Mathematics - Multivariable Calculus - Lecture 9

OK, next Thursday we'll have the first midterm exam.
I know it might come as a surprise to some of you.
I hope not.
All the information is now available on bSpace and also
on my homepage, but there are links between them anyway.
So you could find all the information at either place.
And on Tuesday we'll have a review lecture.
So you are welcome to ask me questions on Tuesday
about the material.
The material for the midterm exam is everything
up to this week.
Everything we studied up to this week.
So next week we'll have review both in sections and in my
Tuesday lecture and then on Thursday we'll have
the exam right here.
Now I requested another room so that we had a little bit more
space, but unfortunately it looks like no rooms
are available.
So we'll just have to use this one and utilize
this as much as we can.
So in a way, it makes things easier so we don't have to
split into two groups, we'll all be here.
So the exam will be just in the usual class time.
We'll start sharp at 3:40 and we'll finish at
5:00, right here.

Any questions?
Yes?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: No, but all questions like that are
addressed so I don't want to waste too much time describing
this because I believe that all the information is available.
In the meantime, I want to go back to the topic which we
started discussing last time.
The topic which we started discussing last time and
namely the differentials.
And already last time I talked about the differentials in the
case of functions in one variable.
I would like to repeat that now and then talk about
differentials of functions in two variables.
First, one variable case.
So in the case of one variable we'll have a function f of x.

And we'll have a graphic of this function.
Graph of this function lives on the plane where in addition to
our variable x we introduce one more variable, which is
responsible for the values of the function.
And usually we denote this second variable by y.
But I would like to sort of depart from this tradition
today and use a different labeling for the second
coordinate because after this I will talk about functions in
two variables where x and y will be to independent
arguments, two independent variables of the function.
And so I don't want to have any confusion between this variable
here, in the case of function one variable and the second
variable for functions in two variables.
So that's why what I want to do is I want to call it z.
Then it will be more close to what will happen for functions
in two variables because for functions in two variables
we'll have x and y, two independent variables and then
z will be responsible for the values.
So x and z and as I've said already many times we don't
have to use x, y or x,z in principle each time you can
use any letters you like.
It's just a tradition to use x and y.
But mine as well today use x and z because then we'll be
able to appreciate the analogy between the one-dimensional
case and the two-dimensional case more.
So the graph of this function then will be given by the
equation z equals f of x.
This will be the graph of this function of f of
x on the zx-plane.

So let's say this is a graph.
Now we pick a point x zero with a particular value of x.
It could be any number, any real number you like and what
we're interested in is we're interested in the tangent line.
Interested in the tangent line to this graph at this point.
That's the tangent line.

So what's so special about the tangent line?

We've talked about this many times before.
The special thing about tangent line is-- well, there
are two special things.
First of all, it's a line.
A line is the simplest curve.
It's given by the simplest possible equations amongst all
curves and just geometrically it is the simplest curve
that you can draw.
That's the first thing and the second thing is that among all
lines, which pass through this point, which pass through this
particular point, this red line approximates the graph in the
best possible way in the small neighborhood of this point.
In other words, I'm not claiming that this lines is
a good approximation to the graph everywhere.
Certainly it's not.
They diverge the farther away we get from this point.
They diverge more the farther we get away from this point.
But just in a small neighborhood of this point
it's actually a very good approximation.
And in fact, when I draw it it's kind of difficult to draw
in such a way to actually insist that they are actually
different because they're so close to each other.
That already gives you the feeling that it's a very
good approximation.
So that's what tangent lines are good for.
They give you linear approximation to your function.
In other words, they capture the essentials of behavior of
this function in a small neighborhood of this point.
To the first order as we say, which already contains
a lot of information.
So therefore, it is useful to write down
the equation for this.
The question for this tangent line or more precisely to think
of this tangent line as a graph of a linear function.
So what is this linear function?
Well we know that since the studies of one variable
calculus, we know that the slope of this tangent line
is given by the derivative of this function.
So the tangent of the angle theta right here is actually
f prime of x zero.

