Mathematics - Multivariable Calculus - Lecture 7

OK, good to see you all.
And you guys, Thursday again, it's one day to go.
I'll try to make it quick, OK.
Or I will try to make it feel like it's quick, not
to bore you too much.
All right, we're actually getting to more and
more interesting stuff.
You can probably tell.
Last time at the end of the lecture I talked about
general curves in space.
And I took its prototype, the simplest curves,
those are lines.
And lines we found convenient to represent
in parametric form.
Which means that you write each coordinate as a function of an
auxiliary variable, which we usually call t, but not
necessarily t, it could be any other letter of your choice.
By default we call t and then each of the three
variables is written as a function of this t.
I'm hearing more people today then I'm used to.
You've spoiled me in the previous lectures.
You've been so quiet, let's keep it this way.
So these are arbitrary functions in, t a priori.
And in the book you can look at various examples.
There are some pretty cool curves that you can
parameterize this way.
For example, the spiral, which goes like this.
And many other ones, I'm not going to go over these examples
because they're fairly easy to you just read about
in the book.
What I want to emphasize is the fact that this is a natural
generalization of the formulas we had for lines, for which
these functions had special form, they
were linear functions.
Functions of degree 1 in the variable t.
I can wait.
Just knock him.
Yeah, thanks.
So that's what we have.
Now I want to look at it in a slightly different way and I
want to think of this as not just a way to parameterize a
curve, but as a function with values and vectors.
Because the point is that when you have three coordinates,
when you have a x,y,z in general you can view them as
coordinates of a point in space, x y, x.
But you could also draw a vector from the
origin to this point.
That's called a position vector of this point.
So there's always these two ways of thinking about the
triple numbers, x,y,z.

This is a point and this, which we also write as xi plus yj
plus zk is the position vector.
And even though we realize that these are two different objects
we use essentially the same information to
represent this object.
And so oftentimes it's convenient to interpret
a triple of numbers in two different ways.
As a point or as a vector.
So if we interpret it as a point, we just think of this
point as traversing this curve in three space.
At each moment t, which actually you can think of as
time if you wish, it will be in a certain position,
and as time goes by it will trace this curve.
But you can also view these three functions as
components of a vector.
So we get a vector-valued function, which we
can denote as r of t.
So that's a vector in which the first component will be f of t,
the second component will be g of t and the third
component will be h of t.
In other words, if this point lies on this curve you can
think of this point, p, moving along the curve so that its
three coordinates are changing as functions of t, f, g, and h.
But you can also think of the vector itself moving so that
the endpoint goes along this curve.
And that's what this function, r of t, represents.
We are more used to thinking of vectors as static objects.
We would usually talk about a vector, so there's
a particular vector.
But nothing prevents us from allowing the vector to
change with time, say.
Or with some variable t.
And then what we get is not a single vector, but we get a
vector for each value of t.
Each value of t gives rise to a particular vector.
Which we simply denote as r of t.
So the convenience of this is that if we think about it in
this way, we can use various operations on vectors
and apply them to this vector-valued functions.
For example, we can add two vector-valued functions.
If we have r1 of t and r2 of t we can take the sum.
So that would just mean taking the sum of the components.
So for example, you'll have f1 of t plus f2 of t and
the same for g and h.
I just want to same time, I'm not going to
write it in detail.
You get the idea.
If you were thinking in terms of points you wouldn't
be able to do that.
It wouldn't make sense because we can not add up points.
We can not add one point to another.
But we can add vectors.
We can use parallelogram rule or triangle rule,
whatever you like.
There is a rule for adding vectors. vectors
can be added up.
So that's why when we convert this into a vector-valued
function, we can use this operation, and sometimes
it's very convenient.
What is perhaps even more important, is that we can
approach this as functions, and apply other operations, which
we normally apply to functions which are valued in numbers
like differentiation and integration.
These kinds of functions can be differentiated and integrated
just like normal functions.
And what it means is simply applying this operation to
each of the three components.
So it's really straightforward.
You don't really have to think about it too much.
You just differentiate the first one, differentiate
the second one and you differentiate the third one.
And this is something which actually has important
applications because derivative of this function will give you
information about tangent lines and tangent vectors
to your curve.
So derivative, that would be r prime of t.

We use the same notation as for normal functions, but I
shouldn't forget this sign, which emphasizes that
it is a vector.
So that would be just taking the first derivative of f, the
first derivative of g and first derivative of h.

So this derivative has a meaning.

I want to emphasize that I want to evaluate it at a particular
point, which I'll call t 0.
There is always this issue with notation.
When I write something like r of t there's the question,
what do I mean by this?
Do I mean a particular value for a particular t or do I mean
all of these values for all values of t as a function of t?
And usually it is clear from the context, but now I
want to emphasize that I take a specific value.
That's why I call it t 0.
I don't want to view this now as a function, I want to view
this as a value of that function, the derivative, at
this particular value, t0.
For example t0 could be equal to 0 like in the example which
we had calculated at the very end of last lecture.
So that would mean that we are evaluating the derivative
at the point t0.
So if we do that this is a specific vector.
This is a specific vector corresponding to the value
of t, which I call t0.
A specific value, particular number.

