Mathematics - Multivariable Calculus - Lecture 10

So, as you know, hopefully, we have our first midterm
exam on Thursday.
That's what it says.
So it's going to be right here, from 3:40 PM to 5:00 PM.
Please arrive early, because there's not enough
space for everyone.
No, I'm kidding.
But, first come, first served.
But, as you see, we are pretty tight on space, and as I told
you, I tried to get a second room, but my request was
denied, unfortunately.
But I think we'll be-- we'll manage, but just-- you realize,
there's so many students, so there are a lot of logistical
problems as it is.
So just to make these problems a little bit easier to
handle, please arrive as early as you can.
Well, obviously you can't come before 3:30 PM, because there's
another class, but if you could arrive shortly after 3:30 PM
and take your seats, that would be great, because then we'll
distribute the exams so we can start exactly at 3:40 PM.
OK?
Now, all the information about the exam is available online,
and I will not go over it now, but I just want to mention
a couple of things.
One is about what's called a cheat sheet.
You will be allowed to have one page of formulas,
standard size, not-- this is standard size.
It has to be handwritten, number one.
It has to be handwritten.
It cannot be typed, it cannot be a photocopy.
It has to be done by you.
That's the point of this.
And it has to be on one side only.
Not on two sides, but on one side only.
If we see that somebody has two sides, or a much bigger sheet
of paper, or something else, we'll have to confiscate it.
I'm sorry, but we have to be fair to everyone.
OK?
So we'll just apply the same rules.
If we see that there is one which does not conform to
the standards, we'll have to take it from you.
OK.
No other material are allowed.
No scratch paper.
If you need scratch paper later on in the exam, you'll ask me
and I'll give you some paper, but it will be just one page of
formulas on one side that you will be allowed to have, and
then I will distribute-- or the TA's will distribute the exams
to you, so you will have the exam on which you will
write your solutions.
OK?
Is that clear?
OK?
No calculators, no books, no nothing.
No other-- no scratch paper.
OK.
Are there any questions about the logistical
aspects of the exam?
Yes?
STUDENT: Are we allowed to put examples on the cheat sheet?
PROFESSOR: You can write anything you want on the
cheat sheet, as long as it's handwritten.
As long as it's written by you.
OK?
Handwritten on one side, and on a standard
sized sheet of paper.
OK.

Any other questions?
STUDENT: Hi.
PROFESSOR: Hi.
STUDENT: Is it going to be curved?
PROFESSOR: What is going to be curved?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: It's more of a terminological
question, I guess.
So it's a question not so much about the exam, about this
midterm exam, but about the grade for the course, right?
Because on
this exam, we're not going to give you a grade, a letter
grade, necessarily.
We'll see.
But the point is that we'll just give you some ranges,
maybe, for the grades, just to give you a ballpark
figure for the grade.
OK?
You'll get the score, so if it would be, say, five or six
problems, so it would be, say, 50 or 60 points maximum, and
you'll know what your score is.
And we'll give you an idea of where you are in,
you know, in the class.
But of course this score will be taken as 20% of your final
score for this course, right?
And so, when we get final score, then at the end of the
semester, we'll look at the final scores for everybody and
we will then derive the grade.
And it will be curved in the sense that there are no
predetermined preset ranges.
The ranges will depend on how everybody's doing, and
how their scores break.
OK?
In that sense, it is curved.

Any other questions about it?
Yes?
STUDENT: Can we use the back of our cheat sheet
as scratch paper?
PROFESSOR: No, if you need scratch paper, just ask me.
Oh, and I should tell you.
At the end of the exam, please put your piece of paper,
the sheet of paper, inside the exam, so we have it.
OK?
I'm sorry?
STUDENT: You mean [INAUDIBLE]?
PROFESSOR: Ah, no.
All right.
Well, you will-- if you absolutely have to have
it back, just write I want to get it back.
I want to keep it for posterity.

All right.
So now, we'll talk about the material for this midterm, so
today is a review lecture, so what I'm going to do is I'm
going to review the material, and then I will-- you
can ask me questions.
So there are various resources for your preparation for
the exam, which are available online.
There is a mock midterm, there are review problems, OK?
And then there are office hours also, which are
organized by your TA's.
And this is another resource, so you can
ask me questions now.
There will also be office hours right after this class, right
here, where you can continue asking me questions
at that time.
But first I want to summarize what we've done so far, and
it's been five weeks-- Yes.
If you really have to ask me now.
STUDENT: When will the review answers be posted online?
PROFESSOR: When?
Tonight.
Tonight the answers for the previous homework set, right?
And for the review problems will be posted tonight
around 6:30, 7:00.
STUDENT: And the mock midterm questions?
PROFESSOR: Mock midterm quiz solutions will not be posted,
but you can ask me about problems from mock midterm
now, for example.
So I want to summarize what we've done, I want to
summarize what we've done in this course up to now.
OK?
Not necessarily in chronological order, but I want
to kind of give you a kind of a bird's eye view of the first
five weeks of this class.
And so the most important point, from my perspective,
to realize here is, what exactly are we doing here?
S we are studying various objects on the
plane and in space.
So in other words, the ambient space-- there is ambient
space-- that's the space where all of those objects live.
OK?
And that could be this-- that's either plane, which we also
denote by R2, or it's a space and we denote by R3.

And in this ambient space, we study objects which
are one-dimensional or two-dimensional.

