine or cosine.

It doesn't even involve, it does involve any function

of anything, but constant function 5 or R in general.

So, that's a good illustration of the advantages of

this coordinate system.

The equations for some important curve, like

the circle, simplifies.

And, for this reason also, various integrals that you get,

if you try to calculate the arc length, the surface area,

so, will also simplify.

So, that the first point.

Why do we need this coordinate system.

It's useful in applications.

The second question we can

ask is how to convert one coordinate system into another.

Because, let's say OK I convinced you that this is a

very useful coordinate system.

And, suppose you are given a curve in the Cartesian

coordinates, and you would like to translate this into the

polar coordinate system.

Or, maybe conversely you are given something in the polar

coordinate system, and you want to translate

back into Cartesian.

So, you need some tools, kind of a dictionary, how to

go from one coordinate system to the other.

And, that's actually done in a straight forward way.

You just have to express the coordinates x and y

in terms of r and theta.

And, conversely you have to express R and theta

in terms of x and y.

And, once you do that, you have a dictionary which will enable

you to go between these two coordinate systems very easily.

So, the dictionary is very simple.

For this, we need to remember how the x and y

coordinates are obtained.

So, this is x and this is y.

And, so now we see very clearly, how to find what x and

y are, because x and y are two sides of this triangle, in

which one of the angles is 90 degrees, or pi over 2, the

right angle, and another angle is theta.

So, for such a triangle, we can find the length of the sides,

by taking the lengths of the long side and multiplying by

cosine and sine of theta.

So, therefore we get this formula: x is r cosine theta,

and y is r sine theta, right.

So, that's it.

What does it mean.

It means that if you're given a point, represented in polar

coordinates, two numbers r and theta, you can find the xy

coordinates of this point.

For example, let's say theta is, in this example, let's

look at this example.

Let's say theta is pi over 4, 45 degrees, and r is 2.

So, this 2, this is pi over 4, right.

So, then you can ask what is x and y, so let's

say 2 and pi over 4.

And, you have to take 2 times cosine theta, and cosine theta

is 1 over squared root of 2.

So, it's going to be 2 over squared root of 2, and

likewise, in this case, sine and the cosine are the same.

So, you end up with squared root of 2, squared root of 2.

A very simple rule.

So, that's one way.

What about the other away if you want to go from

xy coordinates to r theta coordinates.

Well, in this case, we are given the two sides, like this,

and we need to find the lengths of the long side, and we also

need to find the angle.

And, course we can do, because we can use PythagorasTheorem

for the first one.

So, we get r is the squared root of x squared plus y

squared, and theta, to find theta, what we can find

is the tangent of theta.

The tangent of theta is going to be the ratio between

this side and this side.

So, instead of writing what theta is, I'll just write as a

tangent of theta, and that's y

over x.

And, once you know the tangent of theta, you should be

able to find what theta is.

So, at that point, actually, we have to address one question,

which I kind of swept under the rug.

Which is what are the possible, what is the possible range for

r and theta, what are the possible value for r and theta.

So, because if we want to one study, if we want to use it as

a bona fide coordinate system, we better know what are the

possible values of these coordinates.

So what are the possible values of r and theta.

Before answering this, let's ask what are the possible

values of x and y.

It's also a legitimate question.

We never asked it, because it sort of, it was sort of a

given that both x and y can take arbitrary values.

When I say arbitrary, any real number, minus

infinity to plus infinity.

For x likewise for y.

Because, we assume that when we talk about the plane, we talk

about the infinite plane, not just this blackboard,

but the infinite plane.

Which is obtained by expanding this blackboard in all

possible directions, right.

So, x and y, the ranges for x and y is are from minus

infinity to plus infinity.

Not so for the polar coordinates, right for one

thing, r was defined as a distance from the origin.

And, the distance is a positive number, or more precisely

it's a non-negative number.

It could be 0 or positive.

So, r, in this definition, is greater than and equal to 0.

OK.

What about theta.

Theta is the angle, and we know that angles

goes from 0 to 2 pi.

So, it is wise to say that actually theta takes values

between 0 and 2 pi.

Because, if we start looking at theta, if we allow theta

greater than 2 pi or less than 0m what will happen is we will

sort of get double billing.

We will get different ways of representing

the same points, right.

Because, we'll be, the same point, for example, will be pi

over 4, but also 2 pi plus pi over 4 or 4 pi plus pi

over 4, and so on.

And, when I was telling you about coordinate system, one of

the important properties that we wanted it to satisfy was

that it gives us a unique address to a given point.

So, if we allow theta to take arbitrary real values, there

will be instantly many ways to represent the same point.

Because, we could all always then shift theta by 2 pi, and

would get the same point.

So, that's why it's better to specify the range of theta

as being from 0 to 2 pi.

If we want to be pedantic, we should also realize that

actually 2 pi is like 0.

And, so once you get to 2 pi, it's like you're back at 0.

So, you don't want to use the same angle twice, even

if it's just one angle.

So, strictly speaking, it has to be from 0, greater and equal

to 0, but less than 2 pi.

So, this is how it's defined, how it should be defined, in

order to get through an ambiguous answer.

And even then, we actually do have a small ambiguity.

I kind of put it in brackets, because it's really kind of

subtle point, which we are not going to dwell on too much.

But, there is a small ambiguity.

There is a small ambiguity.

If r is equal to 0, if r is equal to 0, we are

actually at the origin.

And, so theta becomes meaningless.

We can't really say at what angle our point is with respect

to the origin, because it is an origin.

So, if r is equal to 0.

Unfortunately my board doesn't look so good, so.

My 0 looks like theta now.

That mean 0 really.

If r is 0, then theta is not determined.

So, there's a subtle point that when I promised you that polar

coordinate system gives you a unique address, assigns a

unique address to each point, it's not exactly, it's not

strictly speaking true.

It does assign a unique address to all points

except the origin.

For the origin, the address will be r equals 0 and

theta could be anything.

So, there are too many addresses if you

assigned to that one.

But, because it's only one point, we're not really going

to worry about it too much.

So, I'm just mentioning it to you, so that you are not

so startled by this one.

You realize it is a fact, but it's not going to be a big

problem for us, just because it's only one point, which

is potentially problematic.

Now, there is kind of a more important point here, which

is that -- in fact, we are going to allow allow.

In our calculations, we are going to allow more

values for r and theta.

We will allow, and this is sort of a technical thing.

What I write here is how the polar coordinates are

defined, strictly speaking.

If we really wanted to define it in an ambiguous, say for

this origin, an ambiguous coordinate system.

But, in our calculations, it will be very convenient to have

certain rules, and to allow points with negative r.

So, this is a rule of convenience for you.

The rule of convenience is that if you have negative r, if r is

negative, we will allow negative r with the

following interpretation, negative values of r.

Instead of explaining it in words, Ill just draw a picture,

because it's much easier to see that once than to

hear it 10 times.

And, many of you probably already know that.

Those who have already tried to the homework exercises for this

chapter, for this section.

The rule is like this, if you have a point, like this, r

theta, then its mirror image -- the point which is obtained

from the other side of the line -- will be equal minus r

theta plus pi, right.

So, we will follow this rule, that negative r, so if r is

positive, you have this point.

But, then if the point was negative r, is going to be a

point for which lies of the opposite end of the of this

line, which is symmetric to this point.

Good, I'm glad, so OK, correct me.

What should I write?

[INAUDIBLE].

That's

right, exactly.

So, if r is negative then -- let's say you are given

a point minus 10 and then pi over 4, OK.

So, where would we plot it.

OK.

So, if it was 10 pi over 4, clearly it would be

somewhere here, right.

But, if it is -- so the way I wrote it is not a good way.

So, let's just, let me just give an example using a

particular values of r and theta.

So, this is a representation for r theta if r is positive.

Let's put it this way.

If r is positive, this is a representation, okay.

And, if r is negative, then the representation will be minus

r, theta plus pi, like this.

So, that's what I meant in this picture, but it was

a little bit ambiguous.

So, in particular, if you 10 pi over 4, it would be here.

But, minus 10 pi over 4 will be here.

This is it going to be the point, pi over 4, which will

be the same as 10 and 3 pi, pi plus pi over 4.

OK.

Yeah, when you write it it's not clear when I say minus r do

I mean the original one, or the minus of the minus of

the negative one.

I mean minus of the negative one.

So, it's not the same r as this one.

If r is positive, r theta is just formed like this.

And, if r is negative, let's just do it like this.

If r is negative, and you take, in other words, here

you take the length r.

And, if r is negative.

You take the lengths negative r, which will be now positive,

and you take the angle, which is theta plus pi.

I think it's clear now.

OK.

You have to explain, you have to be careful when

you try to explain it.

We'll see it now in the calculation.

You will see better what I'm talking about.

Alright.

So, let's see what we can do now with this

coordinate system.

Everything all right?

Yes.

You getting tired.

I guess Thursday afternoon.

You all excited about the long weekend.

Yeah.