And so the equation of the tangent line-- and I'll just--
let's just write it in words of the tangent line at x equal
zero is the following.
y is equal-- sorry, I see I'm writing y because I'm used to
writing y, but like I said, I want to use z. z is equal to
f prime of x zero times x minus x zero plus z zero.
What is z zero? z zero is the value of the
function at this point.
And so z zero of course, is just f of x zero.
But I could actually write here f of x zero also, but I prefer
to write it z zero to make the notation a little
bit less heavy.
And I want to rewrite this also as follows, as z minus z zero
is equal to f prime of x zero times x minus x zero.
I want to emphasize that this is a particular number.
This is a particular number.
This is a number.
Namely, the slope.

And it appears in this formula as the coefficient of
proportionality between the increment in z along this line
and the increment in x along this line.
So the increments in z and x are proportional.
That's what this formula expresses.

Proportional to each other and the coefficient of
proportionality is nothing but the derivative of
f at this point.

So it's important to realize that all of this, everything
that I've done so far is relative to a particular
point, x zero.
It's relative to x zero.
If I choose a different point, let's say some x1, so this
point will live here.
I mean, the point with such x-coordinate will live
here on the graph.
And so the tangent line to this point will be this blue line
passing through this point and surely this blue line has
nothing to do with this red line.
And for a good reason.
Tangent line is useful as much as we want to understand the
behavior of the function in the neighborhood of that point to
which we draw the tangent line.
So the tangent line becomes irrelevant when we start
talking about a point which lies sufficiently far away.
So when you talk about tangent lines you have to specify
the reference point, the initial point.
Tangent line at what point?
There's no such thing as tangent line, the tangent
line to a graph.
There is a tangent line.
There is tangent line for each point on the graph.
That's the first important thing that
you have to remember.
And then once you've fixed x zero, your reference point or
more precisely, your reference point if you look at it on the
plane it's x zero and z zero where z zero is the value
of the function at x zero.
Once you have that then you have the equation for this
tangent line and sure enough it involves the equation x zero
in a very essential way.
It involves it in two places.
First of all, the increment in x is counted relative to x
zero and second of all, the coefficient of proportionality
is the derivative of f at this point x zero, so surely this
coefficient, the slope will be different for different point.
And you can see that.
The slope of the blue line is different from the
slope of the red line.
So this formula really refers to the red line.
So this one.
So now the concept of the differential is just the
concept of this equation that we have written of this
equation which expresses the proportionality of the
increments in the value.
In the argument of the function, the value of the
function under linear approximation.
So the important thing to understand about differential
is that it revolves around the particular choice of notation
for the increments, which I'm now going to explain.
So that all the difficulties in understanding the differential,
in my view, really boil down to understanding the notation.
That's why I'm going to be very careful when I
define this notation.
So we introduce the following notation.

I would like to denote x minus x zero as dx.
This is just a notation, and so I want to emphasis-- by
definition-- so I defined this to be just the difference
between x and x zero.
Now if I write it like this you already see that this notation
is deficient because on the right this expression depends
on the choice of x zero.
When I write x minus x zero it depends on
the choice of x zero.
If x zero is equal to zero this will be just x.
If x zero is 1 this will be x minus 1.
If x zero is 2, it will be x minus 2 and so on.
You know, if it's pi it will be x minus pi.
In other words, on the right-hand side I'm not talking
about a single function in x, but it's a function
which you get.
It's a linear function, it's very simple.
But it's a function which you only get, which you only
specify once you specify x zero.
But my notation on the left does not carry x zero.
That is the first major problem in the notation which we use.
So let me cure that by actually introducing explicitly
on the left-hand side.
So I will keep track of x zero by putting it as an
index in this notation.
So to emphasize that this something that we're doing
for now and we will actually discard later on, I
will put it in yellow.
So let me just introduce notation dx relative to x zero.
I will just denote the following linear function
x minus x zero.
I have the right to do this.
I can introduce any notation I like, so this is notation I
want to introduce for whatever reason.
Likewise, for z I want to do the same. dz also
relative to x zero.

I would like to write as z minus z zero.
Maybe it's better to say relative to z zero, But z zero
of course is determined.
This one since we're talking about a particular function f,
z zero is determined by x zero.
So in fact, I could write as the parameter x zero or
z zero, whatever I like.
But let's write z zero to make it a little
bit more consistent.
This is actually determined by xz.
And that's going to be z minus z zero.
So far it's totally tautological.
There is nothing in this.
It's just the choice of notation.
But if I do that then I can rewrite this formula
in the following way.
Just this formula.
On the left-hand side I recognize what I now call
dz relative to z zero.
And on the right-hand side I recognize dx
relative to x zero.
So the same formula will look like this.
dz relative to z zero is equal to f crime of x zero times dx.

