MAGDALENA TODA: We have any people
who finished the extra credit and are
willing to give it to me today?
I mean, you don't have to.
That's why it's called extra credit.
But I think it's good for extra practice
and for the extra points.
So hold on to it if you cannot give it to me right now.
And I'll collect it at the end of the class.
Today's a big day.
We are starting a new chapter, Chapter 11.
So practically, we are going to discuss all
through this chapter functions of several variables.
And you are going to ask me, wait a minute,
why do we need functions in more than one variable?
Well, we are all functions of many variables.
I was freezing outside, and I was thinking,
I'm a function of everything I eat.
I'm a function of the temperature outside.
Almost everything in our body is a function
of hundreds of factors, actually, thousands.
But we don't have the time and the precise information
to analyze all the parameters that
affect our physical condition every day.
We are getting there.
I'm going to give you just the simple case.
So instead of y equals f of x type of function, one variable,
we are going to look at functions of the types z
equals f of xy.
Can I have many more?
Absolutely I can.
And that's kind of the idea, that I
can have a function in an-- let me
count-- n plus 1 dimensional space
as being of the type xn plus 1 equals f of x1, x2, x3, x4.
Somebody stop me. xn.
Right.
I have many variables.
And that is a problem that affects everything.
Our physical world is affected by many parameters.
In engineering problems, you've already
seen some of these parameters.
Can you give me some examples of parameters you've
seen in engineering classes?
x1, x2, x3 could be the Euclidean coordinates, right,
for the three [? space. ?] But besides those, there was an x4.
It could be?
Time.
Excellent, [INAUDIBLE].
More than that.
I want more.
I want x5.
Who can think of another parameter
that affects physical processes or chemical reactions?
Yes, sir?
STUDENT: Temperature.
MAGDALENA TODA: Temperature.
Excellent.
Another very good idea.
How about x6?
I'm running out of imagination.
But you have a lot more information than me.
Pressure.
Maybe I'm studying a process of somewhere up in the atmosphere.
Maybe I'm in an airplane, and then it
becomes a little bit more complicated,
because I hate the way cabins are pressurized.
I can feel very uneasy.
My ears pop and so on.
We can be in the bottom of the ocean.
There are very many physical parameters
that affect physical processes, chemical processes,
biological processes.
I don't know if this is fortunate or unfortunate,
but I think that was the key to the existence
of the universe in the first place-- all these parameters.
OK.
Let me give you a simple example of a function that
looks like a graph.
This is a graph.
And you say, wait a minute, wait a minute.
Can I have functions of several variables that cannot be
represented as graphs?
Yeah.
Absolutely.
We will talk about that a little bit later.
So if I were to give you an example that you've
seen before, and I would say, give me a good approximation
to a valley that is actually a quadric that we love and we
studied before for the first time.
That quadric is a beautiful object, a valley.
Any imagination, recognition, recollection?
I know I scared you enough for you
to know the equations of those quadrics since some of you
told me we watched all the videos,
we read all the stinking book like never before.
That was kind of the idea.
I didn't want to scare you away.
I wanted to scare you enough to read the book
and watch the videos.
And I'm talking about a valley that you've seen before.
Many of you told me you like the University of Minnesota
website that has the quadric gallery of quadrics.
So you've met this guy before.
They show the general equation.
But I said I like the circular paraboloid.
So they talk about elliptic paraboloid.
Which one do you think I prefer?
The circular paraboloid.
Give me an example of a circular paraboloid.
STUDENT: A flashlight?
Inside.
MAGDALENA TODA: The expression, the mathematical equation.
STUDENT: Oh, sorry.
So it would be x squred plus y squared.
MAGDALENA TODA: Very good.
That's exactly what I had in mind.
Of course, it could be over something, over r.
All right.
That's my favorite.
Now, if I put the flashlight in here just like one of you
said, or the sign on top of the z-axis.
Then I'm going to look at the various-- we discussed
that a little bit before.
So various horizontal planes, they're going to cut.
They're going to cut the surface in different circles,
upon different circles.
We love them, and we use them.
And what did we do with them last time?
We projected them on the floor.
And by floor, I mean the what?
By floor, I mean the xy plane.
Plus this xy plane.
I label it like you like it.
You said you like it when I label it,
so you have the imagination of a table.
This is x and y and z.
And so I gave you an example of a graph cut in with z equals
constant positive or negative?
Well, it better be positive, because for negative, I
have no solutions.
Positive or zero.
Well, for zero I have a degenerate conic.
A degenerate conic could be a point,
or it could be a bunch of lines.
In this case, all those circles-- doo-doo-doo-doo-doo--
a family of one parameter, family of circles.
Like the ones that is-- a dolphin
is now doing that in San Antonio,
San Diego-- to take those old circles
from the bottom of the sea, and bring them different sizes,
and put them together.
So they are very smart.
I love dolphins.
So we'll see 0 [INAUDIBLE] get a point.
That's still a conic.
It's a degenerate circle.
Do you realize that's a limit case?
It's really beautiful.
You know what I mean?
Circle on top of a circle on top of a circle,
smaller and smaller.
All right.
So good.
If I create shadows-- because that's
why you guys wanted the source of light
on top-- of the projections of these circles,
I'm going to have them at the same color.
But dotted lines because I think the book doesn't show them
dotted.
But on my way here, I was thinking,
I think it's more beautiful if I draw them dotted.
And how big is this circle?
Well, god knows.
I'm going to make a purple circle that is, of course,
equal in size, equal in radius with the original purple
circle.
So the dotted purple circle, that's on the ground--
is just the projection of the continuous purple circle.
It's identical in radius.
So for the family of circles on the surface,
I have a family of projections on the ground in the xy plane.
And such a family of projections represents
a bunch of level curves.
We call this family of level curves.
OK?
All right.
So if you think about it, what are level curves?
You view them as being in plane.
Oh, my god.
So I should view them as a bunch of points, a set of points.
If I make it like that, that means
I view this as an element of what?
Element of the xy plane, right, with the property
that f of x and y is a constant.
OK?
In my case, I have a [INAUDIBLE] constant.
In general, I have an arbitrary real constant.
That's a level curve for level C, for the level
C called the level, or altitude would be the same thing.