Well we still have 1/2 hour, but let me take two [LAUGHTER]

but afterwards you go.

So let me, actually let's take a 2 minute break just

to relax a little bit.

I want to tell you something.

It doesn't mean that you should talk to each other.

You still have to listen to me. [LAUGHTER]

It's my time.

So, I want to tell you about an article I read today

in the Forbes magazine.

Not that I read Forbes magazine regularly.

Don't get the wrong idea.

I saw it mentioned on a blog, and the link to this article,

and that's why I read it.

It's available electronically.

And, it's a very interesting article about string theory,

and the controversy about string theory.

And, I'm interested in this, because it's not very far from

my research interest, which involves quantum field theory

and mathematical aspects of quantum field theory.

But, I like this article, because -- remember the first

day when I said that, you know, people think of mathematics and

science as a kind of a something which has been

written in stone and hasn't changed in many years,

which is in fact not true.

And, this article is a very good illustration.

And, I even copied a quote, I quoted a sentence from this

article from the very beginning, which I think

is written very nicely.

So, he writes, the author writes, "Lay people tend

to regard science as a lofty temple inhibited by serene

spoke-like wise men.

Working scientists though will tell you it's more like a stock

market full of fads and fashions, booms and bust."

So, I like this analogy.

Not so much, not so much, I don't like it if you, the

meaning is not say that it's as speculative as

the stock market.

But, I think it's a good illustration of the fact that

it is really a live organism which is constantly changing,

like the stock market.

So, in that sense, I think it's a good analogy, and because

it's Forbes magazine, of course they use that analogy, and

it makes a lot of sense.

Just like the stock market goes up and down, and things change,

likewise in science and mathematics and physics, things

also go up and down, and certain things become

fashionable and then they fall out of fashion.

And, at the end of the day it's the fundamentals, which you

really care about, not the speculation, and not

sort of the, you know, artificial things.

So, I think that it's really cool, and I really like this

analogy with the stock market.

And I suggest, I really recommend this article.

It's very short.

And, it sort of talks about one of the most interesting ideas

of the last, maybe three years in science, the string theory.

So, I'm going to put a link to this article on the bSpace,

on our bSpace page.

And, if you come across any article about math or science,

which you would like to share with other students, please

send me the link and

I'll post it as well.

OK.

All right, now back to boring stuff.

I'm kidding.

It's actually, it's not boring at all.

So, maybe another day we'll take a small break, and I'll

tell you a little bit about what string theory

really is, OK.

But, now let's go back to polar coordinates.

So, let's look at various curves.

And, how to represent those curves using polar coordinates.

We already talked about the circle.

Circle of radius R is represented by the

equation r equals R.

Where, r is one of the two polar coordinates, and

this is the radius.

So, here is another example.

A line passing though the origin with angle theta to the

x-axis also given by a very simple equation, namely now

theta is a constant angle, theta 0 say.

What I mean to say is that this line which forms angle theta 0

to the x-axis is described by the equation theta

equals theta 0.

In the first equation, r equals R, theta is arbitrary.

Well, not exactly arbitrary as we saw it should really

be between 0 and 2 pi.

But, in this equation, r is arbitrary, both

positive and negative.

So, in fact, this is the reason why we introduced this rule,

which may look strange at first sight.

Because, remember, as I said at the beginning, strictly

speaking, r should be non-negative if we follow the

definition polar coordinates.

In which, r is just a distance.

So, it has to be non-negative.

But, if we were to adopt this point of view, then the

equation theta equals theta 0 will actually correspond not

to this entire line, but only 1/2 a line.

Kind of a ray, which goes from the origin to infinity.

Which is fine.

It's a fine geometric object in its own right.

But, if we adopt this rule, if we allow r to be negative, and

we plot points with negative r the way we just discussed, then

not only 1/2 line will be represented in this equation,

which would correspond to positive r, but also the entire

line, the second 1/2 line will correspond to negative

values of r.

So, this is an advantage of this rule, that if we follow

this rule, then we have a nice representation by an equation

of the entire line, like this not just 1/2 line.

OK.

What else can we learn from equations with

polar coordinates.

So, I want to look at a couple of more complicated examples.

I'm sorry?

[INAUDIBLE]

What is?

Arbitrary.

Why is it arbitrary?

The question is why did I say r is arbitrary.

Because, when I write this equation, r is nowhere to be

found in this equation, right.

So, the meaning of this equation is we look at all

points on the plane, whose theta coordinate ix fixed,

if it equals to theta 0.

Theta 0 could be any number between 0 and 2 pi, like pi

over 4, pi over 3, whatever, whatever you want.

Any real number between 0 and 2 pi.

The theta is fixed but r is arbitrary.

When I write this equation, because this equation does not

involve r, this equation means that r can take arbitrary

values within their allowed range.

The allowed range initially was stipulated to be r greater than

0, maybe r greater and equal to 0.

But, eventually we decided that we will allow

negative values of r.

So, that's why -- well actually maybe it's better

to write it like this.

And, on this line, we actually see the part corresponding to r

greater than 0, the part corresponding to r less than 0,

and there is one point which corresponds to r equals 0.

Likewise, when I write this equation, what I'm saying is

that I look at all points on the plane for which the r

coordinate is fixed, it's equal to some number, capital

R, which could be 5, 10, 13, whatever you want.

But, because theta does not appear in the equation, it

means that theta is arbitrary within the allowed

range of theta.

And, what is allowed range.

The Allowed range here is from 0 to 2 pi.

So, that's the meaning of this.

OK.

Now, let's go back to, let's look at more

complicated examples.

So 3, r equals cosine of theta.

So, again a perfectly legitimate equation

involving the two polar coordinates r and theta.

In other words, we're looking at all points on the on the

plane for which, whose r and theta are constrained

by this relation.

It's like writing in the case of Cartesian coordinates

writing an equation x squared plus y squared equals 1.

It's one equation involving our two variables, x and y.

Likewise, this is also one equation involving our two

variables, r and theta.

We have now switched to the polar coordinate system.

So, the question you can be asked is to draw this, too see

what this curve represents.

And, in this case, actually, its easier to understand what

this curve presents by switching back to the

Cartesian coordinates.

This is not to say that this equation is useless.

In fact, on the contrary, you can then use this equation to

compute various things about the curve, as opposed to the

equation and Cartesian coordinates.

But, first we want to visualize them.

What does it mean.

And, here, we have to look back at the dictionary.

At the formulas expressing the polar coordinates in terms of

the Cartesian coordinates.

And, so we could just write r square root of x squared plus y

squared equals cosine theta.

And, then but we know tangent theta, so it's a little bit

fixed, so it's not clear how do we get a good formula.

OK.

So, it doesn't work, so let's look instead at the

geometry of this picture.

So, then let's see what cosine is, cosine theta, what

is it in terms of x

and y.

Well, it's better to say not in terms of x and y, but

in terms of x and r.

Because, this length is x, this length is y,

and this length is r.

So, the cosine theta is the ratio of this

side to this side.

That's why, cosine theta is x divided by r.

Now, let's substitute this in this formula.

So, we get r equals x divided by r.

And, now we can multiply both sides by r.

So, we ge r squared equals x.

OK.

And, now is a good time to express r squared

in terms of x and y.

Because, remember r squared is x squared plus y squared.

So, we substitute here, and so the result is x squared

plus y squared equals x.

And, is already much more manageable.

Let's take x to the left-hand side.

And, let's complete the square.

We can write x squared minus 2 times 1/2, 2 times x

times 1/2 plus 1/2 squared.

So, I introduced an additional term 1/2 squared,

which is one quarter.

On this side and to compensate for this, I also introduced

it on the right-hand side.

So, that I get that through equality.

And, of course I shouldn't forget y squared

as well, you see.

So, I just put tw additional terms on the left- and

right-hand side, which is 1/2 squared.

But, if I do that, then the first three terms

combine into a square.

The first three terms give you x minus 1/2 squared plus y

squared equals 1/2 squared.

And, the end result is already an equation, which is familiar.

The end result is x minus, let me use a different chalk.

So, we get x minus 1/2 squared plus y squared equals

1/2 squared, OK. --

Yes?

[INAUDIBLE]

This was way, this was by way of making

things easier for you.

But, I guess maybe I didn't achieve that goal. 2

times 1/2 is 1, right.

So, that's why, I wanted to write it, you know, I wanted

to use the formula a plus b squared is equal to a squared

plus 2 ab plus b squared.

And, I have x, and I want to have something which is twice,

so that's why I wrote x is 2 times x times 1/2.

Does it make sense?

[INAUDIBLE]

And, then I saw that, so that means x is a and b is 1/2, and

then I put b squared, which is 1/2 squared, which I also

introduced on the other side.

[INAUDIBLE]

Convenient square, that's right.

That's all it is.

So, I'm not cheating, it's all legitimate.

But, yeah, but please check, you know, because sometimes

I can make a mistake.

It works out.

--so, that's the equation we get.

And, now we can draw it, because you see, of course

we're more familiar with the curve x squared plus y

squared is 1/2 squared.