That's the essence of this course, and we've done quite a
bit in this vein already, so let me summarize
what we've done.
First, I want to talk about the one-dimensional objects.
The one-dimensional objects are also known as curves.
So the first object that we have studied is curves, on
the plane and in space.
So, in other words, curves in R2 and R3.

So what did we talk about?
What did we-- how did we study them?
Well, we figured out that there were two major ways
of representing curves.
One is we can either present them in parametric form, or
in other words, parametric representation.

Because a curve is one-dimensional, we need one
extra parameter-- one extra parameter-- say, t, and then,
we write-- represent the curve in the following way.
We write each coordinate as a function of that variable,
that additional variable t.
So x would be some function f of t, and y would be
some function g of t.
If we work on the plane, in R2, that's all, because we only
have two coordinates, but if we work in space, in
three-dimensional space, then we have a third coordinate, so
I'll put it in brackets, that would be in case we have a
third coordinate, that would be a third function.

So that's parametric representation.

So an example of this would be, say, x is equal to cosine t
and y is equal to sine t.
It's one of the first examples we looked at.
This is-- I'm looking at the two-dimensional case.
So there's no third coordinate, so in this case, this
represents a circle in R2 of radius 1, and you see here that
it's important also to keep track of the range
of the parameter t.
So if you really want to get a circle, you have to specify
that t is between 0 and 2pi, say.
So then it's really a circle, otherwise in principle, you
get-- if you don't specify this, and if you allow this
variable to go beyond this range, it means that you'll
have to wrap around the circle many times.
So when you do parametric representation, keep track
of what the limits are for this variable t.
OK, so that's the first one.
The second way to represent a curve is by means
of an equation.

And I'll just first say Cartesian equation in R2.
In the case when the curve is on the plane, it can be
represented by one equation, with respect to the two
coordinates x and y.
Cartesian refers to the standard coordinate system,
which we draw x and y.
Because of the French mathematician and philosopher
Descartes, we call it Cartesian.
So Cartesian equation in R2 example is a circle-- squared
plus y squared equals 1.
At the same circle as the one we represented over there in
the parametric form can also be represented by this one
equation on the plane.
Note that in R3-- in R3, if we are in three-dimensional space,
we would need two equations.
Two equations, because the dimension of the ambient space
is now three, and so to get to a curve, we have to drop the
dimension by 2, so we have to impose two constraints or two
conditions, two equations.
And for this reason, we don't-- we usually do not use this
form, this presentation, two equations.
But sometimes it happens.
So an example is the intersection of two surfaces.
This is-- in the homework there are a couple of exercises like
that, where you're given two surfaces, and each surface,
each is given by an equation.
Each is given by one equation, and so if you have intersection
of two surfaces, it means that you have to impose these two
equations simultaneously.
When you impose these two equations simultaneously, you
will describe the intersection.
An intersection of two surfaces is usually a curve.
So that's why a curve would be then described
by two equations.
So that's essentially how these kind of pairs of equations show
up-- showed up in our studies so far, because we talked about
intersections of two surfaces, and that intersection
being a curve.
But we'll-- I'll talk about this later, when we talk
about more general sources.
OK, so that's Cartesian.
So usually Cartesian equations are in R2
because there's just one.
In R3, it'll be two equations, and normally we would just
prefer to write parametric form, where you just
need one parameter.
It's really a matter of convenience.
OK.
Now, on the plane, we also have another standard coordinate
system, called polar coordinates, and oftentimes,
curves can be represented nicely by using
polar coordinates.

This is also in R2, on the plane.
So, example, I mean, representing curves in
polar coordinates.
So, an example of such a curve would be r equals cosine theta.

Which would be, as we know, it's going to be a circle
which is centered at 1/2 write as 1/2 --.
So that's an equation, that's an equation in Cartesian-- not
in Cartesian coordinates, but in polar coordinates, so it
represents a curve, but using a different system
of coordinates.
And surely we have much more complicated examples, for
example you could put 1 plus cosine theta and it will look
very different, or you can put cosine 2 theta or cosine 3
theta, and there are-- there were multiple exercise in this
direction, in the homework, and we also discussed
them in class.
OK.
So that's another way of representing curves on a plane.
Now next, we look at the v class of curves, which are the
simplest curves, both on the plane and in space, and
those are the lines.
Simplest curves.
In other words, up to now, we're talking about the
general math that's how to represent curves.
Now I'm talking about particular classes of curves.
And the simplest class of curves is lines.
And lines, we learned how to represent in different ways,
and the standard way is a way-- parametric representation--
which we write like this. r is equal to r0 plus vt.
So, t here is a parameter, and r0 is a position vector of a
particular point on this line, right?

So here's a line, and so that's our 0, position vector of a
particular point on this line, and v is a vector
along this line.