So I've done nothing.
I've just used a certain notation for-- just introduced
some new notation for z, for the increment in x and z, but
now the formula starts looking more familiar because you've
seen this formula before in one variable calculus.
But what's usually done is usually we drop the indices so
that it looks like this. dz is equal to f prime of x zero dx
and also we often replace dz by df so we get df is
equal to f prime.
So you recognize this formula, right because you can also use
this to write f prime is equal to df over dx.
But now I hope you can now appreciate this formula more
and understand what it means.
Usually in the textbook they don't really explain what is
meant by dx, what is meant by df?
Or dz?
Which we use here interchangeably because we're
thinking about particular function f for which z
serves as the value.
The point is that dx and dz are nothing but the increments in
the coordinate x and in the value z along the tangent line.
So the equation of the tangent line, which we know is given by
this formula just becomes the old formula that you knew.
That dz is equal to f prime dx or if you wish, df
equals f prime times dx.
The only problem in understanding this formula is
the fact that usually what we can call abused notation.
We abuse notation in a sense that we drop some essential
information from the formula.
This would be to me a much more consistent way of expressing
the fact that what we're doing is just writing down the
formula, the equation for the tangent line.
It is important at least from the outset to indicate the
fact that dx is not an absolute notion.
Like x, x is an absolute notion, it's a coordinate so
x makes sense without the reference to anything else.
It's a particular coordinate. dx is not an absolute relative
to It's a relative notion. dx is defined once you choose
a reference point.
One you choose a reference point x zero then dx
is defined, right?
So then if the reference point is zero dx is just x.
If the reference point is 1 it's x minus 1 and
so on and so forth.
It's just x minus x zero.

Likewise, dz is not an absolute notion.
It is really relative to the reference point.
Once you choose a reference point it's just increment.
Once you realize that and you see that this formula is
nothing but the expression of proportionality of the two
increments along the tangent line, right?
Which is just the equation of the tangent line.
Do you see what I mean by this?
Do you have any questions about this?
In a way you can say, why am I talking about all
of this because we've learned nothing new.
The only piece of essential information is already
available here.
That's the equation of the tangent line.
Something we've known all along.
Well all along since the one variable calculus, right?
Since studying the one variable calculus.
But the reason I explain this is because we're going to use
this notation dx, df and dz and I would like to explain
what it really means.
So now I have explained this, explained what this notation
means and I've explained what the formula, the old formula
that we've known-- df equals x prime dx means-- and it's
just the equation of the tangent line.

Now this expression, f prime of x dx is called a differential.
f prime of x zero dx and I would like to keep insisting on
putting this x zero just to make sure that we understand
that actually this is something which is relative, it's
a relative notion.
It's called the differential of the function f at that
point x equals x zero.

Let me give you an example.
Let's say f of x is x squared minus 3x.

What is the differential?
What is the differential of this function
say for x equals 1.
Well we simply have to take the derivative of this function.
What is the derivative of this function?
It is just 2x minus 3.
We just have to take the derivative of this function
at the point x equals 1.
So this 1 is what I indicate in other formulas by x zero.
This is x zero.
So I simply substitute this.
I have to substitute x zero into this expression like here.
So I get 2 times 1 minus 3, so it's negative 1.
And then I multiplied by dx relative to the point x zero,
which in our case is 1.
So in fact, I could also write it as minus 1, negative
1 times x minus 1.
So the differential of this function is really this linear
function, negative 1 times x minus 1, which is the same as
if you want, it's just 1 minus x, but let's just
leave it like this.
Minus x minus 1.
That's the differential of this function.
But usually what we're going to do is we're going
to abuse the notation.
I mean , from now on, once we understand what the meaning is
we'll actually abuse the notation and we will
drop all the indices.
So when we drop all the indices we'll actually just write
minus dx and say this is a differential of this function
at the point x equals 1.
Yo have to realize that actually in this formula
dx is referring to this particular point.
And if I change the point I will get a different answer.
So change the point.
So let's find a differential for x equals 2.
So now it's 2 which is x zero.
So I have to substitute this value into the derivative.
So I substitute 2 here.
I get 2 times 2 minus 3, so I get 1.
So now f prime of 2 is equal to 1.
Whereas here it was negative 1.
And so I'm going to get 1 times dx at x zero equals 2.