That curve is a circle of course of radius 1/2.

Let me make it bigger.

What's the difference between this equation

and in this equation.

We just shift by 1/2, right.

So, what does it mean geometrically.

Geometrically, it means that we shifted to the right by 1/2.

Why to the right.

Because, for example, here the point with x equals 0 will

correspond to the point with x equals 1/2.

So, that this whole thing will become 0.

So, shifting x by minus 1/2 means shifting everything

to the right by 1/2.

And, when we shift everything to the right by 1/2, it means

the center of the circle, which used to be the origin,

now becomes this point.

Whereas, this point now becomes this point.

And, so it's actually a circle which look like this.

So this is a circle which is represented by this equation.

It's kind of neat, because if you want to write it in

Cartesian coordinates, you get this equation, which is not too

bad, but certainly this looks much more economical in

some sense.

And, so for the purposes of some calculations it

could be very useful.

OK.

What else can we do.

Let's do a small variation on this problem.

So that would be number 4.

A small variation would be r equals cosine of 2 theta.

So, see now we can get away, we can't really get away with

a simple formula for it.

Because, here we used the fact that cosine theta had

such a nice expression.

You can still try it.

Cosine theta is cosine squared of theta minus

sine squared of theta.

So, then you can rewrite cosine squared and sine squared,

but you'll get some very complicated.

So, when you get to a station like this, where you can't

really rewrite it in a nice way by using the Cartesian

coordinate system, you need to try to just kind of understand

qualitatively what this curve looks like.

And, to do that, we should look at this equation as if

r and theta were Cartesian coordinates in some other

world, r and theta are Cartesian coordinates, right.

So, I'll kind of, in a pink world [LAUGHTER], in a pink

world, it just would be like Cartesian coordinates, right.

So, we'll just draw theta and r, and we'll draw this graph in

r and theta, and then we'll try to see what it means in our

world, which is like a yellow world for now, OK.

So, what this is, is just, it's a cosine function except the

period of the cosine function has shrunk to pi

instead of 2 pi.

OK.

Because, already 1 theta is pi, you get cosine of 2 pi, and so

you're back to square one, back to 0.

So, cosine looks like this.

At 0 it's 1, and then it becomes 0 at pi over 2, and

then it becomes negative 1, then it becomes 0 at 3

pi over 2, and then it goes back to 1 at 2 pi.

And, then it continues, like this.

But now, we have two realize that normally if it were just

cosine and function, normally if it were cosine and theta,

this would be pi over 2, OK.

But, because it's 2 theta, it's going to be pi over 4.

And, this will be pi over, normally it would be pi

but now it's pi over 2.

And, this is like 3 pi over 4.

This is pi, and then it continues like this.

So, for example at 5 pi over 4 is going to reach

0 again, and so on.

And, now we would like to plot this on the plane, where r and

theta are polar coordinates.

At least qualitatively, to get a sort of qualitative

understanding what it looks like.

So, we just have to see what the points here correspond

to on this picture.

The first point is a point when theta is 0, and r is 1, right.

So, it's this point in which r is 1, and theta is 0.

Let's plot it on this graph now, on this plane.

So, theta is 0, which means that we are on the x-axis.

All points on the x-axis has theta equals 0, right.

Because, as you remember theta is supposed to be

the angle with an x-axis.

So, if theta is 0, it means you are on the x-axis.

So, you are on the x-axis, and the distance is 1.

So, you start here.

That's your point.

Or, even let's make it even nicer.

Let's start here.

That's our point, that's 1.

What happens next.

What happens next is we are increasing the angle

from 0 to pi over 4.

What is pi over 4.

Pi over 4 is sort of this bisector.

That's pi over 4.

So, when we reach pi over 4, r becomes 0; r becomes 0.

And, r equals 0, no matter what theta is; is as we

discussed the origin.

So, we start here, and then the angles should increase.

Should go from theta equals 0 to theta equals pi over 4.

But, at the same time, r is going to decrease

until it becomes 0.

So, this is going to look like this, roughly.

Because, what else can it look like.

You see, that's the point.

I mean, I used to call it a picture, I'm not insisting

that I know exactly.

See, there is no other way to represent it like this.

If you try to write it in terms of Cartesian coordinates,

you're getting nowhere, You're going to get a very complicated

expression, which is not going to help you.

So, we would can only study it qualitatively, and

that's what it looks like.

And now, we continue.

But, what happens now is that r is a theta, sorry, theta goes

from pi over 4 to pi over 2, but r becomes negative.

That's where our rule become really handy.

Because, if we were, how should I say it, to rigid, and said no

r has to be positive and we don't accept any negative

values, we would say, we would have to say that this

part is not acceptable.

We can only look at the picture in this range,

and this range, and so .

But, not here.

And, what this would mean is that we would not be able to

draw some, represent some very nice pictures by

using polar coordinates.

But, because we've been flexible, and we said we will

accept negative r's, but following a particular rule,

which we discussed, then we'll be able to actually

draw the whole thing.

And, so what this is going to look like is that now

pi is suppose go from pi over 4 to pi over 2.

But, r becomes negative; r becoming negative means, that

we take the absolute value of r, but we shift

the angle by pi.

So, that means that, and the next segment, on this segment,

we look at theta from pi over 4 plus pi to pi over

2 plus pi, right.

So, where is that.

This is pi over 4, pi over 2, pi over 4 plus pi is this,

right, and pi over 2 plus pi is this.

And, is going to go in such a way that r is going

to go to 1 again.

To, well, to negative 1, but we have to take

the absolute value.

So, it's going to be like this.

Now, it becomes positive.

I'm sorry, it still stays negative, and it continues

to 3 pi over 4, and that's like this.

To see this segment is this, this segment is this, this

segment is this, and I think now you can probably

already guess what it's going to look like.

It's going to be like this.

That's right.

Like this, then it goes like this.

Huh?

Not so bad huh. [APPLAUSE]

A good thing to chew on over the long weekend.

So there is one more thing which we need to discuss.

Which is the formula for the surface area for polar curves.

But, it's fairly straightforward.

I'll just write the answer.

Hold on hold on.

We still have three minutes, OK.

So, this is the formula for the area, which is 1/2 r squared d

theta, and that's for a picture which you get by taking a

segment, by taking a sector on the plane, which is bounded by

the curve and by the lines theta equals alpha and

theta equals beta.

The way you derive this formula is exactly the same.

What you need to know is the same as the method which we

used to understand arc length and surface areas for

surface of revolution.

And, the method is to break it into small sectors, and

evaluate the approximate area of which small sector.

And, that's something you can read about in the book.

It's very straightforward.

Now, I want to end with two announcements.

The first announcement is that, as you know, the first homework

assignment is due on Wednesday, because Monday is labor day.

You turn your homework at your section to your GSIs to your

TAs, all right, on Wednesday.

On Wednesday night, after all the sections are over, I will

post the solutions to all the homework problems in this set

on the bSpace page online, OK.

That's the first announcement.

And, the second announcement is about my office hours.

Normally, my office hours are supposed to be in my office in

Evans Hall, but I think we're lucky that there is nobody at

least in the first two lectures.

After the first two lectures, there was nobody here, so we'll

just hold office hours here, in this room, from 5:00

pm to 6:30 pm.

All right.

So, I'll let you go.

Have a good weekend.

OK.

It's good to see you all, and welcome back to

parametric equations.

I want to begin by revisiting what I talked about at

the end of last lecture.

Because, I didn't have enough time to explain it in detail.

I talked about arc lengths of curves, and I wrote a

formula for the arc length.

And, I wanted to tell you a little more about how

this formula is derived.

So, we look at a piece of a curve on a plane, which

is given by, it which is given in parametric form.

And, we'll look at the segment of this curve for

t between alpha and beta.

So, this point would correspond to t equals alpha, and this

point will correspond to t equals beta.

Remember, we discussed last time that in the case of areas,

it was important to keep track of which endpoint was which.

And, so we have to go, when we compute the area, we have

to go from left to right.

It just might happen that the value of t on the left would

be larger than the value of t on the right.

So, then we end up with an integral where the limits

are kind of unusual.

The lower limit is larger than the upper limit.

But, it's no problem, you know, that if you have such an

integral, it's just minus of the integral in which the

limits are switched, they're reversed.

So, there is this subtle point when you do areas.

When you do arc length, there is no such subtly.

It actually doesn't matter.

And you could here t equals theta or t equals alpha.

Either way, you're going to get a good formula for it.

OK.

So, how do we find the arc length of this curve?

The idea, which in fact you will find in many places in

this course, and in all of calculus, is to approximate

the curve by a union of small segments.

Like this.

So, you break it into to many, many pieces.

And, then each of those segments, you want to

approximate by an interval, by a segment of a straight line.

So, that is a crucial idea of all of calculus.

Because you remember, we discussed last time that lines

have the simplest curves.

So, whenever you can approach, approximate well a curve, a

general curve, by segment of a line, you're doing well.