OK?
So these are the two pieces of data that you need to give a
coordinate representation, a parametric representation
for a line.
You need the point, and you need a vector which
goes along this line.
Now, here already we're using vectors.
So this was-- I don't want to separate this as a
separate topic, vectors.
In a way, it's really some technique that we learned to
deal with two-dimensional and three-dimensional space and
objects in two-dimensional and three-dimensional spaces.
It's very convenient, because if you use vectors, you could
add them up, and we saw how, by using these additional vectors,
and also multiplication of vector by a scaler, we can
represent all points along this line in one stroke, so it's by
this formula, our 0 plus vt.
But if you wish, you can write everything in components.
You can write our 0 as some point, as x0, y0, and z0.
Now remember, this is the notation we use for vectors,
these angled brackets, as opposed to the round
brackets for the point.
And it's not just, you know, to be pedantic and to make
everyone's life complicated, but there is, as I've explained
on multiple occasions, there's a big difference between
vectors and points.
For example, points we cannot add up, and
vectors we can add up.
So-- but once you have a point, you can draw a vector from the
origin to this point, and that's the position vector.
So essentially it just means changing the round brackets
by the angle brackets.
And if we write v as abc, then this formula can also be
written in components just like this.
So, the upshot of all this is that, all you need to know to
write down an equation of a line is a point on that line
and a vector along this line.
That's all.
And once you know them, you put them in these very simple
formulas, and that's the parametric representation.
So these are lines.
General curves are more complicated in that these
formulas, which show up on the right-hand sides, are more
complicated than this.
These are the simplest functions and one variable.
Linear functions plus a constant function.
But in general you'll have more complicated expressions, like t
squared or higher powers of t or trigonometric
functions or whatever.

Nevertheless, no matter what your curve is, you can always
approximate this curve by line in the neighborhood
of a given point.
And that's sort of the major-- one of the major ideas that
we've discussed up to now, which is linear approximation.

Linear approximation.
In other words, you can approximate complicated curvy
objects by simple linear ones.
In the neighborhood of a given point, not everywhere
simultaneously, not everywhere at once, but in the
neighborhood of a given point, you can do that.
And what it-- what this amounts to, in the case of curves, is
finding tangent-- tangent vectors and tangent lines.
So for curves, this means for finding tangent lines.
Tangent vectors and tangent lines.
So, general curves.
And I want to emphasize, at a given point.
Because if you change the point, of course the tangent
point is going to be different.
So let's say here, the curve could be something complicated
like this, then this would be the tangent line.
So the curve itself could be given by these equations. x
equals f of t, y equals g of t, and z equals h of t.
And there will be some value t0, which will correspond to
x to point x0, y0, and z0.
That's a point on this curve.
And then you can be asked to write down the equation for
tangent, for the tangent line at this point.
And the way you do this is by taking the derivative of the
vector function, which is obtained by combining
these three functions.
So the tangent vector v would be just x prime of t0, y prime
of t0, and z prime of t0, or in other words, what we
can call r prime of t0.
And once you have this v, then you can write down the equation
of this line, so this is a tangent vector-- this is a
tangent vector-- and a tangent line will then be given by the
equation r is equal to r0, r0 being again, as before, being
the point-- being the position vector of x0, y0, and z0, plus
this v times a parameter.
And this-- at this point, you just-- I want to emphasize that
there is no reason to use the same parameter t, in fact it's
better not to use the same parameter t, because if you use
it, it kind of-- you're kind of implicitly suggesting as though
these were the same parameter as the parameter for the
orgirinal curve, which it is not.
The tangent line and the curve are unrelated, except that just
in the very small neighborhood of this point, they are
very close to each other.
So here, it is much better to emphasize the fact that they
are unrelated, and to use a different parameter.
For example, you could use letter s.
So it's a different, but I want to emphasize, I want
to emphasize that this is a different parameter.
Is that clear?
OK.
Question.
Oh yes.
STUDENT: [INAUDIBLE]
PROFESSOR: Yeah, well, so, so what this means is, this means
the derivative is evaluated at t0.

Right?
That's what I mean.
After that, I take this v, this particular v, and I write
the equation in this form.
STUDENT: [INAUDIBLE]
PROFESSOR: I want to call it s.
You can call it something, anything else, any other letter
you like, but I want to emphasize that it shouldn't be
the same as t, because then it's confusing, because then it
looks like there's some connection between the two
curves, the curve itself and the tangent line.
But there isn't a connection.
Think of this parameter t, say, as a time which-- you know,
a time along the curve.
So let's say there is a bug, which is traveling along this
curve, and that this function, these position functions f, g,
and h, would correspond to the position of that
bug at the time t.
But then there is another bug which is traveling along the
tangent line, and these bugs don't know each other.
The only thing is that they kind of-- they don't
even necessarily come in contact with each other.
Because you see this point, because when one of them was
here, the other one will be somewhere else, because they
just live-- they have different time scales and different time
parameters, you know, different clocks.
So on this curve, this point corresponds to t equals t0.
But upon the tangent line, this point corresponds
to s equals 0.
Because when s is equal to 0, the second term drops out.
The s drops out.
So what you have is just r0.
So the second bug is here when his or her time shows 0.
Whereas the other one is here at the time t0.
So there is no connection between the two.
STUDENT: [INAUDIBLE]
PROFESSOR: The point of tangent line is to have a line which is
the simplest curve, simplest possible curve, approximating
in the best possible way our original curve in the
neighborhood of that point.
But it doesn't mean that, when we parameterize them that the
parameterizations themselves should be related
to each other.
Do you see what I mean?
OK?
Any other questions?
So these are-- these are tangent lines.
And by the way, this includes for example the issue of the
slope, when we talked about parametric curves at the
beginning, when we talked about parametric curves on the plane.
We didn't talk about tangent lines, we talked about the
slope of a tangent line.
But of course a slope of a tangent line can be easily
found in the case of a curve-- in the case of a line on the
plane from these equations.
OK.
So that's more or less the outline of the material that
we learned about curves.
There's one more sort of a subtopic here, which is
applications of curves.
So there are various kinds of quantities we learned how to
compute related to curves.
So there are various integrals, more precisely.
Various integrals.
So you've got arc length, you've got area under the
graph-- under the curve, or-- and area enclosed by some
curves, enclosed area, and also surface, revolution
surface area.