I can just erase 1 of course, just dx and then just in this
case we're going to just write dx dropping the indices.
Suppose this yellow chalk becomes invisible so the answer
just becomes dx, but you have to realize that this is a
differential at the point x equal 2.
And it's different from the differential at the point x
equal 1, which was negative dx.
So what I'm saying is what I drew here on this diagram is
that the differential is the function whose graph
is the tangent line.
The tangent line depends on the choice of the point so
therefore the differential also depends on the choice
of the point.
The coefficients in front of dx is going to be just the
slope, which is a derivative.
Here a negative 1, here 1.
But also dx itself actually has a different meaning.
Here dx is x minus 1 and here maybe I should write here
it's just x minus 2.

Yes?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: The differential is a function of x, which is equal
to-- this is a differential.
It's a function of x which is equal to the derivative of your
original function f at the point x zero times dx, which is
understood as x minus x zero.
That's the differential.
This particular linear function.
It is precisely the function whose graph
is the tangent line.
It's a function which gives the tangent line> You cannot say
a tangent line is the function, right?
You can say tangen line is the graph of a function.
You see what I mean?
I'm trying to be precise, it's not because I want to be
pedantic but there's too much confusion as it is.
So I think to separate different notions.
There's notion of a function.
There's a notion of a graph of a function, right?
That's right.
So you're going to say tangent line is a function.
Function is a rule which assigns to each number
some-- We represent a function by its graph.
This tangent line is a graph of a function.
Which function?
The differential.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: Exactly, there's only one function given which
is f of x, but there's more than one tangent line.
There is a tangent line for each point, which
I have illustrated.
I've drawn two of them.
I've drawn the red one and the blue one.
Yes?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: That's right.
So your right.
So to be absolutely precise here that would be df for x
zero equals 1 is equal to minus dx at x zero equals 1.
df at x zero equals 2 is dx at x zero equals 2.
And if you want I can also do df at x zero
equals 3 for example.
I just need to calculate the derivative at the point 3.
The derivative at the point 3 is 2 times 3 minus
3, which is 3.
So then the answer will be 3 times dx at x zero equals 3.
And so on.
But of course, now you can kind of guess what
this coefficient is.
This coefficient is just 2x minus 3.
So instead of writing an infinite list of answers for
different values of x zero I just write the formula
in one stroke.
I just write one formula, which is responsible for all of this.
I just write df is equal to f prime of x dx.

Yes?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: It's like you know, certain things are allowed
when you become adults.

It's the same reason.
Just to simplify things, but it leads to incredible
confusion in my opinion.
So that's why I am trying to unravel the precise definition
and then explain how do we actually arrive at the formula
which you see in the book.
And I agree and in some sense we should not be allowed.
Or more precisely we should be mindful of this, so whenever we
see this formula that's what we should see.
We should see not a single formula, but a bunch of
formulas which depend on the x zero.
On the choice of x zero.
But this one formula is not just one equation but it's
a collection of equations.
For each value of x zero is the formula which says that df
relative to that x zero is f prime at that x zero times dx
at that x zero you this formula is nothing but just saying z
minus z zero equals f prime of x zero times x minus x zero,
which is the equation of the tangent line.
But we write in one formula, we write down all equations
for all tangent lines.
That's what we do.
That's the point I'm trying to convey to you here and that's
why I give you examples.
So maybe in this case I would actually want to write-- more
precisely in this particular case I will write f prime of x,
I have found it, I have found that f prime of x
is 2x minus 2.
In fact, I will write df is 2x minus 3 times dx.
So more precisely it is this formula which is responsible
for all of this.
Because if you have this formula you will get this one
if you substitute instead of x you substitute value 1.
If you substitute 1 you get minus like I put.
At x equal t you will get 1.
At x equal 3 you will get 3 and so on.
So this one formula will give you all of them.
In other words, I give you this formula you should be able to
substitute any value x zero and you will get the equation of
the tangent line at that x zero.
Does it make sense?
Any other questions?
So there's nothing mysterious about the differential.
The only mysterious thing is that it should actually
be defined relative to a particular point x zero.
And what we have done or people before us have done is that
at some point they decide to drop this from the notation.
So the formula starts looking like this and starts looking
very confusing because what does it mean?
What is df?
What is dx?
It becomes very confusing.
But if you remember that it actually refers to a particular
value, x zero and once you remember that you know that dx
corresponds to x minus x zero and df corresponds to z
minus z zero then the mystery disappears.
There's no mystery.
It becomes an extremely simple and mundane formula.
Just a formula for the tangent line.
The equation for the tangent line.
That's what it is.