You are able to obtain good results.

It's not a good idea to approximate the entire curve

by a straight line segment.

Because, you see that it looks very different from this curve.

And, the length of this, would certainly not be the same as

the length of this curve.

And, in fact, the curve could be much worse, and sometimes

it could be much more wiggly like this.

So, certainly the length does depend on the

shape of the curve.

However, when we break into small pieces, each piece can

be successfully approximated by a straight line segment.

[LAUGHTER]

You know how these days, you know, you have this

advertisements and sublime advertisements so.

I almost feel like, yeah, we should get product placements,

and make some money on this.

Nice watch, you know.

Especially this time, you know, time of financial crisis.

I think university can do really well.

Given that we have, hopefully, a very large worldwide

audience for this.

Because, what else people want to watch, you know, at

home, but multi-variable calculus, [LAUGHTER]

come to think about it.

And, not just before they go to sleep.

Anyway, going back to this.

Let's just blow up one small segment of this curve, so it

would look something like this.

That's just one of those little guys.

And, now we approximate it by a straight line, OK.

And, so then we approximate the length of this segment by the

length of the straight line segment, and that we can

easily compute from the Pythagoras Theorem.

This would be the displacement delta x, displacement in x,

this would be displacement in y, and this would be delta l.

Now, I want to say that this is for particular segment, and

let's say the segments will be numbered.

There will be segments numbered from 1 to N, where N is some

large number; say, a thousand.

On this picture it's about 10 or so.

But, you want to make many, you want to break it

into many pieces.

So, this will be really delta xi and yi, delta li, And, delta

li you find by Pythagoras Theorem to be delta xi squared

plus delta yi squared, square root.

And now, so this is nice, but I would like to rewrite this in a

slightly more convenient form.

You will see why it's more convenient.

I want to divide and multiply by delta ti.

Where, delta ti is the range of our parameter t from

this point to this point.

So, what I do is I divide by this, and I also

multiply by this.

So, you know, here I'm dividing -- I have a squared root but

-- in each square, in each square I divide by delta ti.

So, the net result is that I'm dividing by delta ti.

It's like I'm dividing by delta ti.

But, then at the time I'm multiplying by delta ti, so

the result is the same.

But, this one will be more convenient for us.

And now, the arc length of our curve, which is the sum of the

arc length of this little segments of the curve, can be

approximated by the sum of the arc length of the

straight line segments.

So, we are going to get the sum of this delta li with

I going from 1 to N.

This is to say that we have delta l1, the lengths of the

first segment, delta l2, the lengths of the second segment,

and so on, up to the lengths of the N segment, delta lN.

I hope you remember this notation from the previous

calculus courses, which means the summation

of all those pieces.

OK.

So, I continue the formula, and I substitute this

expression here.

So, what I get is the sum from 1 to N, and here I put the

squared root of delta xi divided by delta ti squared

plus delta yi, delta ti squared delta ti.

And, now the point is, so you this expression, which it

starts approximating the length of the curve, the finer

the partition becomes.

The more number N, capital N becomes.

So, for a thousand pieces, you would get a good approximation,

for a million pieces you'll get an even better approximation.

So, in the limit, when N goes to infinity, this will

actually give us the arc length of the curve.

And, this sum -- this is called, the sum is called the

partial sum, and the integral is defined as the limit of

this kind of partial sum.

And, so if you remember how the integral in one variable is

defined, and this is really going to be an integral in one

variable, because it's just a summation in one variable.

So, this is going to be precisely what we call the

integral from t equal alpha, and t equal beta, which are the

endpoints of this expression, in which you substitute

instead of delta x, delta t, you substitute dx, dt.

This becomes well approximated in the limit, becomes equal to,

it is well approximated for finite end, but becomes equal

to dx, dt, the derivative of x with respect to t.

And, likewise this one becomes, perhaps I should draw it like

this, and this one becomes dy, dt.

So, this whole thing becomes dx, dt squared plus

dy, dt squared, dt.

And, that's the formula which I wrote down at

the end of last lecture.

I explained it very quickly, so I didn't, I skipped this

intermediate steps about summation and taking a limit,

but that's the formula we get by doing this calculation.

If you will, we can write it down even more precisely of

remembering that x is equal to a function f of t, and y is

equal to a function g of t.

So, we can write it as an integral from alpha to beta,

square root of f prime of t squared plus g prime

of t squared, dt.

And, this is a very representative example for many

other things, which we will study in this course.

Because, oftentimes we will try to approximate things, kind of,

we will try to approximate various quantities for curvy

objects like curves or surfaces by sums of the same kind of

quantities for straight objects like straight line intervals or

squares or parallelograms, and things like that.

And, inevitably we end up with an expression like this.

Where you have a summation over all pieces in the partition.

Where you have some expression involving your parameter, in

the case t times delta ti.

And, under good circumstances.

And, in this course, we don't really discuss the sort of the

subtle points here, about what are the conditions

on the functions f and g.

But, let's just say, good circumstances which are

known and well understood.

All the functions, which we will study in this course, will

satisfy those conditions.

Under those conditions, this sum in limit when N goes to

infinity, when partition becomes more and more fine,

becomes an integral like this.

And, it's very easy to read off the expression of the integral

from the expression of this partial sum as you can

see from this example.

So this is a good guiding principal for many other

things that come later.

For example even in this chapter, we discuss also the

area, surface of, a revolution, and also in, Cartesian

coordinates and polar coordinates.

And, the idea is always the same.

Some I'm not going to explain it each time.

I'm going to sort of stress it here in this particular

example, and I want to say that, in those other cases it's

going to work in a very similar way.

But, you know, as always in math or in other subjects,

there are somehow two part of the story.

The first part of the story is to get the formula, and here

I give you an intuitive derivation of the formula.

I give you a rough explanation of why this formula is true.

But, once you derive this formula, you can sort of,

you can forget about this derivation, and you

can just use it.

So, when you go do homework exercises, from a practical

point of view, you don't need to remember every steop

of this derivation.

I think it is worth wild to really understand, to think

about it and to understand how these things work.

Not only for this particular example, but for other problems

or questions which will come next, which will

come in the future.

But, at the end of the day when you do homework, all you

need to know is the formula.

And, you need to know how to work with these formulas.

So, in this case it is very easy to work with the formula,

because normally you are going to be given the

parameterization, or maybe you have to find it yourself.

But, once you have the parmaterization, you have

formula, you just plug things in, and you get an integral.

What kind of integral.

Well, you get a single integral, an integral with

one variable, right.

And, so you get something which was the subject matter of a

single variable calculus.

So, at this point, you have to remember all the tricks and

rules that you learned in Math 1A-1B on how to compute

integrals in one variable.

This is not the subject of this course.

In other words, we already, we used that information

in this course.

So, what we've done here is we have reduced the problem which

arises in multi-variable calculus.

The problem which involves two variables, right, x and y.

It is a curve on the plane.

And, we want to calculate its arc length.

But, what we have done is we have reduced the problem to a

question in single variable calculus.

And, the question really is to calculate an integral

in one variable.

How do you calculate our integral.

Well, at this point you don't need to know anything about

multi-variable calculus.

You switch back to the single variable calculus, and you have

at your disposal all kinds of methods, and tools, and tricks

that you've learned before.

And, so this is a good time to refresh your memory

on this, right.

To look back, in your notes, or in the book, and to see how

to compute those integrals.

But, this is, this will be assumed.

We're not going to dwell on this too much.

There are some standard methods like change of variables, you

know, integration by parts, and things like that.

So, you have to also remember the anti-derivatives of various

trigonometric functions, for example, there is special

functions, like exponential functions, logarithm,

and things like that.

And, so you just take that toolbox of single variable

calculus, and you apply it here, OK.

So, in a way, in this course, and in the homework, and in

the tests, it is not our goals to your knowledge

necessarily on 1B.

In other words, I'm not going to give you very hard integrals

to work on, but still the basic methods you have to know.

And, even if it is a simple integral, you still need to

know some basic methods to be able to compute it.

Alright.

So, that's about arc length.

And next, we have one more calculation, one more integral

expression, which is not for the length, but it's for

surface area, namely surface area, area of the surface

of a revolution.

Which means, that we look at the surface obtained by

rotating a curve about x-axis or y-axis.

Let's start, let's look at the first of the case of x-axis.

So, here, again, we have, let's say we have some curve, a

segment of a curve between some points here, a and b, say.

And, we rotate it about the x-axis.

So, what we get is, let me actually make a slightly

better picture for you.

[UNINTELLIGIBLE]

And, so when rotate each point on the curve makes a circle.

But, for example this point makes a circle.

And, then this point makes a smaller circle,

because it's closer.

And, the curve itself appears many times.

So, it look something like this.

And, then here also you have a circle, here you have

a circle, and so on.

So, you get kind of a cylindrical looking picture.

In fact, you will get a cylinder if your curve were a

straight line, or segment of a straight line, parallel

to the x-axis.

So, the simplest, let me erase it again.