So have you seen these guys in geography?
What in the world are these level curves in geography?
STUDENT: [INAUDIBLE] show the slope
of a-- the steepness of a hill.
MAGDALENA TODA: You've seen topographical maps.
And I'm going to try and draw one of them.
I don't know, guys, how-- excuse me.
I'm not very good today at drawing.
But I'll do my best.
It could be a temperature map or pressure map.
[INAUDIBLE] or whatever.
Now I'll say, this is going to go-- well,
I cannot draw the infinite family.
I have a one-parameter family.
And then I'll-- I'm dreaming of the sea, summer break already.
You see what I'm doing.
Do you know what I'm doing?
That means I'm dreaming of the different depths of the sea.
So for every such broad line, I have the same depth.
The same altitude for every continuous rule.
The same depth for every-- so OK.
I'm not going to swim too far, because that's
where the sharks are.
And I cannot draw the sharks, but I ask you to imagine them.
It's fundamental in a calculus class.
So somewhere here I'm going to have--
what's the deepest-- guys, what's
the deepest point in that?
[? STUDENT: 11,300. ?]
MAGDALENA TODA: And do you know the name?
I know the--
STUDENT: Mariana Trench.
MAGDALENA TODA: Mariana Trench.
STUDENT: Trench.
MAGDALENA TODA: All right.
So these topographical are full of curves.
These are level curves.
So you didn't know, but there is a lot
of mathematics in geography.
And there is a lot of mathematics
in-- oh, you knew it.
When you watch the weather report,
that's all mathematics, right?
It shows you the distribution of temperatures everyday.
That is what we can [INAUDIBLE] also
care about other functions of several parameters, right?
And those functions could be pressure, wind, whatever.
OK.
Speed of the wind.
Something like that.
I did not dare to look at the prediction
of the weather for this place.
This place used to be a beautiful place.
300 days of the year of sunshine.
Not anymore.
So there is something fishy in Denmark
and also something fishy in [INAUDIBLE].
The world is changing.
So if you don't believe in global warming, think again,
and global cooling, think again.
All right.
So unfortunately, I am afraid still
to look at the temperatures for the next few days.
But--
STUDENT: It's going to be 80 degrees on Tuesday.
MAGDALENA TODA: Really? [?
Well, see, I should have looked at it. ?]
[LAUGHTER]
I should gather the courage, because I
knew-- when I was interviewed here
for assistant professor, gosh, I was young.
2001.
And my interview was in mid-February.
And birds were chirping, it was blue skies, beautiful flowers
everywhere on campus.
And I love the campus.
OK.
Give me an example of a surface that cannot be represented
as a graph in its entirety as a whole graph.
You gave me that before, and I was so proud of you.
It was a--
[LAUGHS]
What kind of surface am I trying to mimic?
STUDENT: A saddle.
MAGDALENA TODA: That can be actually a graph.
That's a good example of a graph.
A saddle.
But give me an example of a non-graph that
is given as an implicit form.
So graph or explicit is the same thing.
z equals f of xy.
Give me a non-graph.
One of you said it.
x squared plus y squared plus z squared equals 1.
Why is this not a graph?
Not a graph.
Why is this not a graph?
STUDENT: [INAUDIBLE].
When you move it over to 1, you can't actually--
MAGDALENA TODA: You cannot but you can cut it.
You can take a sword and-- I'm OK.
I don't want to think about it.
So z is going to be two graphs.
So I can split this surface even in a parametric form
as two different graphs.
Different graphs.
If I cut along-- I have this orange, or sphere, globe.
And I cut it along a great circle.
It doesn't have to be the equator.
But you have to imagine something
like the world and the equator.
This is kind of in the unit sphere.
Today I drank enough coffee to draw better.
Why don't I draw better?
I have no idea.
So that's not bad, though.
OK.
So that's the unit sphere.
What does it mean?
It means it has radius how much?
STUDENT: 1.
MAGDALENA TODA: 1.
Radius 1, and we are happy about it.
And it has two graphs.
It's not one graph, it's two graphs.
So this is called implicit equation.
This is your lab from-- I was chatting
with-- instead of studying last night,
I was chatting with you at midnight.
And one of you said, if I had something I hated in calculus,
it was the implicit differentiation.
And I know this is your weak point.
So we'll do a lot of implicit differentiation,
so you become more comfortable.
Usually we have one exercise in this differentiation at least
on the final.
So this is an implicit equation.
And z is going to be two graphs-- 1 minus x
squared minus y squared.
So I have, like, two charts, two different charts.
OK.
The upper hemisphere-- I'm talking geography,
but that's how we talk in geometry as well.
So geography right now is like geometry.
I have a north pole.
Somebody quickly give me the coordinates of the north pole.
STUDENT: 0, 0, 1.
MAGDALENA TODA: 0, 0, 1.
Thank you, Brian.
0, 0, 1.
How about the south pole?
STUDENT: 0, 0, minus 1.
MAGDALENA TODA: 0, 0, minus 1.
And write yourself a note, because as you know,
I'm very absent-minded and I forget
what I eat for lunch and so on.
Remind me to talk to you sometime
at the end of the chapter about stereographic projection.
It's a very important mathematical notion
that also has to do a little bit with geography.
But it's a one-to-one correspondence
between a certain part of a sphere
and a certain huge part of a plane.
Now, we're not going to talk about that now,
because that's not [INAUDIBLE].
That's a little bit harder [INAUDIBLE].
You guys should now see this line, right?
This should be beyond-- in the twilight zone,
behind the sphere.
OK?
So you don't see it.
And who is this? z equals 0.
And so this green fellow should be
the circle x squared plus y squared equals 1
in the xy plane.
Good.
So I have two graphs.
Now, if I were to ask you, what is the domain
and the range of the function?
I'm going to erase the whole thing.
What is the domain and the range of the related function, z,
which gives the upper hemisphere?
Upper hemisphere.
It's a graph.
And square root of 1 minus x squared minus y squared.
You may stare at it until tomorrow.
It's not hard to figure out what I mean by domain
and range of such a function.
You are familiar with domain and range
for a function of one variable.
For most of you, that's a piece of cake.