So these are the various integrals that you can set
up, related to curves.
And of course, this is something you need to know,
that there's various formulas involved here.
OK.
But that takes care of the one-dimensional objects on
the plane and in space.
And as you see, I kind of organized all this material in
under one umbrella topic, whereas in fact we studied this
material in a slightly different way, so not
necessarily in exactly the same way.
Because we first talked about planes, then
three-dimensional space.
But I want to emphasize the fact that actually there is not
so much of a difference, that the way curves are on the
plane and in space is actually very similar.
STUDENT: [INAUDIBLE]
PROFESSOR: Revolution surface.
I didn't want to say the word revolution.
Big Brother is watching.
So I want to be careful.
But you know what I mean.

OK.

Now, next is surfaces.
All right, so these are one-dimensional objects, and
next we talk about surfaces.
Surfaces are two-dimensional objects, and
surfaces live in R3.
Because we can't fit-- you can't fit a basketball
in a, on a plane, in a two-dimensional space.
It has to be, it has to live in a three-dimensional
space, dimension one higher.
So what about surfaces?

So surfaces, as far as the presentation of surfaces,
is given by one Cartesian equation.
They're given by one Cartesian equation, so remember, for
curves, they're given by one parameter, parametric form, or
they would have to use, would have to do two equations in R3.
But now, for surfaces, it's the other way around.
We need the one equation, but if we-- if we want to do a
parametric form, we would have to choose two parameters.
So that means that it's more economical to use Cartesian
equations to describe a surface by an equation, as opposed
to using parameters.
So-- but I'll-- so I'll leave this as a little note, we need
two parameters, two parameters now.
And in fact, this is something we'll do later on in this
course, when we talk about various double and
triple integrals.
We will have to parameterize curves.
But for now, we don't really, we don't really
use this method.
We exclusively param-- represent surfaces by
equations, by a single equation, instead of writing
points on a surface by-- in terms of two parameters.
And so, what comes next is different examples, different
examples of surfaces.
And once again, once again the simplest one, the simplest
class consists of linear ones, which are planes.
Simplest class.
So what do-- what do we need to know to represent a plane?
To represent a plane, we need-- we need a point and a normal
vector, as you remember, we saw in one of the most memorable
images from this course so far.
You have a normal vector to this plane, and
you have this point.
And this is sufficient information to describe all
the points on the plane.
So let me draw it here.
So here's a plane, and let's say this is a normal vector.
STUDENT: [INAUDIBLE]
PROFESSOR: You can't see blue?
Oh!
So now I know when I want to leave a subliminal
message on the board.
I know which chalk to use.
All right.
So which do you see?
The white I hope you see, although today the board is
very white, so it's-- I've got yellow and I've got green--
STUDENT: Green.
PROFESSOR: Green?
Yeah, I think we should-- we're all concerned about the
environment, and so let's use green.
Is that better?
OK.
So we exile the blue-- we exile the blue chalk.
No more.
Only when I want to write something that you don't see.
So we have a normal vector, this is a normal vector,
and again we have a point.
We need to choose a point.
In both cases, lines and planes.
Now this n, this n is usually written as abc, but I don't
want to write it as abc, because we've already
used abc for lines.
I just want to emphasize that this is a different
type of vector.
So let me write n equals def.
So then, the equation of the plane is simply d times x minus
x0 plus e times y minus y0 plus f times z minus z0.
OK?
So that's the equation, which actually we derived by
using that product.
In the first place-- but I don't want to repeat this now,
as there is a very simple way to explain why this is
precisely represents all the points on the plane.
So this is an equation on the variables x, y, and z, where
all other quantities are given.
This is def, our coordinates or components of the normal
vector, so they should be given, and likewise x0, y0, and
z0, are the coordinates of a point, they should
also be given.
You could rewrite this by opening the brackets.

So you could rewrite this by opening the brackets.
So what you could do is you can isolate the terms with x, y,
and z, so you would get dx plus ey plus fz equals
something, defg.
Where g is xe-- I'm sorry, dx0 plus ey0 plus fz0.

So this is a number.
This is a particular number.
So that, the reason I'm writing this is sometimes you could be
asked the following question.
You're-- suppose you're given a plane, let's say you have
a plane 4x plus y plus 3z equals 5.