The simplest example would be if you take this, and then

the surface is just like this, right.

So, this is a kind of wiggly kind of a, like a vase, if you

want to think about it this way, which is turned

on its side.

So, you want to compute the area of this, of this object,

and a cylinder is one example.

Another example, is actually a sphere.

Because, you can think of a sphere as a surface of

revolution over 1/2 a circle.

Because, if you start with 1/2 circle, which has the endpoints

precisely on the x-axis when you rotate it, you're

going to get a sphere.

So, this is kind of a generalization of a sphere and

a cylinder at the same time.

So, it's kind of useful to know what the surface area is.

For instance, this way, you can derive a formula for the area

of a sphere, which is very useful.

So here, the idea, again there are two parts.

One is derivation of the formula, which kind of proceeds

in a very similar way, as here.

And, the second is using the formula.

Once, you get the formula, all you need to do is to substitute

the information you are given, and then use tools and methods

from single variable calculus.

So, what is the formula.

The formula looks like this.

It is the integral from alpha to beta, 2pi, y, and then the

familiar expression, which we have here, for the

arc length, OK.

And, where does this formula come from.

Well, it comes in exactly the same way.

Just like here, I have to break everything into small pieces.

And, so a small piece would be say, I break on the

curve into small pieces.

And, then for each, for each small piece, I'm going to end

up with a little cylinder on this surface, which

looks like this.

It almost looks like a cylinder.

And, so I will approximate the area of the cylinder by taking

the product of the arc length of this curve.

In other words, sort of the height of this cylinder.

And, 2 pi times the radius of this circle, which is the

circumference of the circle as we discussed last time, right.

So, this part comes from the circumference of the circle.

And, this part comes from the arc length of the

segment of the curve.

So, the formula is not at all surprising.

It's just that the formula realizes a very simple fact,

that if you have a cylinder, then the area of the cylinder

is just going to be equal to 2 pi times the radius

of this cylinder.

Which in this case, the radius is y.

That's why you get

2pi, y times the lengths of the side.

The lengths of the side.

And, the lengths of the side is the arc length

that we talked about.

So, it's given by this formula.

That's how you get this.

Once you get it, and you substitute f and g for x and

y, and you get a single variable integral.

Any questions about this.

Yes. [INAUDIBLE]

Do you have to add the sides?

OK.

Good question.

It depends on what is being asked.

First of all, let me repeat the question.

The question is whether in calculating this we have to

add the areas of the top and the bottom of this, right.

That's the question.

So, it depends on what is asked.

If you're asked to calculate.

If you are told that you have to look at the figure which is

which includes both the surface of revolution, and the

top and the bottom.

And, you are asked to calculate the area of the whole thing,

then you have to take the sum of three terms.

One is given by this integral.

And, that is, strictly speaking, is the

area of revolution.

The area of revolution by itself does not include

top and bottom.

But, if you're asked, you can add those two pieces as well.

Any other questions.

[INAUDIBLE]

What if you rotate it with something besides

the axes, right.

That's a good question.

So, here I had talked about rotating about the x-axis.

But, in principle, we could rotate -- well the next level

would be to look at, rotate around the y-axis, right.

But, a formula would be very similar.

It will be instead of 2 pi, y, we will get 2 pi x.

But, now the question is suppose that it's rotated about

a different axis which is neither of these two, but

some other axis like this.

Well, in this case, the formula can also be adopted.

And, but to really do justice to this, you have know the

general rules for changing variables under linear

transformations.

That's the subject of Math 54.

So, in this course we're not going to focus on this kind of

questions -- rotation around lines other than their

coordinate axes.

But, in principle, you could.

And, the way, roughly speaking you do it is by making

a transformation of the whole picture.

By rotating the picture, so that that line becomes one

of the coordinate axis.

And, then doing the calculation using the formula that we got.

There was more question.

You're.

Did you still have a question?

Same question.

OK.

Good.

OK, one more.

[INAUDIBLE]

Why does this formula use the, most of them, this formula in a

special case when x is t and y is equal to f of t, which is as

we discussed, the case of graph of a function, y equals f of x.

This formula becomes the old formula, which we had in

single variable calculus.

So, it is really a generalization.

OK.

So, let's move on.

So, to the next subject, and the next subject

is polar coordinates.

Up to now, we have studied various questions about curves.

And, in all of this discussion, our initial point was, a

certain parameterization of this curve.

In other words, an expression for both x and y, coordinates

of points, which are on this curve, as functions of an

auxiliary parameter, t.

So, what are x and y here? x and y refer to the

coordinates of the point.

So, we are using here a way to parameterize points on the

plane by pairs of numbers, x and y, by their coordinates.

And, as I said already at the first lecture, you should

think of this as a way of addressing those points.

In other words, you can think of this as a unique address

of this point amongst all the points on the plane.

This system of coordinates, x and y is called a Cartesian

coordinate system.

In honor of a French Mathematician,

Philosopher, Descartes.

But, in fact, there are other systems of coordinates, which

in many situations are more convenient and more useful

than the Cartesian system of coordinates.

And, the typical example of a different coordinate system is

polar coordinate system, which we are going to talk about now.

OK.

So, what is the polar coordinate system.

A polar coordinate system is a different way to assign

an address to a given point on the plane.

And, it is defined by a different rule.

So, what is this rule.

Let me do it here.

So, the rule is instead of projecting a point onto the x

and y coordinates -- I still draw those coordinates because,

just because kind of a tribute to the fact how deeply

entrenched this Cartesian coordinate system is in our

minds, because I kind of said, it's like, formatting, think of

it as formatting a disc.

The plane doesn't have any coordinate system, but I kind

of like to draw it, to just indicate that we are viewing it

as a plane on which we are going to draw curves and do

various mathematical calculations.

However, given a point now, we are not going to assign to

it an address by dropping perpendicular lines onto x and

y axes, the way we did before.

But, instead we will measure different characteristics.

Namely, we'll measure the distance to the origin, OK.

And, so we will call this r.

Now, If we just measure this, that's not going to give this

point a unique address.

Because, there are many points on the circle, for which the

distance to origin is equal to r.

In fact, we know precisely what this set is, this

is a circle of radius r.

So, there are way too many.

What we are striving to do on the other hand is to find

a way to assign to each point a unique address.

So, just measuring this by itself is not going to help us.

We need additional information.

And, what gives us additional information, which already

uniquely determines the point, is the angle, which the segment

connecting the origin and our point makes with x-axis.

Let's call this angle theta.

So now you see, as I said, the set of all points, which have

distance are to the origin is the entire circle.

But, within that circle, there's only one point for

which the angle is going to be a particular angle theta.

So, now we've pinned down this point in a unique way,

once we know these two numbers, r and theta.

And, this I call the polar coordinates.

So there are several questions here.

First of all, why do we need another coordinate

system to begin with.

Why can't we be satisfied with the original coordinate

system, the Cartesian coordinate system.

And, the answer to this is that oftentimes if you use the

traditional, the Cartesian coordinate system, you end up,

for example, with various types of integrals.

And, these integrals are going to be single variable

integrals like this, for arc length or surface area.

And, sometimes they're just too hard.

And, even if you are, you had an A plus on 1B, you

wouldn't be able to get a number out of it.

So, even if you apply all the tricks, you're still

unable to solve it.

And, so oftentimes there is a, you should try

a different approach.

And, so oftentimes, the same quantity can be expressed a

different type of integral, right.

And, a way to get to a different type of integral

is to use a different coordinate system.

And, polar coordinate turns out to be very convenient

in many cases.

In many cases it simplifies the answer.

It simplifies the kind of integral that we get.

In fact, just a couple minutes ago, we discussed the

question of rotating around the different lines.

If we were to try to tackle this question by using just the

x and y coordinates, we would not, we will get nowhere.

It would be very difficult.

What I mean to say is that suppose that we were asked to

rotate a curve not around this axis, but around

this axis, you know.

It's a legitimate question.

And, so the answer to this question, the correct answer,

is to realize that in fact, in addition to this coordinate

system, xy, that I kind of draw without thinking on the board,

has as much right to exist as a different coordinate system,

which is obtained by rotating this one by a small angle.

In which, we'll have two different axis, which I

draw with pink chalk.

So, this one I call x prime, and this one

I will call y prime.

And, once you translate it to this coordinate system, the

question becomes exactly identical to the question

we've discussed now, right.

So, this already makes you appreciate the fact that, first

of all there is not a unique coordinate system on the plane,

so it's an illusion that there is a unique coordinate system.

Because, the way I draw it, I draw the horizontal

line, sort of parallel to the floor, right.

But, if I kind of tilt my body a little bit, then it will be

like this, and also if you tilt the floor, right.

So, it's not a good reason.

And, so this tells you already, there is a whole variety of

coordinates you can get by rotating.

All of those coordinate systems though have the same flavor.

They are all Cartesian coordinate systems.

They just rotate one with respect to another

by a certain angle.

This one is sort of radically different.

But, this is already a good illustration that you should

not be stuck with a particular coordinate system.