So numeric value.
And the upshot of all this is that this is a tangent vector.
Find tangent vector to the curve described by these
equations at that point, at a specific point, which I'll
call x0, y 0, and z 0.
Which is f of t0, g of t 0.
And last time at the very end of last lecture we looked at
an example of how this works.

So once you pick a value, t 0, you have a particular
point on your curve.
Let me go back to this picture.
So let's say this is a point, p, so that would be the
point x0, y0 and z 0.
That's the point which corresponds to the
values of t equal t0.
And then for other values it's somewhere along this red curve.
So if you want to calculate the tangent vector to this curve at
this point that's going to look something like this.
So this is precisely this r prime of t0.
Why does derivative give you a tangent vector?

That should be clear from what we've discussed before.
What is the derivative?
The derivative by definition is obtained as follows.
We take the value-- so derivative at t0 is the value
at t 0 plus some small number.
Let's call it h minus r at t 0 divided by h.
So let's say this is a point when t is equal to t0.

And let's say this is a point on the curve.
That's the same red curve, so maybe I should
use the same color.
So there's the point t equal t0.
Our original point, which I call p on that picture.
And this is a point, p prime, which corresponds
to some value t 0 plus h.
But remember, we also have the origin somewhere, so we
have a coordinate system.
I'm magnifying, I blow up a small part of this picture.
So there's a coordinate system somewhere in the background and
there's the origin, the point o, right?
So r of t0 plus h is going to be this vector.

And r of t 0 is going to be this vector.
The vector I have already drawn on this picture.
This one.
And I'm using the standard formula for the derivative.
The derivative is the increment divided by the change
in the parameter.
The increment in the values when you change the parameter
by h divided by the parameter.
So it's like velocity.
You calculate the velocity by taking the difference between
positions and dividing by the time.
It's the same idea.
It's the same formula as for normal functions.
We apply exactly the same formula for vector functions.
So what do we get as a result?
We have to subtract this vector from this vector and we know
how to do this by triangle rule.
It's just the vector connecting the two.
So the point is that the difference between these
two vectors is just the vector connecting them.
Because y is the difference.
Because if you take this plus this you get this.
So that means that if you take this minus this you get this.
So the green vector is the difference, is the numerator.
And I divided by h.
When I divide by h I just shrink it.
Well it depends if I shrink.
If h is very small then actually I multiply
by some large number.
So this is done to rescale it because if I don't do this,
this difference is actually going to disappear.
Because as the point goes closer and closer and closer
this is going to become smaller and smaller and smaller.
And actually what I want to do is I want to take
the limit of this.
What I wrote was an approximation and to make it
more precise I have to say it's a limit when h goes
to 0 of this ratio.
So if I didn't divide by h I wouldn't get
anything meaningful.
I would definitely get 0.
Zero vector because this green thing will shrink
to 0 if h goes to 0.
Because this point is going to be the same. so these
two vectors are going to be the same. r minus r is 0.
So what I do is I rescale by h.
So I adjust for the fact that actually I'm getting
closer and closer.
And when I do that it doesn't necessarily go to 0 anymore.
So in fact it is going to stay finite and as the point gets
closer and closer and closer I'm actually going to get
something, which will be tangent.
Which is what I drew here in this picture.
So that's the reason.
That's the reason why it's a tangent vector.
It's a tangent vector in the following sense.
There are many tangent vectors because if I have one I can
always multiply by something.
By any number.
I can multiply by 5, by 10, by 1 million.
It's still going to be a tangent vector because when I
multiply a vector by something I don't change its direction.
I only change the magnitude.
So if the direction is the same, if it was a tangent
vector to begin with it will remain a tangent vector when
I multiply by something.
So this is slightly inconvenient because you might
get one vector using a particular parameterization of
the curve and someone else may use a different
parameterization and then get a different vector.
They're not are going to be that different because both
will be tangent, so they will both have the same direction.
But one of them could be like this and another
one longer or shorter.
So it oftentimes is a good idea to agree on normalization, so
no matter which result we get we might say, OK, let's find a
vector which has a unique length.
So then what we need to do and that's a vector which is called
t0 is obtained by dividing this tangent vector.
The rho tangent vector that we get by taking derivative and
dividing by its length, by its magnitude.
Because when you divide a vector by its magnitude,
you're going to get something of magnitude 0.
Sorry, magnitude 1.
I was jumping ahead of myself, because what I was going to say
next is that this only makes sense is this is nonzero.