So one more piece of notation to sort of finish with this.
We can also write delta f.
So to make things even more confusing.
There's also notation delta f.
And delta f and again, to be precise we also have to say
that this is relative to some x zero.
This by definition is f of x plus x zero minus f of x.
So it is the increment of the actual function.
It is the increment of the actual function whereas df is
the increment of the tangent line of the linear function
which approximates our function.
You see?
So that's the difference between this big
delta and small d.
This should not be confused with df relative to
the point x zero.

Which is f prime of x zero times x minus x zero.

Let me draw again the picture.
Let me magnify it.
Actually let me not draw the coordinate system, but
let me blow up the small neighborhood of this point.
So here is my graph.
Here is my tangent line.

So here is x zero somewhere and here will be-- I'm sorry, I
wrote something incorrectly. x minus x zero. f of
x minus x zero.
So there is x zero and there is x.

So now I'm trying to confuse you.
I'm sorry.
I was going to draw it on the picture anyway.
So look, this is delta f.
You see what I mean?
This is x zero and this is x.
And x is very close to x zero.

I can measure the value of the function itself, which
is the yellow curve.
I can measure the difference between the values
of the function.
That's this.
Or I can measure the difference between the values of the
linear approximation, the linear function which
corresponds to the tangent line.
That's df.

You see what I mean?
These are different.
Because it doesn't quite go up to here.
They start diverging.
They start deviating from each other.
But the punchline is that they're almost equal.
The difference is negligible.
The closer you get to this point, the closer x and x zero
are the smaller difference is going to become.

That's the punchline, which express the fact that linear
approximation is useful in the following sense that when
you're very close to the point your linear function, whose
graph is a tangent line is almost as good as the
original function.
In case of course, your function is a nice one like
this, so it's kind of a smooth function and such functions for
which this linear approximation works are called
differentiable functions.
So the punchline of all of this discussion is the following.
That there's a large class of functions f, for which the
difference between delta f and df is negligible.
It's very small.
It's much smaller than the different between x and x zero,
so we can actually ignore it when x is very close to x zero.
So the notation is different, but the point is that the two
quantities for differentiable functions are very close.
Which is to say that the linear function, the function which
gives you the tangent line is a good approximation to
the original function.
Let me write this down.

So function f of x is called differentiable at x equal x
zero if delta f at x zero is equal to df relative to x zero
plus-- I promised you not to use epsilon, but let me use
it anyway to indicate that there's something very small.
I will not use delta.