So-- and you can be asked what is a normal vector to this?
What is a normal vector?
So you just look at the equation and you see right away
what the normal vector is, because these are precisely the
coefficients in front of the variables x, y, z.
These are the components of that vector.
So for instance, you could be asked, here is the
equation of a plane.
Write down the equation, a parametric equation for the
line which is normal to this plane and which passes
through a particular point.
So then what should you be thinking?
You should be thinking what do I need to know to write
down an equation of a line?
To write down an equation of a line I need a point and then I
need a vector, the direction vector.
OK.
So the point, let's say, will be given, so then how
do I find out what the direction vector is?
Well the direction-- the line is supposed to be in this
setting, supposed to be perpendicular.
So the direction vector of the line is the normal
vector of the plane.
And now the point is that you can see it right away when
you look at the equation.
It's 4, 413 is the normal vector.
So the equation contains all the information that you need.
That's how you can always-- that's how you should approach
these kind of problems.
To write down the equation of a plane, you need to
know the normal vector.
But conversely, if you already have the equation of the plane,
you can immediately find out what the normal vector is
by just looking at the coefficients.
STUDENT: [INAUDIBLE]
PROFESSOR: What else would you need to do other
than find the parallel?
STUDENT: [INAUDIBLE]
PROFESSOR: Right.
How would you find the line parallel to it?
Well, so to find a line-- just to find a line parallel to a
plane is not a well-defined question, because there are
many parallel lines to a plane passing through a given point.
Let's say, if you have a point somewhere here, and you want to
look, there is a parallel plane to this plane, passing through
this point, so let's say this piece of paper would be--
represent, would be part of that plane.
But inside that plane there are many many lines which you can
always see are parallel to this plane.
So you can't be asked to write down the equation
of a parallel line.
You can be asked to write down the equation of
a parallel plane.
Now, that is actually a good question.
Let's suppose you're asked to write down the equation of a
plane which is parallel to this one, and which passes through--
so a parallel plane passing through the point 1,2,3.
It doesn't matter.

I'm just taking random numbers.
What is the equation of that plane?
Well, since this plane is parallel to the original
one, they share normal vectors, right?
Their normal vectors are the same.
So I might as well use the same normal vectors as for the
original plane, which would be 4, 1, and 3, right?
So these two planes will have to share the left-hand
sides of these equations.
We'll have to have the same left-hand side.
The only thing that could be different is the
right-hand side.
And how do I find the right-hand side?
Well, for that I use the second piece of information.
Second piece of information is that this plane contains
this particular point.
So for this particular point, the right-hand side should be
equal to the expression I get when I substitute the
coordinates of that point.
If I substitute the coordinates of that point, I get 4 times
1 plus 2 plus 3 times 3.
Which is what? 4 plus 2, 6 plus 9, 15-- so the equation is just
4x plus y plus 3z is equal to 15, whereas the original
one was equal to 5.
In fact, so all the planes which have the same left-hand
side but have different right-hand side will represent
planes which are all parallel to each other.
But they will pass through different points.
Only one of them will pass through a particular point, for
example the point 1, 2, 3.
But it's very easy to find the equation of the plane which
passes through that point, by simply substituting the
coordinates of that point.
Yes?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: So, again, finding-- there is not a single
perpendicular plane to a plane.
The condition of being perpendicular would
uniquely define a line.
A line, there is, say, through a given point, there is a
unique-- for a given point on the plane, there is a unique
line, which is perpendicular.
And we just discussed, to find that line you just need to know
the normal vector, but the normal vector, you could
find out right away, right?
From this.
Now, there is another question that you can be asked.
Suppose you intersect two planes, and find out the
equation of the line.
So that's a little bit more tricky.
So here you have to take the cross-product of
the normal vectors.
So there are all kinds of-- Yes?
STUDENT: [INAUDIBLE]
PROFESSOR: If you're given the line?
OK.
So that's very good.
So the-- there's another question also, another
question that can be asked is the relative positions
of line and planes.
So here-- so this particular question I'm asked is about
the relative position of a line and a plane.
So let's suppose you're given the equation of a plane gx
plus-- let's, actually, let's take this one again.
So 4x plus y plus 3z equals 5.
And then suppose you have a line, so it's like 1, let's say
1 plus 2t, and then here will be negative 1 minus
t, z is 2 plus t.
And so these are typical equations for a
line and a plane.
And you can be asked, do this line and plane intersect, or
are they parallel to each other?
Because these are really the only options, right?
If you have a plane and you have some line, the line either
is going to intersect somewhere this plane, or it's going
to be parallel to it.
Or actually, there's one more option, where it actually
may be part of this plane.
How do you find out?
Well, it's very simple.
You simply substitute these parametric equations into this
formula, and you see whether you can find a solution
for t for this equation.
If there are no solutions, it means that they
never intersect.
If there is a solution, if there is a unique solution,
it means they intersect at one point.
And if the equation is satisfied for all values
of t, that means it just belongs to it.
So in this case, we'll have 4 times 1 plus 2t plus y plus--
sorry-- plus negative 1 minus t times 2 plus t equals 5.