Of course, only assuming that r prime of t 0 is nonzero.
It's possible that this vector is 0 to begin with.
So then we can't normalize it, it is 0.
But if it is nonzero we can divide by its magnitude
and get a unit vector.
So this is called a unit tangent vector.
It's still not exactly unique because there
are two of them now.
It could be this one or it could be this one.
But the ambiguity is much less now.
There are two possibilities, so it could be very
useful sometimes.
And at the end of last lecture I used this derivative, this
tangent vector to write down a parametric equation for the
tangent line to this point.
Parametric equations for the tangent line to this point,
to the curve at this point.
And these are very easy to write because we know what the
initial point is, it's just x0, y0 and z 0 and all we need is
the components of a tangent vector.
We have found a tangent vector by taking the derivatives.
That's the top line on this board, so that's going to be f
prime of t 0 and now I have to choose an auxiliary
parameter for the line.
It's better to chose a different parameter then for
the curve because we don't want to indicate that they
kind of go hand in hand.
They don't.
The tangent line we are talking about is this tangent line on
which this tangent vector lies.
It only has something to do with this curve at
this particular point.
A priori, has nothing to do with this curve at this point
or this point or that point.
So there's no reason to call the parameter for this line
also t because t was used as the parameter for
the original curve.
It's better to separate the two things.
We have so many different letters that we can use, so why
not use a different letter?
The first thing that comes to mind is letter s.
Just the next one in the alphabet, so that's why we use
it, but you can use whatever you want as long as it's in the
latin alphabet, I guess.
As long as we can recognize what it is and can recognize
that it's the same one.
So then you get these equations.
And I explained how this works in the particular
example last time.
Are there any questions about this?
Yes?
STUDENT: [INAUDIBLE]
PROFESSOR: Large T is the unit tangent vector.
It's the vector which is obtained by taking the
derivative vector and dividing by magnitude.
Provided that the vector is nonzero.
To be absolutely precise I should say like this.
Does it make sense?
Anything else?
Yes?
STUDENT: [INAUDIBLE]
PROFESSOR: Not really because see I subtract the position.
So the position has no connection to the derivatve.
It's really this vector, but I think what you're saying is, so
the question was about whether this vector has anything to
do with the position vector.
The reason why you might think so is because I draw it
starting from this point, but that's just a matter of
convenience because I would like to say that this
vector is tangent to this curve at this point.
So it's better to draw it here to indicate that.
But it doesn't mean that somehow I have to add the
position vector to this.
This is a vector of itself, right?
I'm choosing to take the initial point for this
vector the point p.
Whereas, I choose as the initial point for the
position vector the origin.
And so as I explained before there's no reason to
chose this of that point.
It depends on the context.
We can choose any point.
In other words, a given vector can be drawn
starting from any point.
I could draw this vector, r prime, starting from this
point, which has nothing to do with this picture at all.
It doesn't negate the fact that this vector is the same as this
vector as long as they're parallel and have
the same magnitude.
You see what I mean?
So just the fact I draw it from here doesn't mean that I have
to add to it the position vector.
That't the derivative vector and that's the
way it's defined.
Does this answer your question?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Yes, tangent vector is a velocity
vector, that's right.
So tangent vector can also be interpreted as
the velocity vector.

That's a good point.
OK, so let's move on.

One more thing before we move on.
We have discussed now in detail differentiation of
vector-valued functions.
And as I explained, differentiation of a
vector-valued function like this simply means
differentiating the first component, the second
component and the third component separately.
You can also integrate vector-valued functions and
this is something which will be useful also for us a little
bit later in this class.
And again, to integrate a vector-valued function you
simply integrate the first component, the second
component, and third component.
So it's very straightforward.
By the way, in the homework assignment which was posted
before for this particular section, 13.2, there was a bug
in the html file and I prefer to say it's a bug, but
actually I made a mistake.
So I made a mistake and I cut a couple of problems
from the slide.
So I put them back now, but please use the updated version
of the homework assignment.
I haven't made any other changes, just in 13.2.
This particular section about derivatives of vector-valued
functions Don't worry.
There's just a couple of exercises which
are very simple.
I just didn't want you to get the idea that-- what's
this turmoil about?
Look there were only exercises about derivatives, about
tangent vectors for plain curves and I did't want you to
get the idea that you don't need to know about derivatives
for things in three-dimensional space, so that's all.
So let's calm down and let's move on.
Derivatives, OK.

So the next topic we need to discuss is functions of
two and three variables.
By the way, one other thing I remember now.
For plane curves there is a connection to what
we discussed at the beginning of this course.
By a plane curve I mean a curve for which everything happens on
the plane instead of in space.
So let's say in r2 you would have also vector-valued
functions like this, which would correspond to curves on
the plane in the same way in which the three component
vector-valued functions correspond to curves in space.
And then again, the same thing happens, which is that the
tangent vector could be found by taking derivatives
at a particular point.
It could be found by taking derivatives
of these components.
Now in what sense is this related to what
we discussed before?

What we discussed before was not so much the tangent vectors
because at that time we didn't talk about vectors.
We discussed the slope.
So we had a curve now on the plane and we talked about
the slope, which is that tangent of the angle theta.
And we found the slope and this is again, a point t equal t 0
and we found the slope as x prime-- sorry, y prime divided
by x prime, which whatever, dy-- if you want dy/dt divided
by dx/dt which was g prime of t 0 divided by f prime of t 0.
And now I just want to show you that everything is consistent
because now we actually give a more detailed information.
Nit just the slope, but actually the tangent vector
itself and what I'm saying is that in this vector, this
vector has two components.
One of them, the horizontal one is f prime of t0.
And the vertical one is g prime of t0.
So in fact, the tangent vector has these two components. f
prime of t 0 and g prime of t 0.