That oftentimes, to get an answer or to get a good

solution or to get a better approach your problem, it

is advantageous to try a different coordinate system.

So, here we try something which is different, and the advantage

of this is that equations of curves simplify, of certain

curves simplify when you use this coordinate system.

And, of course the curve for which equation simplifies

is the circle.

Circle, I recall can be parameterize using the

traditional Cartesian coordinate system in

the following way.

We write x is cosine t, and y is sine t.

That's not too bad.

But, see we are using two trigonometric functions.

By the way, this is for a circle of radius 1, but if you

want a circle of radius R, let's call it R capital

to distinguish it.

So the circle of radius R.

For example, a circle of radius 5 would have x, 5 cosine t and

y, 5 sine t.

That's not too bad, but this has some telemetric functions,

which are not, not kind of not elementary functions.

On the other hand, in polar coordinate system, the equation

we can try to write equations for this curve as well, for

this circle as well, using this new coordinate system.

And, the equation will simply be r equals R.

So, this is in polar coordinate system.

So, it's the same curve, the circle, but represented in two

different coordinate systems.

Here, we use cosine and sine.

Here we use nothing.

It's just r equals to R capital.

Keep in mind that hear, the small r and the big R play

completely different roles.

This r is one of the two polar coordinates.

This is coordinate like x or y.

And, this R is a number.

Because, I'm asking you to write an equation for

the circle of radius R.

So, for example it's a number which could be equally to any

number you want, like 5, for example.

So, in this case the equation would read just R equals 5.

So, it's this type of equations that we get when looking at

a circle from the point of view of polar coordinates.

And, surely this is a much simpler equation than this one.

It doesn't involve sine or cosine.

It doesn't even involve, it does involve any function

of anything, but constant function 5 or R in general.

So, that's a good illustration of the advantages of

this coordinate system.

The equations for some important curve, like

the circle, simplifies.

And, for this reason also, various integrals that you get,

if you try to calculate the arc length, the surface area,

so, will also simplify.

So, that the first point.

Why do we need this coordinate system.

It's useful in applications.

The second question we can

ask is how to convert one coordinate system into another.

Because, let's say OK I convinced you that this is a

very useful coordinate system.

And, suppose you are given a curve in the Cartesian

coordinates, and you would like to translate this into the

polar coordinate system.

Or, maybe conversely you are given something in the polar

coordinate system, and you want to translate

back into Cartesian.

So, you need some tools, kind of a dictionary, how to

go from one coordinate system to the other.

And, that's actually done in a straight forward way.

You just have to express the coordinates x and y

in terms of r and theta.

And, conversely you have to express R and theta

in terms of x and y.

And, once you do that, you have a dictionary which will enable

you to go between these two coordinate systems very easily.

So, the dictionary is very simple.

For this, we need to remember how the x and y

coordinates are obtained.

So, this is x and this is y.

And, so now we see very clearly, how to find what x and

y are, because x and y are two sides of this triangle, in

which one of the angles is 90 degrees, or pi over 2, the

right angle, and another angle is theta.

So, for such a triangle, we can find the length of the sides,

by taking the lengths of the long side and multiplying by

cosine and sine of theta.

So, therefore we get this formula: x is r cosine theta,

and y is r sine theta, right.

So, that's it.

What does it mean.

It means that if you're given a point, represented in polar

coordinates, two numbers r and theta, you can find the xy

coordinates of this point.

For example, let's say theta is, in this example, let's

look at this example.

Let's say theta is pi over 4, 45 degrees, and r is 2.

So, this 2, this is pi over 4, right.

So, then you can ask what is x and y, so let's

say 2 and pi over 4.

And, you have to take 2 times cosine theta, and cosine theta

is 1 over squared root of 2.

So, it's going to be 2 over squared root of 2, and

likewise, in this case, sine and the cosine are the same.

So, you end up with squared root of 2, squared root of 2.

A very simple rule.

So, that's one way.

What about the other away if you want to go from

xy coordinates to r theta coordinates.

Well, in this case, we are given the two sides, like this,

and we need to find the lengths of the long side, and we also

need to find the angle.

And, course we can do, because we can use PythagorasTheorem

for the first one.

So, we get r is the squared root of x squared plus y

squared, and theta, to find theta, what we can find

is the tangent of theta.

The tangent of theta is going to be the ratio between

this side and this side.

So, instead of writing what theta is, I'll just write as a

tangent of theta, and that's y

over x.

And, once you know the tangent of theta, you should be

able to find what theta is.

So, at that point, actually, we have to address one question,

which I kind of swept under the rug.

Which is what are the possible, what is the possible range for

r and theta, what are the possible value for r and theta.

So, because if we want to one study, if we want to use it as

a bona fide coordinate system, we better know what are the

possible values of these coordinates.

So what are the possible values of r and theta.

Before answering this, let's ask what are the possible

values of x and y.

It's also a legitimate question.

We never asked it, because it sort of, it was sort of a

given that both x and y can take arbitrary values.

When I say arbitrary, any real number, minus

infinity to plus infinity.

For x likewise for y.

Because, we assume that when we talk about the plane, we talk

about the infinite plane, not just this blackboard,

but the infinite plane.

Which is obtained by expanding this blackboard in all

possible directions, right.

So, x and y, the ranges for x and y is are from minus

infinity to plus infinity.

Not so for the polar coordinates, right for one

thing, r was defined as a distance from the origin.

And, the distance is a positive number, or more precisely

it's a non-negative number.

It could be 0 or positive.

So, r, in this definition, is greater than and equal to 0.

OK.

What about theta.

Theta is the angle, and we know that angles

goes from 0 to 2 pi.

So, it is wise to say that actually theta takes values

between 0 and 2 pi.

Because, if we start looking at theta, if we allow theta

greater than 2 pi or less than 0m what will happen is we will

sort of get double billing.

We will get different ways of representing

the same points, right.

Because, we'll be, the same point, for example, will be pi

over 4, but also 2 pi plus pi over 4 or 4 pi plus pi

over 4, and so on.

And, when I was telling you about coordinate system, one of

the important properties that we wanted it to satisfy was

that it gives us a unique address to a given point.

So, if we allow theta to take arbitrary real values, there

will be instantly many ways to represent the same point.

Because, we could all always then shift theta by 2 pi, and

would get the same point.

So, that's why it's better to specify the range of theta

as being from 0 to 2 pi.

If we want to be pedantic, we should also realize that

actually 2 pi is like 0.

And, so once you get to 2 pi, it's like you're back at 0.

So, you don't want to use the same angle twice, even

if it's just one angle.

So, strictly speaking, it has to be from 0, greater and equal

to 0, but less than 2 pi.

So, this is how it's defined, how it should be defined, in

order to get through an ambiguous answer.

And even then, we actually do have a small ambiguity.

I kind of put it in brackets, because it's really kind of

subtle point, which we are not going to dwell on too much.

But, there is a small ambiguity.

There is a small ambiguity.

If r is equal to 0, if r is equal to 0, we are

actually at the origin.

And, so theta becomes meaningless.

We can't really say at what angle our point is with respect

to the origin, because it is an origin.

So, if r is equal to 0.

Unfortunately my board doesn't look so good, so.

My 0 looks like theta now.

That mean 0 really.

If r is 0, then theta is not determined.

So, there's a subtle point that when I promised you that polar

coordinate system gives you a unique address, assigns a

unique address to each point, it's not exactly, it's not

strictly speaking true.

It does assign a unique address to all points

except the origin.

For the origin, the address will be r equals 0 and

theta could be anything.

So, there are too many addresses if you

assigned to that one.

But, because it's only one point, we're not really going

to worry about it too much.

So, I'm just mentioning it to you, so that you are not

so startled by this one.

You realize it is a fact, but it's not going to be a big

problem for us, just because it's only one point, which

is potentially problematic.

Now, there is kind of a more important point here, which

is that -- in fact, we are going to allow allow.

In our calculations, we are going to allow more

values for r and theta.

We will allow, and this is sort of a technical thing.

What I write here is how the polar coordinates are

defined, strictly speaking.

If we really wanted to define it in an ambiguous, say for

this origin, an ambiguous coordinate system.

But, in our calculations, it will be very convenient to have

certain rules, and to allow points with negative r.

So, this is a rule of convenience for you.

The rule of convenience is that if you have negative r, if r is

negative, we will allow negative r with the

following interpretation, negative values of r.

Instead of explaining it in words, Ill just draw a picture,

because it's much easier to see that once than to

hear it 10 times.

And, many of you probably already know that.

Those who have already tried to the homework exercises for this

chapter, for this section.

The rule is like this, if you have a point, like this, r

theta, then its mirror image -- the point which is obtained

from the other side of the line -- will be equal minus r

theta plus pi, right.

So, we will follow this rule, that negative r, so if r is

positive, you have this point.

But, then if the point was negative r, is going to be a

point for which lies of the opposite end of the of this

line, which is symmetric to this point.

Good, I'm glad, so OK, correct me.