So we open the brackets, we get 4 plus 8t minus 1 minus
t plus 6 plus 3t equals 5.
So we get-- did I make a mistake?
STUDENT: [INAUDIBLE]
PROFESSOR: 4 plus 8t?
That's right.
I hope you do better than I do on Thursday. 4 plus 8t.
OK.
Now we-- so this-- for t, we get these three terms,
so that gives us 10.
And 4 minus 1 plus 6 gives us 9.
Equals 5, so 10t is equal to negative 6, so there is a
unique solution, which is negative 6/10.
What did I do?
OK.
It's one of those days.
So minus-- negative 4/10.
But I'm glad that you are paying attention.
That's a good sign.
Negative 2/5.
So what could happen is that it could happen that all
the terms with t disappear.
It could happen that they all cancel each other out, and then
you get a number on the left-hand side and a number on
the right-hand side, and then two things could happen.
They're either the same, in which case it means that the
equation is satisfied for all values of t, which means that
the line belongs to the plane, or it may be saying something
like 5 equals 6, which is wrong, which is false, and
therefore the equation is not satisfied for any value of t.
That means that they're parallel.
Someone was asking, wanted to ask a question.
Yes?
I answered your question?
OK, good.
So-- but as you see, the most likely generic case in a
generic case, there will be some non-zero coefficients in
front of t, and then there will be unique solutions.
So generically, a line and a plane will intersect, but
sometimes it could happen that they cancel out.
All right, so where were we?
I talked about the simplest class of surfaces, namely
planes, and there is questions that could be asked about
planes, but these are not the only surfaces that
we have studied.
We have also studied quadric surfaces, surfaces which are
given by quadratic equations.
So the next, the next example is here, is quadratic surfaces.
And for quadric surfaces, we have a equation like this,
except now we allow second powers.
Squares, x squared, y squared, also mixed
products like xy or yz.
So what you need to know here is that we can break-- that all
of those quadric surfaces break into several major groups, and
what-- how to tell whether a given equation describes a
surface in that group, and what are sort of the qualities
and features of that group?
So you've got here ellipsoids, and you got hyperboloids of two
different types, and you've got paraboloids of two
different types, OK?
So you have to be able to tell by looking at the equation
as to what it represents.
Qualitatively.
You don't have to necessarily to-- you know, you will not be
asked to draw a-- what's it called-- a hyperbolic
paraboloid.
OK, that's not-- we don't-- that's not the goal to have,
to test your drawing skills.
The goal is to see that you realize-- you understand the
difference between the different equations that are
described by the surfaces.
And what are the salient features of different
quadric surfaces.
All right.

By the way, when I talked about quadric surfaces, I did not
mention two important class-- two important groups
of quadric surfaces.
So in addition to the ones before, in addition to the
ones discussed before, we have this-- cylinders.
We have these two classes, one is called cylinders.
And cylinders are surfaces which are described by
equations in which one of the variables is not involved.
So you could have, for example, x squared plus
y squared equals 1.
So that this variable z is not present in the equation.
And it's very easy to represent this.
You just look-- you just draw the corresponding curve on the
plane, which is-- which corresponds to these two
coordinates which are involved-- in this particular
case it's a circle-- and then you take this, you think of
this as a frame, and then you move that frame parallel to the
z axis, and the surface you get by sweeping, you know, which
will be swept by this frame, by this curve, will
be your surface.
So that's-- in this case, that's exactly the cylinder.

So that's why they're called cylinders, even though the
original frame doesn't necessarily have
to be a circle.
It could be a hyperbola, for example, or a parabola,
so you'll get sort of parabolic surfaces.
Think of them as kind of cookie cutters.
You just have a certain shape that you want to cut, and then
you make a cylinder of that shape.
So that's what those cylinders are.
And the second one is-- second class of surfaces, which we
didn't talk about in class but I also want to mention because
they're also important, are the cones.
The simplest example of a cone is given by its equation z
squared equals x squared plus y squared.

And that's given by this picture.
Well, it's sort of self-explanatory, because
it is called a cone and it looks like a cone.
Except it's sort of a double cone.
When you draw it like this, it's a double cone in the sense
that z could be both positive and negative, and there is a
basic symmetry between the upper half of this cone and
the lower half of this cone.
If you flip the sign of z, the equation will not change,
because the square will kill that sign anyway.
So that's why it has these two parts.
So these are the cones.
OK.
So planes and quadric surfaces are the two classes of surfaces
we have studied in greater detail than the more general
surfaces, and for more general surfaces, we have discussed
linear approximation.
So again, I want you to see-- I would like you to know, to see,
and to appreciate this analogy.
For curves, we talk about linear approximation of
curves by tangent lines.
And for surfaces, we talk about linear approximation of general
surfaces by tangent planes.
And just like in the case of-- just like in the case of
tangent lines, we have a very efficient method for writing
down the equation of a tangent plane.
So, tangent planes.
So what you need to know here is that, if you have a graph of
a function and two variables, z equals f of xy, and you have a
point, x0y0, f of x0y0-- which we'll call z0-- then you should
be able to write down the equation of the tangent plane
for the surface at this point.
And the way you do it is actually very similar to the
equation for the tangent lines, we talked about this last time,
so the equation specifically looks like this.

Where we have-- so we have two partial derivatives of the
function f, f sub x and f sub y, evaluated at this point.
This is the equation of a-- this is the tangent plane.
Follow the tangent plane to the graph.
OK?
You can also be asked to write down the equation
of a normal line.
Of a normal line to the graph.
What do I mean by a normal line?
That's the line which passes through the same point and
which is perpendicular to this tangent plane.
Which I-- and we just talked about how to write down
equations of normal lines.
And see, for normal lines, what you need to do, is you need to
keep track of the coefficients in front of x, y, and z.
So what are these coefficients?
Let me actually do it right here.
So you see, in this, this is a nice way to write it, but if I
want to write it in the way we usually write, I have to put
all variables on the same side.
So, instead of writing like this, I would have to write f
sub x, x minus x0, plus f sub y, y minus y0, and then I
would have to take the other guy to this side also.
So that means the coefficient here is negative 1.
So what is the normal vector then?
That's this green vector that we talked about earlier.
It's just f sub x, x0y0, f sub y, fx0y0, and negative 1.

So that's a normal vector.
And now you can write down the equation of a normal line by
using this as a direction vector for that
normal line, right?
What do we need to write down equation of a line?
We need to know the starting point, and we need to know
the direction vector.
Well, the starting point is given, it's x0, y0, and z0,
where z0 is again just the value of the function f x0y0.