But then of course, what is the slope?
What is the tangent of this angle?
Well this is a triangle with a right-angle.
So the tangent by definition is this side divided by this side.
And lo and behold you get back that formula.
So what do we get know?
This more precise information about what the tangent vector
is is really consistent with the old information, which was
just about the tangent of this vector also known as slope.
In a three-dimensional space we can't really talk about the
slope because you know, now our vector has three components.
So we can talk about ratios of different pairs of components.
There's not a particular pair of components that is special.
But in two-dimensions there are only two coordinates and
therefore we can talk about the ratio of the second one to the
first one, which is what we call the slope.
So thinking in terms of vectors kind of allows us to generalize
the analysis we did for plane curves to the case of
curves in three space.
OK, that was my last comment on this subject and now I want
to talk about functions in two and three variables.
So the next topic is functions in two and three variables.

So in fact, I would like to put it in a more general context.
Up to now we've talked about different kinds of functions.
We have talked about functions in one variable, which we
usually write like this, f of x.
Like x squared or cosine of x or e to the x.
This is a function in one variable.
And today we talked-- and a little bit last time-- we
talked about vector-valued functions, which we
write as r of t.
And these vector-valued functions may take values in
vectors in two-dimensional space or in vectors in
three-dimensional space.
So in both of these cases the word function is used, which
begs the question as to whether these objects really are
similar and in what way?
What are functions?

And this is really very elementary and very intuitive,
but we have to say it once.
Once and for all.
So that there is no ambiguity left.
So the question is, what is a function?
What do we mean by a function in mathematics, in general?

You know, of course in our everyday life we
use the word function.
You know, we say our appetite is a function of how
which we've eaten before.
So what does it suggest?
It suggests a logical connection.
You know, if I ate a lot before I'm not so hungry.
If I haven't eaten all day, you know, I was
busy then I am hungry.
So each cause has a certain result.
So there's a certain rule, a certain logical connection and
that's exactly what we mean by a function in mathematics.
In mathematics, a function is a rule which takes each element
of what we call the domain of the function and transforms it
to an element of what we call the range of the function.
And the point is that we can choose as domain and range,
we can choose all kinds of objects.
And each time we make a choice we get particular functions.
For instance, to get a function 1 variable we choose as the
domain, R, the set of real numbers.
And as the range we also choose R, the same
set of real numbers.
So it's kind of boring in a way because we choose the
simplest set in both cases.
Not the simplest one, but perhaps one of the
most obvious ones.
So then the rule takes the following shape.
Given an element of R, which we'll call x, so
it could be any number.
Zero, 1, whatever-- pi.
This rule tells us, what is the value of the function.
So that's going to be another number.
Well in general, it's going to be an element of the range of
this set, which we call the range.
But range now is R, again.
So the result of the rule should be an element of --
So in other words, when we write the formula f of x,
what do we mean by this?
We mean that we have created a rule, which assigns to
any value, x, we assign a certain value, f of x.
For example, say f of x is x squared.
So that means I can ask you-- I can give you x and you will
tell me immediately what f od x is.
Say x is 0, it's 0.
If x is 1, it's 1.
If it's 2, it's 4.
But it doesn't have to be an integer.
Anything-- pi.

For any value I can substitute it into this function
and get this.
In other words, writing this formula is just a very
economical way to describe this rule.
Instead of making a table where for each value of x, I will
write the value of f of x, and actually I wouldn't be able to
do it because there are infinitely many values.
Instead of making this table I just do it in a single formula.
Before I can use this formula to convert any given value in
the domain into a value in thel range by simply squaring it.
So one formula takes care of all elements in the domain,
which is the set of real numbers.
That's what we mean when we say function f of x.
It's not one number.
It's not even a rule, which assigns to one number
another number.
It's a rule which assigns to any number here, which
is R, some number here.
Is that clear?
Now, when we write certain functions sometimes the
domain is not necessarily the entire R.
For example, sometimes we write f of x is 1/x.
So in this case we cannot say the domain is R because the
value x equals 0 is forbidden.
This rule does not tell us what the value is at x equal 0
because at x equal 0 to get the rule from this formula we would
have to calculate 1 divided by 0.
And we know that 1 divided by 0 is meaningless.
There's no such number 1 divided by 0.
So we have to make some adjustments.
And the simplest adjustment is to say that domain for this
function is not R, but is R without this point 0.
So all nonzero numbers.

But the function doesn't have to have as a domain R
and the range to have R.
A function could have any domain and any range.
For instance, let's say the domain is the set of all
students in this class.
And the range is letters A, B, C, D and F.
Then the function is a grade.
The final grade.
Well it's actually a little more because there's like
a minus, a plus, b plus.
But let's simplify.
So A, B, C, D and I don't know why there's E because it's
never used, so they say F.
Anyway, we'll hope that we don't have so many
values like this.
So in principle any function can happen, right?
In principle.
And personally, I wish that everybody gets an A.
It's possible.
If everybody gets great results on the test and everything.
Everybody will get an A.
Now historically, that's not what has happened.
So realistically I don't expect that.
So this is a function that's going to be-- sometimes people
call this a curve because you can look at the scores and then
you can try to draw the curve looking at the number of people
getting certain-- and so on, right?
So it already suggests some connection to functions, but
just think about it in a simple way.
Each student gets a grade.
So that's a function.
Each element in the domain, which is a student
gets a grade.
Which is a letter from A, B, C, D, F.
So that's a function, too.
We don't know yet what this function is.
We will find out by the end of December.
But this is also a good illustration just
by way of example.
That anything where you have a certain input and then you have
a rule which transforms this input into some output
is a function.
Now in this course we look at domains and ranges
which are R to the n.
So in other words, what we're going to do now and what's
different between this course and the previous calculus
course, which had to do with single variable calculus is
that we are going to allow not just R going to R, but
we will look at R to the n going to R to the m.
Where n and m take values 1, 2, and 3.
You have a question.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Say again.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Oh, this one.
The question is about this notation.
Is this notation familiar?
[INAUDIBLE]
No, OK.
So I'm glad that you asked me about this.
So this means belongs to.