What should I write?

[INAUDIBLE].

That's

right, exactly.

So, if r is negative then -- let's say you are given

a point minus 10 and then pi over 4, OK.

So, where would we plot it.

OK.

So, if it was 10 pi over 4, clearly it would be

somewhere here, right.

But, if it is -- so the way I wrote it is not a good way.

So, let's just, let me just give an example using a

particular values of r and theta.

So, this is a representation for r theta if r is positive.

Let's put it this way.

If r is positive, this is a representation, okay.

And, if r is negative, then the representation will be minus

r, theta plus pi, like this.

So, that's what I meant in this picture, but it was

a little bit ambiguous.

So, in particular, if you 10 pi over 4, it would be here.

But, minus 10 pi over 4 will be here.

This is it going to be the point, pi over 4, which will

be the same as 10 and 3 pi, pi plus pi over 4.

OK.

Yeah, when you write it it's not clear when I say minus r do

I mean the original one, or the minus of the minus of

the negative one.

I mean minus of the negative one.

So, it's not the same r as this one.

If r is positive, r theta is just formed like this.

And, if r is negative, let's just do it like this.

If r is negative, and you take, in other words, here

you take the length r.

And, if r is negative.

You take the lengths negative r, which will be now positive,

and you take the angle, which is theta plus pi.

I think it's clear now.

OK.

You have to explain, you have to be careful when

you try to explain it.

We'll see it now in the calculation.

You will see better what I'm talking about.

Alright.

So, let's see what we can do now with this

coordinate system.

Everything all right?

Yes.

You getting tired.

I guess Thursday afternoon.

You all excited about the long weekend.

Yeah.

Well we still have 1/2 hour, but let me take two [LAUGHTER]

but afterwards you go.

So let me, actually let's take a 2 minute break just

to relax a little bit.

I want to tell you something.

It doesn't mean that you should talk to each other.

You still have to listen to me. [LAUGHTER]

It's my time.

So, I want to tell you about an article I read today

in the Forbes magazine.

Not that I read Forbes magazine regularly.

Don't get the wrong idea.

I saw it mentioned on a blog, and the link to this article,

and that's why I read it.

It's available electronically.

And, it's a very interesting article about string theory,

and the controversy about string theory.

And, I'm interested in this, because it's not very far from

my research interest, which involves quantum field theory

and mathematical aspects of quantum field theory.

But, I like this article, because -- remember the first

day when I said that, you know, people think of mathematics and

science as a kind of a something which has been

written in stone and hasn't changed in many years,

which is in fact not true.

And, this article is a very good illustration.

And, I even copied a quote, I quoted a sentence from this

article from the very beginning, which I think

is written very nicely.

So, he writes, the author writes, "Lay people tend

to regard science as a lofty temple inhibited by serene

spoke-like wise men.

Working scientists though will tell you it's more like a stock

market full of fads and fashions, booms and bust."

So, I like this analogy.

Not so much, not so much, I don't like it if you, the

meaning is not say that it's as speculative as

the stock market.

But, I think it's a good illustration of the fact that

it is really a live organism which is constantly changing,

like the stock market.

So, in that sense, I think it's a good analogy, and because

it's Forbes magazine, of course they use that analogy, and

it makes a lot of sense.

Just like the stock market goes up and down, and things change,

likewise in science and mathematics and physics, things

also go up and down, and certain things become

fashionable and then they fall out of fashion.

And, at the end of the day it's the fundamentals, which you

really care about, not the speculation, and not

sort of the, you know, artificial things.

So, I think that it's really cool, and I really like this

analogy with the stock market.

And I suggest, I really recommend this article.

It's very short.

And, it sort of talks about one of the most interesting ideas

of the last, maybe three years in science, the string theory.

So, I'm going to put a link to this article on the bSpace,

on our bSpace page.

And, if you come across any article about math or science,

which you would like to share with other students, please

send me the link and

I'll post it as well.

OK.

All right, now back to boring stuff.

I'm kidding.

It's actually, it's not boring at all.

So, maybe another day we'll take a small break, and I'll

tell you a little bit about what string theory

really is, OK.

But, now let's go back to polar coordinates.

So, let's look at various curves.

And, how to represent those curves using polar coordinates.

We already talked about the circle.

Circle of radius R is represented by the

equation r equals R.

Where, r is one of the two polar coordinates, and

this is the radius.

So, here is another example.

A line passing though the origin with angle theta to the

x-axis also given by a very simple equation, namely now

theta is a constant angle, theta 0 say.

What I mean to say is that this line which forms angle theta 0

to the x-axis is described by the equation theta

equals theta 0.

In the first equation, r equals R, theta is arbitrary.

Well, not exactly arbitrary as we saw it should really

be between 0 and 2 pi.

But, in this equation, r is arbitrary, both

positive and negative.

So, in fact, this is the reason why we introduced this rule,

which may look strange at first sight.

Because, remember, as I said at the beginning, strictly

speaking, r should be non-negative if we follow the

definition polar coordinates.

In which, r is just a distance.

So, it has to be non-negative.

But, if we were to adopt this point of view, then the

equation theta equals theta 0 will actually correspond not

to this entire line, but only 1/2 a line.

Kind of a ray, which goes from the origin to infinity.

Which is fine.

It's a fine geometric object in its own right.

But, if we adopt this rule, if we allow r to be negative, and

we plot points with negative r the way we just discussed, then

not only 1/2 line will be represented in this equation,

which would correspond to positive r, but also the entire

line, the second 1/2 line will correspond to negative

values of r.

So, this is an advantage of this rule, that if we follow

this rule, then we have a nice representation by an equation

of the entire line, like this not just 1/2 line.

OK.

What else can we learn from equations with

polar coordinates.

So, I want to look at a couple of more complicated examples.

I'm sorry?

[INAUDIBLE]

What is?

Arbitrary.

Why is it arbitrary?

The question is why did I say r is arbitrary.

Because, when I write this equation, r is nowhere to be

found in this equation, right.

So, the meaning of this equation is we look at all

points on the plane, whose theta coordinate ix fixed,

if it equals to theta 0.

Theta 0 could be any number between 0 and 2 pi, like pi

over 4, pi over 3, whatever, whatever you want.

Any real number between 0 and 2 pi.

The theta is fixed but r is arbitrary.

When I write this equation, because this equation does not

involve r, this equation means that r can take arbitrary

values within their allowed range.

The allowed range initially was stipulated to be r greater than

0, maybe r greater and equal to 0.

But, eventually we decided that we will allow

negative values of r.

So, that's why -- well actually maybe it's better

to write it like this.

And, on this line, we actually see the part corresponding to r

greater than 0, the part corresponding to r less than 0,

and there is one point which corresponds to r equals 0.

Likewise, when I write this equation, what I'm saying is

that I look at all points on the plane for which the r

coordinate is fixed, it's equal to some number, capital

R, which could be 5, 10, 13, whatever you want.

But, because theta does not appear in the equation, it

means that theta is arbitrary within the allowed

range of theta.

And, what is allowed range.

The Allowed range here is from 0 to 2 pi.

So, that's the meaning of this.

OK.

Now, let's go back to, let's look at more

complicated examples.

So 3, r equals cosine of theta.

So, again a perfectly legitimate equation

involving the two polar coordinates r and theta.

In other words, we're looking at all points on the on the

plane for which, whose r and theta are constrained

by this relation.

It's like writing in the case of Cartesian coordinates

writing an equation x squared plus y squared equals 1.

It's one equation involving our two variables, x and y.

Likewise, this is also one equation involving our two

variables, r and theta.

We have now switched to the polar coordinate system.

So, the question you can be asked is to draw this, too see

what this curve represents.

And, in this case, actually, its easier to understand what

this curve presents by switching back to the

Cartesian coordinates.

This is not to say that this equation is useless.

In fact, on the contrary, you can then use this equation to

compute various things about the curve, as opposed to the

equation and Cartesian coordinates.

But, first we want to visualize them.

What does it mean.

And, here, we have to look back at the dictionary.

At the formulas expressing the polar coordinates in terms of

the Cartesian coordinates.

And, so we could just write r square root of x squared plus y

squared equals cosine theta.

And, then but we know tangent theta, so it's a little bit

fixed, so it's not clear how do we get a good formula.

OK.

So, it doesn't work, so let's look instead at the

geometry of this picture.

So, then let's see what cosine is, cosine theta, what

is it in terms of x

and y.

Well, it's better to say not in terms of x and y, but

in terms of x and r.

Because, this length is x, this length is y,

and this length is r.

So, the cosine theta is the ratio of this

side to this side.

That's why, cosine theta is x divided by r.

Now, let's substitute this in this formula.

So, we get r equals x divided by r.

And, now we can multiply both sides by r.

So, we ge r squared equals x.

OK.

And, now is a good time to express r squared

in terms of x and y.

Because, remember r squared is x squared plus y squared.

So, we substitute here, and so the result is x squared

plus y squared equals x.

And, is already much more manageable.