That was even pre-calc wasn't it?
It was in Calc 1.
So most of you had algebra and pre-calc.
Now, what is the domain of such a function?
Domain of definition has to be a set of points, x and y in plane
for which the function is defined.
If the function is impossible to be defined
for a certain pair, x, y, you kick that couple out
and you say, never come back.
Right?
So what I mean by domain is those couples that we hate.
Who we hate?
The couples x, y for which x squared plus y squared is how?
What existence condition do I--
STUDENT: [INAUDIBLE].
MAGDALENA TODA: Yeah.
You see this guy under the square root
has to be positive or 0.
Right?
Otherwise, there is no square root in real numbers.
That's going to be in imaginary numbers,
and you can take a walk, because we
are in real calculus in real time as well.
So x squared plus y squared must be how?
Less than or equal to 1.
We call that a certain name.
This is called a closed unit disk.
Please remember, I'm teaching you a little bit more
than a regular Calc 3 class.
They will never make a distinction.
What's closing with this?
What's opening with this?
Everything will come into place when you
move on to advanced calculus.
If I don't take the boundary-- so everything inside the disk
except for the boundary, I have to put strictly less than 1.
That's called open unit disk.
For advanced calculus, this is [INAUDIBLE].
All right.
This is just a parentheses.
My domain is the closed one.
What is the range?
The range is going to be--
STUDENT: [INAUDIBLE].
MAGDALENA TODA: The altitude starts having values from--
STUDENT: Negative 1 to 1.
STUDENT: 0 to 1.
MAGDALENA TODA: So I'm 0 to 1.
I'll only talk about the upper hemisphere.
I should even erase, because I don't want it.
So say it again, guys.
STUDENT: 0 to 1.
MAGDALENA TODA: 0.
Open or closed?
STUDENT: Open.
STUDENT: Closed.
STUDENT: Closed, closed.
MAGDALENA TODA: Closed to?
STUDENT: 1 closed.
MAGDALENA TODA: 1 closed.
Yes.
Because that is the north pole.
I've been meaning to give you this example.
And give me the other example for the lower hemisphere.
What's different?
The same domain?
STUDENT: It ranges from--
STUDENT: Negative 1.
STUDENT: Negative 1 to 0.
MAGDALENA TODA: Closed internal, right?
When we include the endpoints, we call that closed interval.
It has a certain topological sense.
You haven't taken topology, but very soon,
if you are a math major, or you are a double major, or some
of you even-- they want to learn more about topology,
you will learn what an open set is versus a closed set.
Remember we called this closed.
This is open.
And if it's closed here and open there, it's neither.
OK?
Don't say anything about that.
OK.
To be closed, it has to be containing both endpoints.
I'm going to erase this.
And this was, of course, 11.1.
We are in the middle of it.
In 11.1, one of you gave me a beautiful graph to think about.
And I'm going to give you something to do,
because I don't want you to get lazy.
I'm very happy you came up with the saddle.
All right.
We drew such a saddle.
And I did my best, but it's not hard.
It's not easy to draw saddle.
When I am looking at the coordinates, x, y, z,
I have z equals minus y squared will look down.
Maybe I made it too fat.
I'm really sorry.
And down.
This continues.
OK?
And then what other thing did I want to point out?
I want to point out-- do you see this?
This should look a little bit more round.
It doesn't look round enough here.
STUDENT: Your'e drawing a saddle, right?
MAGDALENA TODA: No, I'm drawing just the section
z equals minus y squared.
So I took x to be 0.
And the purple line should be on this wall.
I know you guys have enough imagination.
So this is going to be z equals minus y
squared drawn on yz wall.
I've done this before, but I'm just reviewing.
What if it's y0?
Then I have to draw on that wall.
And I have to draw beautifully, which I am not-- don't always--
I can't always do.
But I'll try.
I have z equals x squared drawn on that wall.
If I start drawing, I'll get fired.
That I have this branch.
I should go through that corner and go out of the room
and continue with that branch.
All right?
This is curved like that in this direction.
And this other is curved like this.
So if the guy is going to put his feet,
where is the butt of the writer going to sit?
He is here.
And these are his legs.
And these are his cowboy boots.
OK.
Do they look like cowboy boots?
No, I apologize.
STUDENT: Looks like socks.
MAGDALENA TODA: Yeah.
They look more like Christmas socks.
But anyway, it's a poor cowboy.
Let's lower the saddle a little bit.
He cannot see the horse, OK?
So the saddle.
If I cross the saddle, this is the saddle.
And these are his hands.
And he is holding his hat.
This is [INAUDIBLE].
And with one hand is on the horse.
I don't know.
It's very [INAUDIBLE].
So what I'm trying to draw looks something like this.
Right?
Eh.
Sorry.
More or less.
It's an abstract picture.
Very abstract picture.
So with this in mind, if I were to look at the level curves,
I'm going to ask you, what are the level curves?
Oh, my god, what are the level curves?
You already have them in your WeBWorK homework.
But for one point extra credit, I
want you to draw them on the floor.
Draw the level curves.
Remember what those were?
They were projections of the curves on the surface
at the intersection with z equals c planes.
You project them on the ground.
What do you think they are?
Think about it.
What are these?
If I take c, what if c is positive?
What if c is 0?
What if c is less than 0?
What am I going to have?
Your imagination gives you c equals 1, Magdalena.
Let's draw that.
OK.
Well, I'll try.
a and b would be 1, right, guys?
So a and b would be 1.
This is a square.
These would be the asymptotes.
So very, very briefly, the hyperbola
would be this one-- x squared minus y squared equals 1,
right?
If I have the last case for c equals 1,
I'm going to have-- c equals negative 1-- I'm
going to have the conjugate.
Are you guys with me?
So I'll have an a squared, asymptotes, conjugate.
What if I have different level c? c equals 1/2. c equals 2.
c equals pi. c equals-- what are they?
I'm going to get families of hyperbolas,
trenches that look like that.
Standard trenches and conjugate trenches.
A multitude of them, an infinite family of such hyperbolas,
an infinite family of such hyperbolas.
I wanted to draw it.
What do I get when c is 0?
What are those?
STUDENT: Don't you get, like, [INAUDIBLE]?
