Mathematics - Multivariable Calculus - Lecture 3

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.
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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
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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.