Where epsilon of x goes to zero as x goes to zero.
In other words, this piece is negligibly small.
We see because this already goes to zero.
And this certainly goes to zero-- sorry, when
x goes to x zero.
So this goes faster than x minus x zero.
It goes to zero faster than x minus x zero.
So this is precisely what I indicated by that
little difference.
Let me point out this little piece.
This is what is epsilon of x times x minus x zero.
This little one.
This tiny little piece is something that goes to zero
faster than x minus x zero because this already goes to
zero and this goes to zero so it goes as a square.
It roughly goes as a square of x minus x zero.
So let me unravel this for you.
In other words when we're saying a function is
differentiable that's the expression, delta f is df plus
this, so what is delta f?
I explaned that delta f is just f of x minus f of x zero,
so let me just spell out what that formula means.
Let me call this star.
So for star is just saying the following.
This and now I want to spell out what df at x zero is.
And df at x zero is this, right?
So it's f prime x zero times x minus x zero plus epsilon
over x x minus x zero.
Now do you remember Taylor series?
This gives you a good perspective on this formula.
Do you remember Taylor series?
What?
You don't want Taylor series?
How about just the first two terms in Taylor series?
I will not ask you for more than that.
Taylor series, the idea was that the difference, f of x is
equal to f of x zero plus f prime of x zero times x minus x
zero plus 1/2 f double prime x zero times x minus x zero
squared plus and so on.
So I just want to explain to what we're doing now.
We are just looking at the first two terms
of the Taylor series.
We're just on focusing on the first two terms of the Taylor
series because this one term is this.
OK, now it's on the left-hand side, but big deal.
We can just rewrite it like this.
Just take it to the other side.
So you see this matches this.
This term matched this.
In the Taylor series we then have all the higher derivatives
and higher powers of x minus x zero, second, third, and so on.
And now I have taken all of this stuff and denoted it
by this one expression.
Because see the main point is that the powers of x minus x
zero, which show up in this term and the next term
are 2, 3, 4 and so on.
They're higher than 1.
So I can sort of chip of the first power, x minus x zero and
what will remain, for example, here it's like second power, so
I just split x mius x zero squared into this and one
more x minus x zero.
So in other words, this thing goes to zero by itself.
You see what I mean?
What I'm doing roughly is let's take this term, 1/2 f double
prime of x zero times x minus x zero squared.
What I'm doing is just I'm writing it like this.
I've done nothing.
I just wrote the square as a product.
After this I take this piece and call it epsilon.
So then what I get is epsilon times x minus x zero.
The point is that this guy by itself goes to zero.
It goes by itself to zero because this is finite.
This is just some expression for example, for the function
which I had which was the first derivative was 2x was minus 3,
so second derivative would be 2.
So this is just some number, which is 2.
And it's 1/2 of 2 so it's just 1.
But then there is x minus x zero. x minus x zero is
something that goes to zero as x approaches x zero.
But in addition I have one more power x minus x zero because I
started out with a second power.
The only term for which this will not be the
case is the first term.
In the Taylor series the first term will have x minus
x zero to the power 1.
And that's the 1 which I retain and then everything else, I
said everything else is really negligibly small
compared to this term.
So that's the idea.
That's the main point of our calculation.
Now before we tried to write it as an infinite series which
involves all powers- sorry, all derivatives of the function.
It involves no derivative at all.
The first derivative, the second derivative and so on.
What we are saying now is that let's just keep the constant
term, which is f of x zero and let's keep the first derivative
term, which gives rights to the linear term in x.
Let's just keep this and everything else we'll just put
in the bag and call it epsilon of x times x minus x zero.
It's something very small times x minus x zero.
So it's negligibly small compared to
the first two terms.
The special thing about the first two terms is that they
give you a linear function, which is the function whose
graph is the tangent line.
So writing down this formula means that you approximate
your original function f of x by a linear function.

This one plus some really, really small, negligibly
small error term.
That's what we've done.
Questions?
Yes?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: That's right.
So I'm talking about functions in one variable because I'm of
the opinion that before you do complicated thing you should
start first with a simple thing, And this is the
case of one variable.
So I wanted to explain everything in excruciating
detail for functions in one variable.
So now we'll have to talk about functions in
two variables, right?
But the point is that everything is going to work
in exactly the same way.
And I believe that you first have to understand what happens
for one variable and only then you'll be able to fully
understand what happens for functions in two variable.
So let me explain now what happens for functions
in two variable.
So what can I erase?
I want to keep this and I kind of want to keep everything.
Let me erase this.
So now we have a function in two variables, x and y.
So we have f of x,y.
So see, aren't you glad I haven't used y in the
previous calculations.
We had x and z.
Now we have x, y, and z.
So now we're going to have a graph of this
function, but in space.
And that's going to be a surface.
Let me try to draw this.
I will draw it in a slightly different way than last time
because I think it will be a little bit more clear,
but you tell me.
Like this.
I want to draw it like this.
Now we have a point an now I want to say that there
are two curves here.
Which actually last time I drew them with red and blue, but OK.
Let's not worry about this.
So this is one curve.

That's another curve.
Maybe I do it like this.
It's better like this.
Te class is never satisfied with these drawings.
I think this is better.
You see what I mean?
It's of a surface like this, but it's concave.
Last time I drew it convex like this and now it's concave.
I think it's a little bit easier on the eyes so to speak.
So this of course, all lives in the three-dimensional space.
So there is a three-dimensional coordinate system
as usual, x, y, z.
And this corresponds to a particular point with
coordinates x zero and y zero.
You see?