MAGDALENA TODA: They get-- very good.
Why?
x squared minus y squared equals 0 would lead
me to y equals plus/minus 1.
And who are those y equals plus/minus 1?
Exactly.
But exactly the first bisector, which is y equals x.
They are [? then the ?] function.
And the other one, y equals negative [? x. ?] So these
are the asymptotes.
So I'm going to get a-- you guys have to do this better than me.
Sorry.
These are all hyperbolic trenches.
They are all going to infinity like that.
And I'm sorry that I'm giving you
a little bit too many hints.
This is part of your homework, your WeBWorK.
I shouldn't talk too much about it.
Any questions so far?
Is this hard?
Yes, sir?
No.
STUDENT: So [? spherically, ?] if you had z
equals y squared minus x squared,
it's that same picture, just flipped?
MAGDALENA TODA: What would it be?
It would be the poor saddle-- or cowboy--
STUDENT: Would be upside down.
MAGDALENA TODA: --would be upside down.
Or projected in something like a mirror.
I don't know how to say.
It would be exactly upside down.
So the reflection of that.
So you take all the points.
If you have-- I don't know.
It's hard to draw a reflection in three dimensions.
But--
STUDENT: No, I understand.
MAGDALENA TODA: Practically every curve
would be upside down with respect to the floor.
OK.
All right.
I'm going to erase in one.
And you say, well, you've taught us about these things,
like the domain and range.
But what about other notions, like continuity and stuff?
Let me move on to 11.2.
Limits of functions of the type z equals f of xy.
So what do you remember about the limit
of a function of one variable?
Comparison.
What about the limit if you take [? z's, ?] I don't know.
I should look stunned.
And I should be stunned.
Of a function of y equals f of x of one variable.
When do we say that f has a limit at a?
STUDENT: When the [INAUDIBLE] approaches from the right
and the left to the same value.
MAGDALENA TODA: Actually, that was the simpler definition.
Let's think a little bit deeper.
We say that f has a limit L at x equals a.
That's kind of the idea, left and right limits.
But not both of them have to exist, you see.
Maybe only the limit from the left or limit
from the right only exists.
If, for any choice of values of x, closer and closer, closer
and closer to a, we get that F gets closer and closer to L.
And this "any" I put in.
My god, I put it in a red circle thing,
because one could get subsequencies of a sequence.
And for that subsequence thing, things
look like I would have a limit.
And then you say, well, but in the end,
I don't have a limit, because I can get another subsequence
of the sequence.
And for that one, I'm not going to have a limit.
Can you give me an example of some crazy function that
does not have a limit at 0?
Example of a crazy function.
No.
No, don't write "crazy."
Of a function f of x that is not defined at 0
and does not have limit at 0, although it
is defined for values arbitrarily close to 0.
Moreover, I want that function to be drawn without-- I
want the function to be drawn without leaving
the paper when I draw.
[INAUDIBLE]
So something that would be defined on the whole 0
infinity except for 0 that I can draw continuously
except when I get to 0, I get some really bad behavior.
I don't have a limit for that function.
You are close to that.
Sine of 1/x.
STUDENT: I said y equals 1/x.
MAGDALENA TODA: y equals 1/x.
Very good.
Let's see.
STUDENT: Oh, yeah. [INAUDIBLE].
MAGDALENA TODA: Yeah, yeah.
Both are excellent examples.
So let's see.
This guy is a very nice function.
How do we draw him, or her?
Well, it's a her, right?
It's a she.
It's a function.
No, no.
In English, it doesn't make any sense,
but if I think French, Italian, Spanish, Romanian-- now
I speak both Italian and Romanian--
we say it's a she, it's a feminine.
So as I approach with values closer and closer and closer
to 0, what happens to my poor function?
It blows up.
OK.
So I have limit of 1/x from the right and from the left.
If I take it from the left, I don't care.
Let's take it only from the right.
OK?
It's close to 0.
That's going to blow up, right?
And I restrict it.
So let's say, if I want the domain to be containing
[? both, ?] that's also fine.
So if you guys want, we can draw the other one.
This goes to paradise.
The other one, I'm not going to say where it goes.
But it's the same idea, that as you approach 0
with closer and closer and closer values,
it's going to blow up.
It's going to explode.
This is a beautiful function.
How beautiful [INAUDIBLE].
Beautiful with a bad behavior near 0.
So I'm not going to have a limit.
No limit.
Some people say, limit exists and is infinity.
But does infinity exist?
Well, this is a really philosophical,
religious notion, so I don't want to get into it.
But in mathematics, we consider that unless the limit is
finite, you cannot have a limit.
So if the limit is plus/minus infinity, there is no limit.
Could the limit be different or different subsequences?
This is what I wanted to point out.
If you try this guy, you are in real trouble on that guy.
Why?
You can have two.
If you have a graphing calculator, which
I'm going to be opposed to you being used in the classroom,
you would probably see what happens.
Sine is defined on all the real numbers.
But you cannot have a value at 0,
because the 1/x is not defined at 0.
Imagine you get closer and closer to 0 from both sides.
I cannot draw very beautifully.
But as 1, this is plus 1 and this is minus 1.
I'm going to have some behavior.
And how many of you have seen that on a computer screen
or calculator?
You've seen.
Yeah, you've seen.
By the way, did you see the Lubbuck High?
Was it in high school you saw it the first time in Calc 1
or pre-calc?
STUDENT: [INAUDIBLE] Algebra 1 with Mr. West.
[INAUDIBLE]
MAGDALENA TODA: So I'll try-- oh, guys, you
have to be patient with me.
I'm not leaving the poor board with the tip of my pencil.
I'm not leaving him.
I have continuity.
As I got closer to this, I still have the [INAUDIBLE] property.
Anyway, it's OK.
I'm not leaving this.
I am taking all the values possible between minus 1 and 1.
So on intervals that are smaller, smaller,
I'm really taking all the values between minus 1 and 1,
and really rapidly-- [INAUDIBLE].
When I'm getting closer to 0, I'm not going to have a limit.
But as somebody may say, but wait.
When I have a sequence of values that
is getting closer and closer to 0,
is that no guarantee that I'm going to have a limit?
Nope.
It depends.