Or is an element of the set.
And this means the same, but now in this direction.
So I just switched it.
Usually we write like this.
I did this to make it look nicer.

Otherwise if I switched them I'd have to draw a
line, a curve this is.
But it's the same.
The three ends is to where it belongs.
This means x belongs to R to the n and not
the other way around.
And here f of x belongs to the R to the m.
It's clear now?
OK, good.
So the domain and range that we're going to look at in this
class are usually going to be R to the n and R to the m,
which means R, R2, R3.

Sometimes we'll have to look at some subset here.
We'll have to remove something because function for example,
may not be well defined at a certain point.
Like this function.
This function 1/x is not well defined when x is equal to 0
or more precisely, this rule does not tell us what
to do at x equal
0.
So that means this rule is really only defined for the
domain, R without the point 0.
Something like this could happen here as well.
So now, if we stand on this point of view that there are
these functions which have as domain, R to the n and as
range, R to the m for some n and m, 1, 2, 3, which
could be arbitrary.
Not necessarily equal then we can describe all the previous
examples as sort of objects under the same umbrella.
Because for instance, vector-valued functions we can
now interpret as functions in which the domain is R and
that's where our variable t is going to live.
And the range, is R2 or R3.
So in other words, in the case of R2 you will have t going to
f o t and g of t, an element of R.
See you can also do this sign in this way.

And then you get an element in R2.
And likewise, in R3 each t will go to f of t, g of t
and h of t, which is in R3.
So again, the point in R will go to a point in R#.
Note that there is a small difference which I already
mentioned earlier, which is that you can treat f of t and g
of t as components of a point or as components of a vector.
So now I'm really standing on the point of view that I'm
getting a point in R2 instead of a vector.
But because these are so closely related we
can think about both scenarios as functions.
It just becomes really the matter of what do
I mean by R, R2 or R3?
D I interpret R2 as a set of points or as a set of vectors?
And in fact, it's better to think of it as a set of
vectors, but we oftentimes think of it as a set of points.
But points and vectors are given by the same information,
by two components anyway.
So that's why here I'm really thinking of R@ as being
collections of two numbers, x,y.
First coordinate, second coordinate.
So I'm not worrying about the fact whether these are
components of a point or components of a vector.
I know that in each particular situation in any given context
I will be able to interpret that in the right way.
I'm just using the fact that it has two components.
So vector-valued function now becomes an example of a
function because it's just the rules assigning now to a single
variable, to a single number, which we usually call t.
A pair of numbers like here or a triple of numbers like here.
Does it makes sense?
OK, so that's what we mean by function with
values in vectors.
So in this case our domain is R, it's always R.
But the range is R3 or R2, you see?
So we go sort of from the smallest one to the two
big ones, R2 and R3.
And now we can understand very easily what are functions
in two and three variables.
It's sort of the opposite situation where the domain is
R2 of R3, but the range is R.
So in that sense it's no more difficult to understand
as a concept than vector-valued functions.
Or even just ordinary functions from R to R.
It's all part of the same story, of the same
concept of functions.
So going back to the title of this topic: functions in
two and three variables.
By a function in two and three variables I simply mean
the rule in which the domain is R2 or R3.
If I want two variables it will be R2.
If I want three variables it will be R3 and the range is R.
So maybe I'll write it here.

Let's just write like this.
R2 to R and R3 to R.
So then how to express such a rule.
The domain is now say R2, so it's a set of all pairs of
points and the range is just R.
So it means that each x,y has to go to some number.
So we have to denote this number in some way.
And the way we denote is say f of x,y.
For example, I can write f of x,y is equal to x
squared plus y squared.
What do I mean by this?
I mean that whenever you give me some values x,y I can
substitute them into this formula and I get a number.
You give me two numbers, but I give you one number because
I do something with them.
I take a square root and y squared and I take the
sum, so I get one number.
So it is a rule which assigns to this one number.
That is an element of R.
That's why writing this formula can be interpreted as giving
the function in two variables.
Two variables because our point and R2 has two components, x
and y, which we think of as two variables.
So again, I could try to make a table.
You know, different values of x,y and the corresponding
values of f.
So for example, I start with 0,zero I get 0.
I start with 1,0 I get 1.
You know, I start with 1,1 I get 2.
But of course, I can not make the table for the entire R2
because there are infinitely many points in that table.
So instead I simply write this formula.
So these kind of functions appear in
real life all the time.
And a typical example could be a map of temperatures
or atmospheric pressure.
In this case you can think of you know, some domain being--
you can draw a map of a certain region, say the Bay Area and
then you plot it on the xy-plane so that each point
in the Bay Area gets coordinates x and y.
And then you say like on television when you watch the
news, they show you this and they write a temperature like
85 degrees or whatever.
Here and then maybe here it's 90 and then here they
say something else, 83.
And then San Francisco would be like 55.
No, I'm joking.
On a day like this maybe 70.