And now I want to draw the analog of the tangent line.
So this yellow thing is the analog of the curve of the
graph of the function.
When I say yellow thing I mean of course, the whole thing.
The surface.
You see what I mean?
So now, before I had the tangent line because
it was a curve.
So for a curve it makes sense to talk about a tangent line.
But for a surface like this it makes more sense to talk
about the tangent plane.
It is two-dimensional so in your linear approximation
you should use a linear surface, but not a curve.
A linear surface, which is what we call a plane.
So in fact I brought something to illustrate this even more.
This should play the role.
I tried to find a basketball in the math department, but
finding a basketball in the math department is like-- well,
you complete the sentence.
It was difficult.
this is what I found.
So this is the surface and I just want to explain what
the tangent plane is.
The tangent plane is a plane-- let's say if I pick a point,
the tangent plane is a plane which is the closest
plane to this surface.
So that's the first point I want to make.
The second point I want to make is that tangent plane is going
to change when you change the point of contact.
If you're interested in the tangent plane at this
particular point that's the plane you have.
But if you want at this one, it's going to be this one.
So it's exactly the same thing as we had for
curves for tangent lines.
Tangent line depends on the choice of the point.
Let me draw this tangent line for you.
So you see to indicate, to give sort of more shape and volume--
well, not volume but really the two-dimensional aspect to this,
perspective to this, I drew these two curves.
What are these curves?
These are exactly the same curves that I drew last time.
This is the curve of intersection.
This is intersection with the plane which goes like this.
Which is parallel to the yz-plane.
So it's really the plane where we fix the x-coordinate.
x equal x zero.
You will see that what I'm doing now is very similar to
what I did last time, but there are small differences.
For example, the graph I was using was convex
in a different way.
Also, instead of x zero, y zero I used notation a,b and so on.
But otherwise it's very similar to what I did on Tuesday.
And so what I do is I can cut my plane, vertical
plane like this.
I can cut the graph by this vertical plane.
A vertical plane, which is parallel to yz.
And that's the curve I get.
This is a curve which lives on the graph, on the surface,
on the vase if you want.
So this is like part of the vase and this
lives on the vase.
Then there's the perpendicular one, which I get by cutting
it by a plane, which is parallel to the xz-plane.
That's this one.
The only thing is I confused them.
I think what I was trying to say is that this is the curve
you get by cutting with a vertical line parallel to zy.
And now this one is what I get by cutting with a
plane parallel to xz.
So this one, this curve is intersection with the
plane y equals y zero.
So the graph is complicated.
The graph is a surface, but on this graph for my point x zero,
y zero I have drawn two curves, which in some sense are
perpendicular to each other.
These are the curves of intersection of the graph
with two natural planes.
Two vertical planes.
One plane is parallel to the blackboard.
That's this one.
And the other one is parallel to this one, which
kind of has an angle.
Which really, since I'm just giving it a three dimensional
perspective, but I'm drawing it like this.
But in fact, you should realize that it's a plane which should
be perpendicular to the blackboard, this xz-plane.
So I get two curves.
Now curves is something I can handle because we now know
everything about curves.
We don't know everything about tangent lines to curves.

So these two curves have tangent lines for sure,
just like this curve has a tangent line at our point.
So let me draw two tangent lines.
One will be the tangent line to this curve of intersection and
the other one will be the tangent line to this
curve of intersection.
I'm kind of trying to separate them a little bit.
So that's why you see it as though there's some distance
between, but there shouldn't be.
Just to make it more visible.
And now this two planes, these two lines in fact they span a
plane, which looks like this.
That's the plane, which is kind of slightly
underneath the graph.
That's what I was illustrating when I put the vase here and
that would be the tangent plane.
So on this tangent plan I would have two lines.
These two lines, which are tangent to the intersection to
the curves on the graph which you get by intersecting
with the vertical planes.
You see what I mean?
Is this clear?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: That's right.
It's the tangent to that particular point.
Exactly.
If I take a different point it'll change.
So because you know, imagine that there is a sphere here.
So that would be tha tangent line, but if I take a different
one it will be like this.
It will get tilted in a different way.
And not just like this, but also like this.
In all possible ways.
So everything I say is relative to a point.
Relative to a particular point, which I now
call x zero, y zero.
So what I need to do is I need to write down the equation
for this tangent plane.
Just like I wrote down here the equation for this tangent line
and then I will say that this tangent line approximates the
graph in a very nice way if the function is differentiable.
And the equation for this tangent plane will be called
the differential of the function at this point.
So it will be exactly parallel to what we did in the
one variable case.
So the immediate task at hand is to write down the equation
for the tangent plane.
Fortunately we know everything we need to know about
equations for a plane.
And we're going to use it now.
Because we have previously studied the question of writing
down the equation of a plane when we know two vectors
which belong to this plane.
We know how to do it.
First of all, we know that to write down the equation of a
plane we need to know a normal vector to the plane
as well as a point.
And surely we have a point.
That's x zero, y zero, So we need a normal vector and we
could calculate a normal vector by taking cross product
of these two vectors.