And the direction vector is just these numbers. fx of x0y0,
fy of x0y0, and negative 1, and then you have to chose a
coordinate along this time-- along this normal line.
And so let's again use s as before as a second--
as some coordinate.
In this case you could use t, because we haven't
used t in this setup yet.

So this is the equation of a normal line.
Or a parametric representation, more precisely,
of a normal line.
So that's basically, that's most of it, that's like 90%
percent of the material that we've done.
And then, there are what we also did last week.
Last week we started to discuss various aspects of the--
just general aspects of differential calculus.
And the first topic we discussed here were
limits, was limits.
So I kind of have to-- it doesn't quite fit in either
surfaces or lines, so I would have to start-- I would have
to start a different topic, that would be three.
And so that would be elements of the differential calculus.
This is something which we will continue to work on for
the next couple of weeks.
And here we have limits, we talked about limits, different
aspects of limits, and continuity, and we also talked
about partial derivatives, and finally we talked about
the differential.
Although the differential, the notional differential, is of
course, as I explained in great detail last time, it's very
closely related to the notion of the tangent plane.
Tangent plane is a graph of the differential.
So if you understand tangent planes, you understand the
differentials as well.
So that's roughly the summary.
That's the summary of the material we've done so far.
And now we have a few minutes left, so you
can ask me questions.
Any questions?
STUDENT: [INAUDIBLE]
PROFESSOR: Will there be epsilon delta
proofs on the exam?
The answer is no.
There will be no epsilon delta proofs.
But you have to know, as you can see on the mock midterm,
there is a question about showing that the limit
does not exist.
So you should be able to give an argument why
it does not exist.
Yes?
STUDENT: What about proving that the limit does exist?
PROFESSOR: What about proving that the limit does exist?
That would be only in the case when it can be reduced
to one-dimensional case.
And where you could use one of the results from
one-dimensional calculus.
OK.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: On the mock midterm.
Number four.
OK.
So it will take-- so it will take me some time, so maybe I
will-- let me ask let me see if somebody has sort of a
shorter question, and then I will get back to this.
Is that OK?
All right.
So you were asking.
STUDENT: I don't know if mine's any shorter, but could you give
an example of showing where the limit does exist?
Like, showing where [UNINTELLIGIBLE]
PROFESSOR: Oh.
Well, there is one problem-- so the question is what kind of
problems for existence of a limit should we know?
A good example is, a problem on the homework, in that section,
where the limit can be written in polar coordinates.
So the function could be written in terms of x squared
plus y squared, and then it becomes something like r
squared times logarithm of r, so that's exactly the kind
of example I have in mind.
Nothing else.
STUDENT: [INAUDIBLE]
PROFESSOR: No.
No [UNINTELLIGIBLE]
there.
OK?
Go ahead.
Will there be true and false questions?
Possibly.

s represents a parameter of-- s is a parameter.
But it could be-- you could use letter t, for example.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Hyperbolic?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Oh, hyperbolic trigonometric functions.
No, you don't need specifically to know that.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: I'm sorry?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Polar drawings?
Would you-- the question is whether you would have to
sketch polar drawings.
Polar drawings are fair game, because we studied them in
class, right, and there have been lots of them in homework.
Of course, I will not ask you to do something incredibly
complicated, but something very basic like r equals cosine
theta, r equals cosine 2 theta, r equals 1 plus cosine theta,
is something that you should definitely be able to do.
Right?
OK?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Say again?
STUDENT: [INAUDIBLE]
PROFESSOR: Yeah.
Actually, you know what, guys, I don't know.
Maybe it's not a good idea.
Maybe you would rather me talking-- I thought that, you
know, you might be interested in some of the questions your
colleagues have, you know?
Because some of them might open something for you as well.
But we cannot do it if people talk to each other.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Right.
Oh, you mean for limits.
Right, so for the for limits I've already explained that you
could have some-- on the homework you have this function
x squared plus y squared times something like this, maybe not
exactly but something very similar.
Is that what you're asking?
So in this case, you should be able to explain why there is a
limit when x and y go to 0.
And the way you argue is by saying, in terms of polar
coordinates, this is r squared times logarithm of r squared.
And so it effectively-- so this limit when xy goes to
0, 0 means that r goes to 0.
So effectively, answering this equation is equivalent to
answering the same question, but with this function,
one variable.
And therefore, you could use the methods of
one-variable calculus.
And the method of one-variable calculus here is
L'Hopital's Rule.
So you could you do it by L'Hopital's Rule.
This is something you need to know.
In principle, since this kind of problem was on the
homework, it is fair game.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: What exactly is perimeter?

Parameter?
What exactly is a parameter?
In the parametric curves?
So the equation is, what exactly is a parameter
in a parametric curves?

Well, so to answer this, we look at a general curve,
say in R3, in space.
What does it mean to parameterize it?
Well, for me, the easiest way to think about
it is as follows.
So think of this curve as a rope which is somehow
hanging in the air.
Here is a curve.
Right.
It would be complicated like this.
But if it is a rope, you could always just stretch it.
You just stretch it out and make it into a line.
You can always just-- straighten it out, not
stretch it, not stretch it, straighten it out.
That's what I mean.
Straighten it out, make it a straight line.
When you make it into a straight line, you could
introduce measurement-- measurement on it.
You could just pick a point and say, this is point 0, and after
that, you'll have, you know, one inch, two
inches, and so on.
So you could put measurement on it.
But if you could do that to this curve, when you straighten
it out, it means that so you could just mark each point,
each point will have a certain distance from a given point 0,
so that's the parameter t.
Then you think of-- then you put it back where it were, but
now each point has a parameter.
So that's what I mean by a parameter.
Do you see what I mean?
This has got-- this is like parameterizing.
We are not surprised by the fact that we have measuring
instruments, that they are parameterized.
In other words, when we measure things, each point on this
little ruler has a marker there.
But that's because usually we mark things on straight lines.
So here we deal with something which is not straight.
What I'm trying to explain is that there isn't so much of a
difference between the two, because even if you have a
complicated curve, if you can always straighten it out.
And then you can view it, you could apply the same procedure
to it as to a straight line.
And so then you can mark each point by a certain number.
That's this axial coordinate.