So what is it?
You assign a value to each point so you are
defining a function.
Each point on this map, each point in this region x,y
corresponds to certain temperatures so to each point
you assign a temperature.
So that's an example of a function.
Likewise, you can think of the landscape, you can think of the
map of this area and you can look at that height
of each point.
The height of this point compared to the sea level.
So then again, the height gives you a function.
For each point x,y you get the particular height.

This actually leads to a nice idea of trying to interpret
functions or visualize functions in two variables
in terms of graphs.
Of course, for functions in one variable this
is a very old idea.
Functions in one variable we've been drawing
graphs for a long time.
It's almost reflex.
If I say y equals f of x-- say x squared or let's just say f
of x equals x squared so you immediately think
about this picture.
So this is the picture which you draw on the
plane by introducing additional variable.
So this is your x and there's an additional variable you
introduce to accommodate the value of the function.
And then for each value you plot the point on this plane
by taking as the x-coordinate the value of x and as the
y-coordinate you take f of x.
Of course here again there's a clash of notation, which
you have to realize.
I mean, it's sort of again very clear from the context,
but I would like to mention it anyway.
When we talk about functions in one variable we talk
about their graphs.
And the graph of a function in one variable
lives on the plane.
So even though in your original problem the domain is
one-dimensional, it's just x.
There's only one variable.
You are forced to introduce a second variable just because
you also want to accommodate the value, the range, the
element in the range.
And we usually call this coordinate y.
So we write the graph as y equals f of x,y-- sorry,
y is equal to f of x.
But this y has nothing to do with this y.
Because when I talk about functions in two
variables I talk about functions in x and y.
Why?
Because x and y is like the two variables go next to each other
in the alphabet, so it's natural.
Once I say x, I have to say y.
You know, you'd be surprised if I said function of x and a.
Or something.
It's like, what's wrong with you.
Why is he talking like this.
So once I say x the next one you say y.
But know in this context y has nothing to do with the domain.
This is the domain.
It only has one variable.
And this y corresponds to the range.
But now when we talk about functions in two variables
we use the same notation, x and y to indicate the two
variables in the domain.

So now to draw a graph of this function we have to go to the
three-dimensional space because we have to introduce one more
variable which corresponds to the range.
The range in both cases is R, but here I denote it by y
because I use just x, so the next letter that comes is y.
Now I already use x and y for my domain.
So I have to use a different coordinate, different
name for the range.
And usually I say z because it's the next letter
in the alphabet.
So for instance, if I choose as my function in two variables
this function, f of x, y equals x squares plus y squared, what
kind of picture do I get?
It will just be a surface given by the equation x equals
x squared plus y squared.
And fortunately we have already discussed this surface so
we know what it looks like.
This is an example of an elliptic paraboloid,
which looks like this.
And I kind of want to indicate some more curves on it just
to give it more shape.
So that's a paraboloid.
So this is the graph of the function, which is
a paraboloid, elliptic.
Likewise, you can also talk about functions
in three variables.
So here I talk about function in two variables, but I can
also talk about function in three variables.
So in this case domain is R3 and the range is R.

x,y,z, right?
So of course, this is also meaningful because for example,
if you talk about barometric pressure, if you talk about
atmospheric pressure.
Well for temperatures, we usually just measure
temperature of the surface, but for pressure it makes sense
to also look higher.
So we're really talking about a point in space and we assign to
a point space the pressure.
So it's really a function in three variables.
Really for each point in this classroom there's
a particular value.
The temperature also.
For example we could measure temperature you know, at each
point in this classroom and there will be some
small variations.
If you wanted to draw a graph of this function though, you
would have to introduce one more variable and now we've run
out of letters because z is the last letter of the alphabet.
So then we start from the beginning so now we can say
a is equal to f of x, y, z.
So the graph is now going to be in four-dimensional space.
But since we can not really visualize our four we're not
going to talk about this.
What we can talk about though, are the so-called level
curves and level surfaces.
So when I draw this graph I almost feel the urge to draw
this additional circles because you know, when you look for far
away this kind of gives it a certain depth to the picture
and it makes you understand what I'm what talking about.
In fact, it looks even better if I draw a few more.
And so I am actually indicating, I'm kind of
thinking that they're transparent so I'm indicating
also the backside of those circles, but they're not as
visible as the front ones.
So what are these circles?
This is something that we've learned when we talked about
quadratic surfaces of which of course elliptic paraboloid
is just one example.
When we talk about quadratic surface we learned that a very
efficient way to understand the structure of these quadratic
surfaces it to cut like or slice them planes parallel to
a given coordinate plane.
So in particular we could slice it by a plane
z equals something.
Cut by the plane z equals k.
So when we cut this graph by this plane we get a curve
because it's intersection of two surfaces now.
And the intersection is a curve.
What is this curve?
Well look at this equation and look at this equation.
Now in this equation this is a number. k is a number.
So k for example, could be 1, 2, 3, 4, whatever.
That's different from x, y, z which are variables, which can
take arbitrary values here.
Now if you look at this equation and you couple it with
this equation what you get is of course, x squared plus
y squared equals k.
And that's one of those circles.