If you say "any," it has to be for any choice of points,
any choice of points that you go closer to 0.
Not for one sequence of points that
is getting closer and closer to 0.
For example, if your choice of points is this,
choice of points.
Getting closer to 0.
[INAUDIBLE] xn equals 1 over 2 pi n.
Isn't this going to 0?
Yeah.
It then goes to infinity.
This sequence goes to 0.
What is it?
1 over 2 pi?
1 over 4 pi?
1 over 8 pi?
1 over 16 pi?
1 over 32 pi?
1 over 64 pi?
This is what my son is doing to me.
And I say, please stop.
OK?
He's 10 years old.
He's so funny.
Now, another choice of points.
Ah.
Somebody-- all of you are smart enough to do this.
What do you think I'm going to pick?
1 over what?
And when [? other ?] something that goes to 0
then goes to infinity.
And I know that your professor showed you that.
pi over 2 plus 2 pi n.
Doesn't this go to 0?
Yes.
As n gets bigger and bigger, this is going to 0.
However, there is no limit.
Why?
Well, for the first sequence, as xn goes to 0, f of xn
goes to-- what is sine of-- OK, I
am too lazy to write this down.
Sine of 1 over 1 over-- of 1 over 1 over 2 pi?
STUDENT: It's the sine over 2 pi.
MAGDALENA TODA: This is sine of 2 pi n.
And how much is that?
STUDENT: 0.
MAGDALENA TODA: 0.
So this is a 0.
And this is a-- this converges to 0.
So I say, oh, so maybe I have a limit, and that'll be 0.
Wrong.
That would be the rapid, stupid conclusion.
If somebody jumps [? up, ?] I picked some points,
I formed the sequence that gets closer and closer to 0.
I'm sure that the limit exists.
I've got a 0.
Well, did you think of any possible choice?
That's the problem.
You have to have any possible choice.
F of yn sine of 1 over pi over 2 plus 1 over 1
over-- Magdalena-- pi over 2 plus 2 pi n.
So we saw that this was 0.
What happens to sine of 1 over 1 over sine of pi
over 2 plus 2 pi n?
And where does this go?
It then goes to infinity.
This sequence goes to 0.
What is f of the sequence going to?
To another limit.
So there is no limit.
What's the limit of this subsequence?
It's a constant one, right?
Because look, what does it mean pi over 2 plus 2 pi n?
Where am I on the unit trigonometric circle?
[INTERPOSING VOICES]
Always here, right?
Always on the sort of like the north pole.
So what is the sine of this north pole?
STUDENT: 1.
MAGDALENA TODA: Always 1.
So I get the limit 1.
So I'm done because there are two different limits.
So pay attention to this type of problem.
Somebody can get you in trouble with this kind of thing.
On the other hand, I'm asking you,
what if I want to make this a function of two variables?
So I'll say, one point extra credit.
I'm giving you too much extra credit.
Maybe I give you too much-- it's OK.
One point extra credit-- put them together.
Does f-- do you like to do the f?
I used big F, and then I changed it to little f.
This time I have a function of two variables-- little
f with xy-- to be sine of 1 over x squared plus y squared.
Does this function have a limit at the point 0, 0?
So when I approach 0, 0, do I have a limit?
OK.
And you say, well, it depends how I approach that 0, 0.
That's exactly the thing.
Yes, sir.
Oh, you didn't want to ask me.
And does f of xy equals-- let me give you
another one, a really sexy one. x
squared plus y squared times sine of 1
over x squared plus y squared.
Have a limit at 0, 0?
I don't know.
Continuous it cannot be, because it's not defined there.
Right?
For a function to be continuous at a point,
the function has to satisfy three conditions.
The function has to be defined there at that point.
The function has to have a limit there
at that point of the domain.
And the limit and the function value have to coincide.
Three conditions.
We will talk about continuity later.
Hint.
Magdalena, too many hints.
This should remind you of somebody
from the first variable calculus.
It's a more challenging problem.
That's why I gave it to extra credit.
If I had x sine of 1/x, what would that look like?
STUDENT: x times--
MAGDALENA TODA: x times sine of 1/x.
When I approach 0 with-- so if I have-- I
don't ask for an answer now.
You go home, you think about it.
You take the calculator.
But keep in mind that your calculator can fool you.
Sometimes it can show an image that misguides you.
So you have to think how to do that.
How about x times sine of 1/x when--
does it have a limit when x goes to 0?
Is there such a limit?
Does it exist?
So if such a limit would exist, maybe we
can extend by continuity the function x times sine over x.
What does it mean?
Like, extend it, prolong it.
And say, it's this 4x equals 0 and this if x is not 0.
So this is obviously x is different from 0, right?
Can we extend it by continuity?
Think about the drawing.
Think about the arguments.
And I think it's time for me to keep the promise I made
to [? Aaron, ?] because I see no way.
Oh, my god, [? Aaron, ?] I see no way out.
The epsilon delta definition of limit.
[? Right? ?] OK.
So what does it mean for a real mathematician or somebody
with a strong mathematical foundation and education
that they know the true definition
of a limit of a function of, let's say, one variable?
The epsilon delta, the one your dad told you about. [INAUDIBLE]
try to fool you when avoid it in undergraduate education.
People try to avoid the epsilon delta,
because they think the students will never, never understand
it, because it's such an abstract one.
I think I wasn't ready.
I wasn't smart enough.
I think I was 16 when I was getting ready for some math
competitions.
And one professor taught me the epsilon delta and said,
do you understand it?
My 16-year-old mind said, no.
But guess what?
Some other people smarter than me,
they told me, when you first see it,
you don't understand it in any case.
So it takes a little bit more time to sink in.
So the same idea.
As I'm getting closer and closer and closer and closer
to an x0 with my x values from anywhere around-- left,
right-- I have to pick an arbitrary choice of points
going towards x0, I have to be sure that at the same time,
the corresponding sequence of values is going to L,
I can express that in epsilon delta.
So we say that.
f of x has limit L at x equals x0 if.
For every epsilon positive, any choice of an epsilon positive,
there is a delta.
There exists-- oh, OK, guys.
You don't know the symbols.
I'll write it in English.