Anything else?
STUDENT:
PROFESSOR: That's a kind of a metaphysical question.
Philosophical question.
Are they going to-- are the problems going to be
harder than the homework?
On average, I think they will be about the
same as the homework.
At least I aim for them to be about the same.
But that-- that is something in the eye of the beholder, what
is harder and what is not.
They're not going to be much harder, in any case.
Same kind of problems.
I think that the mock-- the reason why I posted the mock
midterm is precisely to give you an idea of what kind of
problems I would consider putting on the exam.
So I think that gives you an idea.
And speaking of which, let me go back to the question
about number four.
Number four, I haven't forgotten.
Number four or the mock midterm.

I'm very proud of this question, because it combines
so many different things.
It's too bad I cannot use it now on the midterm.
But maybe something similar.

I'm certainly not putting this problem on the midterm
now,after I explain it.

OK.
So let's calm down.
So, do you want me-- do you want to hear the
solution or not?
STUDENT: Yes.
PROFESSOR: OK.
But then, let's just-- let's calm down a little bit.
All right.
So sketch-- you have to first sketch the surface.
So it's-- see, I'm going to do it in real time, because of
course I don't remember.
I don't remember what the surfaces look like, I'm
just trying-- I'm going to just try to guess.
So first of all, I see that there is one plus and two
minuses, and so-- but I don't remember, which one is it?
Is it the one with-- the one with one part or with one part?
So how do I tell?
The way I would tell-- of course, you could-- you will
have-- one way to do it is to write it, all of this,
on your cheat sheet.
But then you-- then you will be using very valuable space for
this, whereas in fact you could very easily deduce this
by just looking at it.
And what I see is, if you take the y squared and the z squared
to the right-hand side, so this is 1 plus y squared
over 4, right?
So what you see is that this is greater than or equal to 1.
All right?
It's greater than or equal to 1.
So that means that if x is between negative 1 and
1, there is no chance that I can solve this.
So that means there is a gap.
That there is a break.
The x equals 0 plane separates two pieces, two
pieces of this graph.
And now I see that this is a hyperboloid with two parts.
So the x equals 0 plane is the yz plane.
It's the yz plane.
So that means this is one like this, which will be exactly the
point 1, and then the second one which is at negative
1, and we'll add this.

So that's the bas-- that's the way I would draw this picture.
Now, well there is this 1 over 4, but just to sketch it,
it doesn't really matter.
So that's the first thing you need to do.
Second, we want to compute the surface area of the part of the
surface bounded by the planes x equals 1 and x equals
square root of 5/2.
So here is x equals 1, and x equals square root of 5/2
is something which is just above 1.
And why did I choose square root of 5/2?
Because then, if I look at this plane, x equals square root of
5/2, I will get an intersect that plane and
this hyperboloid.
I will get a circle, and what is that circle?
So it will be square root of 5/2 squared equals 1 plus y
squared over 4-- well, we can just write it here.
So this will be like 5/4 equals this, which means y squared
over 4 plus z squared over 4 equals 1/4.
So that means y squared plus z squared equals 1.
So that's the circle.
And now, to compute the area, I want to represent this as the
area-- as the surface of revolution.
Right?
So, now revolution of what?
It will be revolution of this curve in the xz plane.
So I just have to draw this curve.
Of course, you have to remember that I have to look at it, not
from this perspective, but from the back of the blackboard, so
that x goes from left to right.
If I look like this, x goes from right to left, but I want
to go from left to right.
So actually, it's going to start at 1, and it's
going to go like this.
So in fact, what I want to do is I want to set, say-- if you
want, it could be, maybe it's better to do an xy plane.
Let's do it in xy plane.
So to do it in xy plane, I have to put z equals
0 in this formula.
So I put z equals 0, and so what I get is x squared minus
y squared over 4 equals 1.
So that's actually a hyperbola.
That's a hyperbola.
It's part of the hyperbola.
So what I'm saying is that, I want to go from 1 to square
root of 5/2, and then the formula for the surface area is
going to be 2 pi y time square root of dxdt squared
plus dydt squared.
So this will be from-- well, I have to take from alpha to
beta, where I would have to write some parameterization for
it, I would have to write some parameterization for this
curve, and as x is some f of t, and y is some g of t, and so t
would be from some alpha to beta, and-- right?
And then I would have to take this integral.
Does everyone agree with this?
OK.
So now, the only thing, I think I'm afraid I'm out of time,
so I'm saved by the bell.
But-- so I feel as though I should let people go, but I
will be happy to continue and explain the rest, after a
five-minute break, while-- during the office hours.
So let's stop here, and good luck on Thursday, so I'll see
you on Thursday at 3:30, the exam starts at 3:40, and
now we have office hours.