When you slice it the set of intersection will be a curve
and that curve will be given by this equation, which indicates
that this is a circle of radius square root of k.
And these circles we get my looking at all the points on
the graph which share the same value of z.
So in other words, it's kind of like the same level.
If you think of this as being the landscape, so there is sort
of a hole in the ground and this would be all the points
which are the same height or sort of at the level
below the surface.
So in other words, what this is, it immediately leads to
the terminology, level curve.
This is all the points on the grpah which are the same level.
And saying level just means fixing the value of z.
Likewise, when you have a function in three variables you
can also fix this value a.

So you say a is equal to k.
And what you get is an equation like this, f of
x, y, z is equal to k.
So what it is now is a surface.
And we call it the level surface For this function.
In other words, for each value of k for example, 1, 2, 3, 4
and so on we can look at all points x, y, z which share the
same value with respect to the function f.
So that's now a surface just like before it was a curve.
But now we have an extra variable so we would get
the surface and that's called a level surface.
So you can look at various examples of this.
So that's roughly what we need to know about functions in two
and three variables from the sort of qualitative
point of view.
And now we would like to understand them from a more
quantitative point of view.
What can we do with this function?

And of course our goal really is to do derivatives and
integrals of these functions.
And that's really the crux of this course, it's really
understanding derivatives of these functions and integrals
of these functions.
And you see right away that it's going to be more
difficult than before.
Because for instance, if we think just in terms of two
variables-- for example we can differentiate with respect to x
or we can differentiate with respect to y.
And in fact there's more because we could even
differentiate in some direction which is kind of in between.
It's kind of a combination of x and y.
You see?
So there are many more choices and we have to understand much
better what derivative means in this context.
Same with integrals, but first we'll talk about
differentiation.

Now when we define differentiation and I kind of
talked about it earlier when I talked about the derivative
of vector function.
The derivative is defined in terms of a limit.
When you take the change in position divided by
the change in time.
When you get the velocity for example.
You the limit.
So before we can talk about derivatives we have
to talk about limits.
We have to understand what limits mean in the context
of functions in two and three variables.
And this is a somewhat obscure subject because to really truly
understand it we have to work really hard because we have to
give a precise definition of the limit and things like that.
We're not going to do that.
We're not going to prove theorems about limits.
If you like you can read about in the book.
But what I'd like you to understand is the kind
of qualitative picture.
In other words, the main ideas as to what the intuitive idea
of the limit is and what kind of examples, you know, what
are the possible scenarios?
What are possible situations when limits exists
or doesn't exist.
And that's what explain.

So let me start with functions in one variable.
If you have a function in one variable let's look at the
graph of this function.
Let's say this is a graph of a function.
You pick a particular point x, y.
So you want to see what happens with the value of the function
in the small neighborhood of this point as you
approach this point.
You want to see what happens.
Can you hear me?
Yes?
OK, tell me if you stop hearing because I have the red light,
so I hope it works for another 15 minutes otherwise we'll have
to see if I can find a spare battery.

We would like to see how the value of this function
behaves in the neighborhood of this point.
In other words how predictable the function is.
As we approach this point we can trace the values of this
function and we can try to see whether they also tend to the
value on the function at the point x0.
Now of course on this picture it's clear that yes because
it's kind of smooth.
I drew it in one stroke, which is pretty much to say that it
is predictable function or what we call more precisely,
continuous function.
But that's not the only possibility.
I could draw something like this.
I could draw instread something like this.
Where you go like this and then you go like this.
So in other words, the function jumps here.
So let's say the value to the left of this point.

The values kind of behave in a nice way.
If I start from a point below x0 and I start moving towards
x0 the value will move towards the value of the
function at x0.
But I want to indicate that this circle means that
there's no point here.
This branch only takes care of the points which are larger,
greatest than this one.

So on this side if I start approaching this point I
get values on this branch.
But then, when I get to this point there is a jump.
The value's here and not here.
So in this case we will say that the limit of this function
does not exist because it exists on one side, but it
doesn't exist on the other side.
In other words, the limit of the function exists, but it's
not equal to the value given.
So the function is not continuous.
There is another option.