For every epsilon positive, no matter
how small-- put parentheses, because you
are just [? tired-- ?] no matter how small,
there exists a delta number that depends on epsilon.
So that whenever x minus x0 is less than delta,
this would imply that f of x minus L,
that limit I taught you about in absolute value,
is less than epsilon.
What does this mean?
I'm going to try and draw something
that happens on a line.
So this is x0.
And these are my values of x.
They can come from anywhere.
And this is f of x.
And this is L. So it says, no matter-- this
says-- this is an abstract way of saying,
no matter how close, you see, for every epsilon positive,
no matter how close you get to the L.
I decide to be in this interval, very tiny epsilon.
L minus epsilon.
L plus epsilon L. You give me your favorite epsilon.
You say, Magdalena, pick something really small.
Big epsilon to be 0.00001.
How about that?
Well, if I really have a limit there,
an L at x0, that means that no matter how much you shrink
this interval for me, you can be mean and shrink it
as much as you want.
I will still find a small interval around x0.
[? But ?] I will still find the smaller
interval around x0, which is-- this would be x0 minus delta.
This would be x0 plus delta.
So that the image of this purple interval fits inside.
You say, what?
So that the image of this purple interval fits inside.
So f of x minus L, the distance is still that, less than xy.
Yes, sir?
STUDENT: Where'd you get epsilon [INAUDIBLE]?
MAGDALENA TODA: So epsilon has to be
chose no matter how small.
STUDENT: [INAUDIBLE].
MAGDALENA TODA: Huh?
Real number.
So I'm saying, you should not set the epsilon to be 0.0001.
That would be a mistake.
You have to think of that number as being as small as you want,
infinitesimally small, smaller than any particle in physics
that you are aware about.
And this is what I had the problem understanding--
that notion of-- not the notion of, hey, not
matter how close I am, I can still
get something even smaller around x0 that fits in this.
That's not what I had the problem with.
The notion is to perceive an infinitesimal.
Our mind is too limited to understand infinity.
It's like trying to understand God.
And the same limitation comes with microscopic problems.
Yeah, we can see some things on the microscope,
and we understand.
Ah, I understand I have this bacteria.
This is staph.
Oh, my god.
But then there are molecules, atoms, subatomic particles
that we don't understand, because our mind is really
[? small. ?] Imagine something smaller
than the subatomic particles.
That's the abstract notion of infinitesimally small.
So I'm saying, if I really have a limit L there,
that means no matter how small I have this ball around it,
I can still find a smaller ball that
fits-- whose image fits inside.
All right?
The same kind of definition-- I will try to generalize this.
Can you guys help me generalize this limit notion
to the notion of function of two variables?
So we say, that f of xy has the limit L at x0y0.
What was x0y0 when I talked about-- what
example did I give you guys?
Sine of 1 over x squared plus y squared, right?
Something like that.
I don't know.
I said, think of 0, 0.
That was the given point.
It has to be a fixed couple.
So you think of the origin, 0, 0, as being as a fixed couple.
Or you think of the point 1, 0 as being as a fixed couple
in that plane you look at.
That is the fixed couple.
If-- now somebody has to help me.
For every epsilon positive, no matter how small,
that's where I have a problem imagining infinitesimally
small.
There exists-- I no longer have this problem.
But I had it enough when I was in my 20s.
I don't want to go back to my 20s and have-- I mean,
I would love to.
[LAUGHTER]
To go having vacations with no worries and so on.
But I wouldn't like to go back to my 20s
and have to relearn all the mathematics.
Now way.
That was too much of a struggle.
There exists a delta positive that depends on epsilon.
What does it mean, depends on epsilon?
Because guys, imagine you make this epsilon
smaller and smaller.
You have to make delta smaller and smaller,
so that you can fit that little ball in the big ball.
OK?
That depends on epsilon, so that whenever-- now,
that is a big problem.
How do I say, distance between the point xy and the point
x0y0?
Oh, my god.
This is distance between xy and x0y0 is less than delta.
This would imply that-- well, this
is a function with values in R. This is in R. Real number.
So I don't have a problem.
I can use absolute value here.
Absolute value of f of the couple xy minus L
is less than epsilon.
The thing is, can you visualize that little ball,
that little disk?
What do I mean?
Being close, xy is me, right?
But I'm moving.
I'm the moving point.
I'm dancing around.
And [? Nateesh ?] is x0y0.
How do I say that I have to be close enough to him?
I cannot touch him.
That's against the rules.
That's considered [INAUDIBLE] harassment.
But I can come as close as I want.
So I say, the distance between me--
I'm xy-- and [? Nateesh, ?] who is
fixed x0y0, has to be smaller than that small delta.
How do I represent that in plane mathematics?
STUDENT: Doesn't [INAUDIBLE]?
MAGDALENA TODA: Exactly.
So that delta has to be small enough so
that the image of f at me minus the limit is less than epsilon.
Now you understand why all the other teachers avoid
talking about this [? one. ?] So I
want to get small enough-- not too close-- but close enough
to him, so that my value-- I'm f of xy-- minus the limit,
the limit-- I have a preset limit.
All around [? Nateesh, ?] I can have different values,
no matter where I go.
My value at all these points around [? Nateesh ?] have
to be close enough to L. So I say,
well, you have to get close enough to L.
Somebody presents me an epsilon.
Then I have to reduce my distance to [? Nateesh ?]
depending to that epsilon.
Because otherwise, the image doesn't fit.
It's a little bit tricky.
STUDENT: So is this like the squeeze theorem kind of?
MAGDALENA TODA: It is the squeeze theorem.
STUDENT: Oh, all right.
MAGDALENA TODA: OK?
So the squeezing-- I ball into another [? ball ?] [? limit. ?]
This is why-- it's not a ball, but it's a--
STUDENT: A circle.
MAGDALENA TODA: Disk.
A circle, right?
So how do we express that in Calc 3 in plain?
This is the [? ingredient, ?] distance d.
So Seth, can you tell me what is the distance between these two
points?
Square root of--
STUDENT: [INAUDIBLE].
MAGDALENA TODA: x minus x0 squared plus y minus y0
squared.
Now shut up. [? And I ?] am talking to myself.
STUDENT: Must be less than delta.