Let's take x to the left-hand side.

And, let's complete the square.

We can write x squared minus 2 times 1/2, 2 times x

times 1/2 plus 1/2 squared.

So, I introduced an additional term 1/2 squared,

which is one quarter.

On this side and to compensate for this, I also introduced

it on the right-hand side.

So, that I get that through equality.

And, of course I shouldn't forget y squared

as well, you see.

So, I just put tw additional terms on the left- and

right-hand side, which is 1/2 squared.

But, if I do that, then the first three terms

combine into a square.

The first three terms give you x minus 1/2 squared plus y

squared equals 1/2 squared.

And, the end result is already an equation, which is familiar.

The end result is x minus, let me use a different chalk.

So, we get x minus 1/2 squared plus y squared equals

1/2 squared, OK. --

Yes?

[INAUDIBLE]

This was way, this was by way of making

things easier for you.

But, I guess maybe I didn't achieve that goal. 2

times 1/2 is 1, right.

So, that's why, I wanted to write it, you know, I wanted

to use the formula a plus b squared is equal to a squared

plus 2 ab plus b squared.

And, I have x, and I want to have something which is twice,

so that's why I wrote x is 2 times x times 1/2.

Does it make sense?

[INAUDIBLE]

And, then I saw that, so that means x is a and b is 1/2, and

then I put b squared, which is 1/2 squared, which I also

introduced on the other side.

[INAUDIBLE]

Convenient square, that's right.

That's all it is.

So, I'm not cheating, it's all legitimate.

But, yeah, but please check, you know, because sometimes

I can make a mistake.

It works out.

--so, that's the equation we get.

And, now we can draw it, because you see, of course

we're more familiar with the curve x squared plus y

squared is 1/2 squared.

That curve is a circle of course of radius 1/2.

Let me make it bigger.

What's the difference between this equation

and in this equation.

We just shift by 1/2, right.

So, what does it mean geometrically.

Geometrically, it means that we shifted to the right by 1/2.

Why to the right.

Because, for example, here the point with x equals 0 will

correspond to the point with x equals 1/2.

So, that this whole thing will become 0.

So, shifting x by minus 1/2 means shifting everything

to the right by 1/2.

And, when we shift everything to the right by 1/2, it means

the center of the circle, which used to be the origin,

now becomes this point.

Whereas, this point now becomes this point.

And, so it's actually a circle which look like this.

So this is a circle which is represented by this equation.

It's kind of neat, because if you want to write it in

Cartesian coordinates, you get this equation, which is not too

bad, but certainly this looks much more economical in

some sense.

And, so for the purposes of some calculations it

could be very useful.

OK.

What else can we do.

Let's do a small variation on this problem.

So that would be number 4.

A small variation would be r equals cosine of 2 theta.

So, see now we can get away, we can't really get away with

a simple formula for it.

Because, here we used the fact that cosine theta had

such a nice expression.

You can still try it.

Cosine theta is cosine squared of theta minus

sine squared of theta.

So, then you can rewrite cosine squared and sine squared,

but you'll get some very complicated.

So, when you get to a station like this, where you can't

really rewrite it in a nice way by using the Cartesian

coordinate system, you need to try to just kind of understand

qualitatively what this curve looks like.

And, to do that, we should look at this equation as if

r and theta were Cartesian coordinates in some other

world, r and theta are Cartesian coordinates, right.

So, I'll kind of, in a pink world [LAUGHTER], in a pink

world, it just would be like Cartesian coordinates, right.

So, we'll just draw theta and r, and we'll draw this graph in

r and theta, and then we'll try to see what it means in our

world, which is like a yellow world for now, OK.

So, what this is, is just, it's a cosine function except the

period of the cosine function has shrunk to pi

instead of 2 pi.

OK.

Because, already 1 theta is pi, you get cosine of 2 pi, and so

you're back to square one, back to 0.

So, cosine looks like this.

At 0 it's 1, and then it becomes 0 at pi over 2, and

then it becomes negative 1, then it becomes 0 at 3

pi over 2, and then it goes back to 1 at 2 pi.

And, then it continues, like this.

But now, we have two realize that normally if it were just

cosine and function, normally if it were cosine and theta,

this would be pi over 2, OK.

But, because it's 2 theta, it's going to be pi over 4.

And, this will be pi over, normally it would be pi

but now it's pi over 2.

And, this is like 3 pi over 4.

This is pi, and then it continues like this.

So, for example at 5 pi over 4 is going to reach

0 again, and so on.

And, now we would like to plot this on the plane, where r and

theta are polar coordinates.

At least qualitatively, to get a sort of qualitative

understanding what it looks like.

So, we just have to see what the points here correspond

to on this picture.

The first point is a point when theta is 0, and r is 1, right.

So, it's this point in which r is 1, and theta is 0.

Let's plot it on this graph now, on this plane.

So, theta is 0, which means that we are on the x-axis.

All points on the x-axis has theta equals 0, right.

Because, as you remember theta is supposed to be

the angle with an x-axis.

So, if theta is 0, it means you are on the x-axis.

So, you are on the x-axis, and the distance is 1.

So, you start here.

That's your point.

Or, even let's make it even nicer.

Let's start here.

That's our point, that's 1.

What happens next.

What happens next is we are increasing the angle

from 0 to pi over 4.

What is pi over 4.

Pi over 4 is sort of this bisector.

That's pi over 4.

So, when we reach pi over 4, r becomes 0; r becomes 0.

And, r equals 0, no matter what theta is; is as we

discussed the origin.

So, we start here, and then the angles should increase.

Should go from theta equals 0 to theta equals pi over 4.

But, at the same time, r is going to decrease

until it becomes 0.

So, this is going to look like this, roughly.

Because, what else can it look like.

You see, that's the point.

I mean, I used to call it a picture, I'm not insisting

that I know exactly.

See, there is no other way to represent it like this.

If you try to write it in terms of Cartesian coordinates,

you're getting nowhere, You're going to get a very complicated

expression, which is not going to help you.

So, we would can only study it qualitatively, and

that's what it looks like.

And now, we continue.

But, what happens now is that r is a theta, sorry, theta goes

from pi over 4 to pi over 2, but r becomes negative.

That's where our rule become really handy.

Because, if we were, how should I say it, to rigid, and said no

r has to be positive and we don't accept any negative

values, we would say, we would have to say that this

part is not acceptable.

We can only look at the picture in this range,

and this range, and so .

But, not here.

And, what this would mean is that we would not be able to

draw some, represent some very nice pictures by

using polar coordinates.

But, because we've been flexible, and we said we will

accept negative r's, but following a particular rule,

which we discussed, then we'll be able to actually

draw the whole thing.

And, so what this is going to look like is that now

pi is suppose go from pi over 4 to pi over 2.

But, r becomes negative; r becoming negative means, that

we take the absolute value of r, but we shift

the angle by pi.

So, that means that, and the next segment, on this segment,

we look at theta from pi over 4 plus pi to pi over

2 plus pi, right.

So, where is that.

This is pi over 4, pi over 2, pi over 4 plus pi is this,

right, and pi over 2 plus pi is this.

And, is going to go in such a way that r is going

to go to 1 again.

To, well, to negative 1, but we have to take

the absolute value.

So, it's going to be like this.

Now, it becomes positive.

I'm sorry, it still stays negative, and it continues

to 3 pi over 4, and that's like this.

To see this segment is this, this segment is this, this

segment is this, and I think now you can probably

already guess what it's going to look like.

It's going to be like this.

That's right.

Like this, then it goes like this.

Huh?

Not so bad huh. [APPLAUSE]

A good thing to chew on over the long weekend.

So there is one more thing which we need to discuss.

Which is the formula for the surface area for polar curves.

But, it's fairly straightforward.

I'll just write the answer.

Hold on hold on.

We still have three minutes, OK.

So, this is the formula for the area, which is 1/2 r squared d

theta, and that's for a picture which you get by taking a

segment, by taking a sector on the plane, which is bounded by

the curve and by the lines theta equals alpha and

theta equals beta.

The way you derive this formula is exactly the same.

What you need to know is the same as the method which we

used to understand arc length and surface areas for

surface of revolution.

And, the method is to break it into small sectors, and

evaluate the approximate area of which small sector.

And, that's something you can read about in the book.

It's very straightforward.

Now, I want to end with two announcements.

The first announcement is that, as you know, the first homework

assignment is due on Wednesday, because Monday is labor day.

You turn your homework at your section to your GSIs to your

TAs, all right, on Wednesday.

On Wednesday night, after all the sections are over, I will

post the solutions to all the homework problems in this set

on the bSpace page online, OK.

That's the first announcement.

And, the second announcement is about my office hours.

Normally, my office hours are supposed to be in my office in

Evans Hall, but I think we're lucky that there is nobody at

least in the first two lectures.

After the first two lectures, there was nobody here, so we'll

just hold office hours here, in this room, from 5:00

pm to 6:30 pm.

All right.

So, I'll let you go.

Have a good weekend.