The value could go to infinity.
In this case we'll also say that the limit doesn't exist.
For example our favorite example for this
is a hyperbola.
You know, y equals 1/x.
No limit as x goes to 0.
And that's not all because another thing that can happen
is the function we know goes to infinity, but it can
oscillate like crazy.
For example, a typical example, the function sine 1 over x.
Around 0 this function doesn't have a limit because as you get
closer to 0, 1/x goes to infinity and the function
starts oscillating faster and faster so you never know, you
can not say what is the limiting value becuase somehow
it's everywhere between minus 1 and 1.
So again, no limit at x equal 0.
In other words, we see that there is sort of the most
favorable scenario on some sense, which is the
most typical one.
What we are going to use mostly in this course.
But at the same time you have to realize that that's not
the only possible scenario.
It's really a favorable one, but it's quite possible to
have different situations.
It's quite possible to have situation of a jump.
That the limit from one side is not equal to the limit
from the other side.
There's a situation where the value goes to infinity, so
again, there's no limit.
Or the function oscillates in a way that makes it impossible to
say what is the limit when it approached a particular point.
That's just the case of function in one variable.
So in the case of a function in two and three variables there
are even more scenarios.
And what I would like to do is to explain just a couple of
them so that you get an idea of the way those functions
could behave.
So first of all, the favorable scenario still remains.
All the functions in two variables that we looked up to
now, for example, the quadratic functions, they're all
well-behaved in the sense that they're predictable.
As you move, as you take points x, y in a small neighborhood of
a given point x0, y0 and you take the value of the functions
f of x, y then when x and y tends to x0, y0 the value of
this function will also tend to the value at x 0, y0.
So in this case we'll say that the function has a limit at
x0, y0 and is continuous.
And functions like polynomial functions for example, and all
kinds of functions involving sine and cosine and the
exponential function, are all continuous.
You get in trouble when you start dividing by something
that could become 0, which of course you see already
in the example of the one-dimensional case.
When you divide by something which could become 0
you're in trouble.
But in the case of two and three variables there are
sort of more possibilities.
There are more possibilities to divide by 0.
And the typical examples which you will see in the book and on
the homework involve division by some expression, some
polynomial, which becomes 0 at the given point.
So a typical example is the function f of x, which is equal
to, say, x squared divided by x squared plus y squared.
So the question, if you ask, is whether this
function has a limit.

Does it have a limit as x, y tends to 0, 0?

You see if for example, if you had x squared plus y squared
plus 1 you don't have to think about this really because
this is actually never 0.
It's never 0 because you add 1.
For sure it's not 0 in a small neighborhood of this point.
Even plus any small number will do, so then there's no issue.
This doesn't become 0.
And therefore you can be assured by various theorems,
which can be proved in a fairly straightforward way that
actually this has a limit and this function is continuous.
The only reason why we have to worry about this is because we
have an expression which actually becomes
0 at this point.
So we can't really divide.
We can't literally divide by this.
We can not literally divide by this value.
It's 0, so we can not divide by 0.
If here you have something like x squared last 1, again, you
don't have to think about it too much.
Or even for simplicity let's just say 1 divided by x
squared plus -- This will be exactly like this.
I mean, the thing just blows up.
Because when x and y go to 0, this becomes very small, and
you take 1 divided by this, so it becomes very large.
So when x, y goes to 0, 0 this goes to infinity.
So clearly there's no limit.
So again, that's an easy case.

In other words, if the numerator doesn't go to 0 and
denominator goes to 0, the thing goes to infinity.
I mean, the value of the function goes to infinity,
so there's no limit.
So the only subtlety is when both numerator and denominator
go to 0 like in this case.
So in this case sometimes it could be that it has a limit
and sometimes it could be that it doesn't.
So I have just enough time to tell you what happens in this
particular case and then you will see in the book and on the
homework you will see other example very similar to this.

So I'm going to show you that this function actually
does not have a limit.
So here's how I'm going to do. that.
I'm going to approach the point 0, 0, the origin
in two different ways.
So approach in two different ways.

The first way I'm going to approach it is I'm going to
say I'm going to approach it along this, along
the y-axis like this.
So in other words, in the first way I will say that x, y is
0, y and then y goes to 0.
So this way I eliminate one of two variables.
Namely, x and I get a function in jsut one variable y.
So I kind of go back to the one-dimensional case, which
I can analyze more easily.
So let me see what do I-- I'm sorry, I made a mistake--
it's f of x, y.
I have a function in two variables, but now I
convert it into a function in one variable.
So I set x equal 0.
So what do I get?
I get 0 squared divided by 0 squared plus y squared.

If y is nonzero this is nonzero.
So it's 0 divided by y squared.
When I'm approaching 0, I'm approaching 0,
but I am not at 0.
I'm looking outside of the point 0, but very, very close.
So outside of the point 0 this is very small,
but still nonzero.
So that's why I'm allowed to take the ratio of this.
Because y is not zero.
It's not 0, but it's approaching 0.
So this is OK and it gives me 0.
So along this ray, along this path I get the value 0.
So that's the first calculation.
And the second calculation I will go along this path.
Path number two.
And now I will do the opposite.
I will say x, y is x0.
And now I will take x to 0, but not equal to 0
and see what I get.
So I get f of x 0 is equal to x sqaured divided by x
squared plus 0 squared.
So that's x squared divided by x squared.
Now if I have such an expression, x squared divided
by x squared and I know that x is nonzero I can take the
ratio and it's just 1.
So the upshot of this calculation is that the result,
the value of the function depends on the path along which
I'm approaching my point.
You see in some sense it's very similar to what
happens in this picture.
If I approach from this side I get this value.
If I approach from this side I get this value.
But if my domain is just one-dimensional that's the
worst-- in some sense-- that can happen because the only
ambiguity I have when approaching the point is from
this side or from this side.
But when the domain is two-dimensional there
are many more choices.
I can approach it along this path, along this path,
along this path.
In fact there are many other paths as well.
And so what you need to see for the function to have a limit is
that along each of these paths you're going to get
the same result.
I have now shown you that along one of them you get 0 and along
the other one you get 1.
So you get different results.
That means that the function does not have
a limit at this point.
So we're out of time, so I'll let you go and we'll
continue next week.