[LAUGHTER]
MAGDALENA TODA: Less than delta.
So instead of writing this, I need
to write that I can do that in my mind.
OK?
All right.
This is hard.
We need to sleep on that.
I have one or two more problems that are less hard-- nah,
they are still hard, but they are more intuitive,
that I would like to ask you about the limit.
I'm going to give you a function.
And we would have to visualize as I get closer to a point
where I am actually going.
So I have this nasty function, f of xy
equals xy over z squared plus y squared.
And I'm saying, [INAUDIBLE] the point is the origin.
I choose the origin.
Question.
Do I have a limit that's-- do I have a limit?
Not [? really ?] for me.
Does f have a limit at the origin?
You would have to imagine that you'd draw this function.
And except you cannot draw, and you really don't care to draw
it.
You only have to imagine that you have some abstract graph--
z equals f of xy.
You don't care what it looks like.
But then you take points on the floor,
just like I did the exercise with [? Nateesh ?] before.
And you get closer and closer to the origin.
But no attention-- no matter what path I take,
I have to get the same limit.
What?
No matter what path I take towards [? Nateesh-- ?]
don't write that down-- towards [? z0y0, ?] I have to get
the same limit.
Do I?
Let's imagine with the eyes of your imaginations.
And [? Nateesh ?] is the point 0, 0.
And you are aspiring to get closer and closer to him
without touching him.
Because otherwise, he's going to sue you.
So what do we have here?
We have different paths?
How can I get closer?
Either on this path or maybe on this path.
Or maybe on this path.
Or maybe, if I had something to drink last night-- which
I did not, because after the age of 35,
I stopped drinking completely.
That's when I decided I want to be a mom,
and I didn't want to make a bad example.
So no matter what path you take, you can make it wiggly,
you can make it any way you want.
We are still approaching 0, 0.
You still have to get the same limit.
Oh, that's tricky, because it's also the same in life.
Depending on the path you take in life,
you have different results, different limits.
Now, what if I take the path number one, number two, number
three possibility.
And number [? blooie ?] is the drunken variant.
That is hard to implement in an exercise.
Imagine that I have limit along the path one.
Path one.
xy goes to 0, 0 of xy over x squared plus y squared.
Do you guys see what's going to happen?
So I'm along the-- OK, here it is.
This line, right, this is the x-axis, y-axis, z-axis.
What's special for the x-axis?
Who is 0?
STUDENT: x.
STUDENT: yz.
MAGDALENA TODA: y is 0.
So y is 0.
Don't laugh at me.
I'm going to write like that because it's easier.
And it's going to be something like limit
when x approaches 0 of x over x squared.
STUDENT: It's 1/x.
MAGDALENA TODA: Times 0 up.
Oh, my god.
Is that-- how much is that?
STUDENT: 0.
MAGDALENA TODA: 0!
I'm happy.
I say, maybe I have the limit.
I have the limit 0.
No, never rush in life.
Check.
Experiment any other paths.
And it's actually very easy to see where I can go wrong.
If I take the path number two, I will get the same result.
You don't need a lot of imagination
to realize, hey, whether she does it for x
or does it for y, if she goes along the 2, what
the heck is going to happen?
y is going to shrink.
x will always be 0.
Because this means a point's like what?
0,1.
0, 1/2.
0, 1/n, and so on.
But plug them all in here, I get 0, 1/n times 0.
It's still 0.
So I still get 0.
Path two.
When I approach my-- xt goes to 0, 0.
The poor [? Nateesh ?] is waiting for an answer.
I still get 0.
Let's take not the drunken path, because I
don't know [? it unless ?] the sine function.
That is really crazy.
I'll take this one.
What is this one, in your opinion?
Is that going to help me?
I don't know, but I need some intuition.
Mathematicians need intuition and a lot of patience.
So what is your intuition?
The one in the middle, I'm going to start walking on that, OK,
until you tell me what it is.
STUDENT: y [INAUDIBLE].
MAGDALENA TODA: y equals x is the first bisector
or the first quadrant.
And I'm very happy I can go both ways.
y equals x.
x [INAUDIBLE].
So limit when x equals y, but the pair xy goes to 0,0.
I'm silly.
I can say that, well, Magdalena, this
is the pair xx, because x equals what?
Let me plug them in.
So it's like two people.
x and y are married.
They are a couple, a pair.
They look identical.
Sometimes it happens.
Like twins, they start looking alike,
dressing alike, and so on.
The x and the y have to receive the same letter.
And you have to tell me what in the world the limit will be.
STUDENT: 1/2.
MAGDALENA TODA: 1/2.
Oh, my god.
So now I'm deflated.
So now I realize that taking two different paths,
I show that I have-- on this path, I have 1/2.
On this path, I have 0.
I don't match.
I don't have an overall limit.
So the answer is, no overall limit.
Oh, my god.
So what you need to do, guys, is read
section 11.1 and section 11.2.
And I will ask you next time-- and you can lie,
you can do whatever.
Did the book explain better than me,
or I explain better than the book?
This type of example when the limit does not exist.
We are going to see more examples.
You are going to see examples where the limit does exist.
Now, one last thing.
When you have to compute limits of compositions of functions
whose limit exist-- for example, you
know that limit is xy goes to x0y0 of f
of xy [INAUDIBLE] limit of xy go to x0y0 of gxy
is L-- L-- L-- M-- M.
How are you going to compute the limit of alpha f plus beta g?
This is in the book.
But you don't need the book to understand that.
You will already give me the answer,
because this is the equivalent thing to the function of one
variable thing in Calc 1.
So if you would only have f of x or g of x,
it would be piece of cake.
What would you say?
STUDENT: [INAUDIBLE].
MAGDALENA TODA: Right.
Alpha times L plus beta times M. Can you also
multiply functions.
Yes, you can.
Limit of fg as xy goes to x0 or y0-- will be LM.
How about-- now I'm going to jump to conclusion, hoping
that you are going to catch me.
You are going to catch me, and shout at me,
and say, ooh, pay attention, Magdalena,
you can make a mistake there.
I say it's L/M when I do the division rule, right?
Where should I pay attention?
STUDENT: M [INAUDIBLE].
MAGDALENA TODA: Pay attention.
Sometimes you can have the-- right?
And this also has to exist as well.
STUDENT: [INAUDIBLE].
MAGDALENA TODA: So one last-- how many minutes
have I spent with you?
I've spent with you a long number of hours of my life.
No, I'm just kidding.
So you have one hour and 15, a little bit more.
Do I have a little bit more?
Yes.
I have 15 minutes.
I have--
STUDENT: So we get out at--
[INTERPOSING VOICES]
MAGDALENA TODA: 50.
Five more minutes.
OK.
So I want to ask you what you remember about some
of your friends, the trig functions involved in limits.
Why did we study limits at the point
where the function's not defined?
Well, to heck with it.
We don't care.
The function is not defined at 0.
But the limit is.
And nobody showed you how to do the epsilon delta
to show anything like that.
OK.
Can you do that with epsilon delta?
Actually, you can do everything with epsilon delta.
But I'm not going to give you any extra credit.
So I trust you that you remember that.
1!
How about-- let me-- OK.
I am so proud of you.
Let me challenge you more.
Tangent of ax over bx.
x go to 0.
I asked this to a girl from Lubbock High.
She was in high school.
She knew the answer.
STUDENT: Oh, I can't disappoint everybody in getting this.
STUDENT: Is it 1/a?
Oh, I can't remember.
MAGDALENA TODA: Tell me what to do to be smart.
Right?
I have to be doing something smart.
She-- can you give me hint?
I'm your student and you say, well--
STUDENT: ba--
STUDENT: It's 0.
STUDENT: It's [INAUDIBLE].
MAGDALENA TODA: Um, it's a what?
STUDENT: b/a?
MAGDALENA TODA: I'm not [INAUDIBLE].
I don't think so.
So what should I do?
I should say, instead of bx-- that drives me nuts.
This goes on my nerves-- bx.
Like, maybe I go on your nerves. bx is ax, right?
If it were ax, I would be more constructive,
and I knew what to do.
I say replace bx with ax, compensate for it,
and divide by bx.
And I was trying to explain that to my son,
that if you have a fraction a/b, and then you write a/n
times n/b, it's the same thing.
Gosh, I had the problem with him.
And then I realized that he didn't do simplifications
in school.
So it took a little more hours to explain these things.
This is fourth grade.
I think I remember doing that in fourth grade.
Third grade, actually.
So these two guys disappear.
I haven't changed my problem at all.
But I've changed the status, the shape of my problem
to something I can mold, because this goes to somebody,
and this goes to somebody else.
Who is this fellow?
It's a limit that's a constant-- a/b.
Who is this fellow?
STUDENT: 1.
MAGDALENA TODA: 1.
Because tangent of x/x as x goes to 0 goes to 1 exactly
like that.
So limit of sine x over cosine x, that's tangent, right?
Over x.
You do it exactly the same.
It's limit of sine x/x times 1 over cosine x.
That's how we did it in high school.
This goes to 1.
So it's 1.
So thank you, this is 1.
I know I took a little more time to explain than I wanted to.
But now you are grown up.
In two minutes, you are going to be finishing
this section, more or less.
What if I put a function of two variables,
and I ask you what the limit will be,
if it's the same type of function.
So you say, oh, Magdalena, what you doing to us?
OK, we'll see it's fun.
This one's fun.
It's not like the one before.
This one is pretty beautiful.
It's nice to you.
It exists.
xy goes to 0, 0.
So you have to imagine some preferable function
in abstract thinking.
And you want it in a little disk here.
And xy, these are all points xy close enough to 0, 0,
in the neighborhood of 0, 0.
OK.
What's going to happen as you get closer and closer
and closer and closer with tinier and tinier and tinier
disks around 0, 0?
You're going to shrink so much.
What do you think this will going to be,
and how do I prove it?
STUDENT: [INAUDIBLE].
MAGDALENA TODA: Who said it?
You, sir? [INAUDIBLE] going to go to 1.
And he's right.
He has the intuition.
A mathematician will tell you, prove it.
STUDENT: Um, well, let's see here.
MAGDALENA TODA: Can you prove?
STUDENT: You could use the right triangle proof,
but that would probably take way more [INAUDIBLE].
MAGDALENA TODA: x and y are independent.
That's the problem.
They are married, but they are still independent.
It's a couple.
However, we can use polar coordinates.
Why is polar coordinates?
Well, in general, if we are in xy, it's a pair.
This is r, right?
So rx is r cosine theta.
y is r sine theta.
And I can get closer and closer to the original.
I don't care.
What happens about x squared plus y squared,
this is r squared.
And r is a real number.
And as you walk closer and closer to the original
without touching it, that r goes to 0.
It shrinks to 0.
So that r squared goes to 0 but never touches 0.
So this becomes limit as r goes to 0, the radius of that disk
goes to 0.
Sine of r squared over r squared.
But r squared could be replaced by the real function, t,
by the real parameter, lambda, by whatever you want.
So then it's 1.
And then Alexander was right.
He based it on, like, observation, intuition,
everything you want.
It was not a proof.
On a multiple-choice exam, he would be a lucky guy.
I don't want you to prove it.
But if I want you to prove it, you have to say,
Magdalena, I know polar coordinates,
and so I can do it.
And one last question for today.
Guys, I'm asking you, limit xy goes to 0, 0.
You will see some of these in your WeBWorK for Chapter 11
that's waiting for you, homework 3.
Tangent of 2 x squared plus y squared over 3
x squared plus y squared.
What is that?
2/3.
STUDENT: 2/3.
MAGDALENA TODA: Am I asking you why?
No, enough.
OK.
[INAUDIBLE] I gave you everything
you need to show that.
x squared plus y squared, again, is Mr. r squared.
It's OK.
I taught you that. a/b. a is 2, b is 3.
Is it hard?
It is not easy, for sure.
Calc 3 is really difficult compared to other topics
you are probably taking.
But I hope that I can convince you
that math, although difficult, [INAUDIBLE] Calc 3,
is also fun.
OK?
All right.
So I need attendance and I need the extra credit.
STUDENT: Yeah, [INAUDIBLE].
MAGDALENA TODA: Before you go, you need to sign.