Which go along these two lines.
And what are those two vectors?
Those two vectors are the tangent vectors to those
curves, which we can easily find from the one
variable calculation.
That' what we're going to do now.
Let me explain what-- actually it would be more like this way
because it will be this way and this way.
It will be in the direction of x and y rather than in
the opposite direction.

So we want to write down the equation for the tangent plane
to the graph at that point x zero, y zero and z zero, where
by the way z zero of course, is again f of x zero, y zero.
That's the value of the function.
So what is the equation?
We know that the equation is going to look like these, a
times x minus x zero plus b y minus y zero plus z times z
minus z zero equals zero.
Where a, b, c should be a normal vector.
How do I find a normal vector?
I take these two vectors.
Let's call them r1 and r2 and I think their cross product.
See so now you can actually apply the knowledge which we
have acquired earlier about equations for a plane.

Now we can appreciate why it was important to actually learn
these techniques about you know, planes and lines and
so on and cross product.
So we have to figure out what r1 and r2 are.
So lot let me talk about r1.
But see r1 is something which I'll find by using a curve.
This curve is obtained, I recall, by intersecting
the graph with the plane y equals y zero.
In other words, it's what I called on Tuesday, freezing
the second variable.
Intersecting with this plane means that we freeze the
second variable, y. y is fixed it's y zero.
So effectively the problem which three-dimensional because
we have x, y, and z becomes two-dimensional because only
the unfrozen variables participate.
Namely x and z so that's why now I'm going to draw that
curve on the x and y plane.
And it's going to look like this.
Where is my yellow-- I lost my yellow chalk.
Oh yes, I know.
Here it is.
Just for consistency I want to draw it with yellow, so it's
going to look like this.
Because I'm looking at it now from that angle so that x goes
this way and z goes this way.
So that this curve, this is the curve I'm drawing,
it's decreasing.
That's why I draw it like this.
And this is the tangent vector, which I'm interested in.
This is my r1.
It's going to be a tangent vector.
We have to find a formula for this r1, right?
But we know how to find tangent vectors to parametric curves.
How to do this?
Well first of all, what is this curve.
This curve is z equals f of x, y zero.
We have frozen y.
We have just set y equals y zero.
So this function which by the way last time I
used to indicate it by writing this is red.
So the function effectively becomes a function in
one variable only.
Namely, x.
Let me call this g of x because the function's one variable
only so let's just call it g of x.
So this is a graph of the function z equals g of x.
And by the way, this is the notation I used on Tuesday.
Now I want to convert this into parametric form because we
learned how to do tangent vectors when we have
parametric form.
So let's do parametric form, but for graphs it's very easy.
parametric form is x equals t and z equals g of t.
So I have this vector r of t, which is t g of t.
That's the vector value function which corresponds
to this parametric curve.
Now I know that this vector r, which is a tangent vector, r1,
is just a derivative of this.

Is just r prime at x zero.
My point corresponds to t equals x zero.
This is our point.
So this is the formula we've learned before, which
now comes very useful.
Namely, the formula for the tangent vector to
a parametric curve.
I have converted my curve into parametric curve in two
variables with this parameterization and now I can
use this and what do I find?
The first derivative is 1.

This is a prime.
The prime means derivative with respect to t.
I'm just recalling.
Derivative with respect to the t variable.
So the derivative of t is 1.
And the derivative of g of t is g prime of t.
But t is x zero, so it's g prime of x zero.
Now remember we agreed that the derivative of g at x zero is
the first partial derivative of f.
Because that's how we define the first partial derivative.
First partial derivative was defined by taking that function
g which we get by freezing y and taking the derivative.
So what this is is just 1 and f sub x at x zero, y zero.

This is what I needed.
I have found our 1.
This is not exactly through because our 1 is a vector in
three space and now I have done the calculation in two space
where I kind of ignored the y variable.
But in fact, the y variable should be here.
It's actually not like this, but it's going into the
blackboard to comply with the rule.
What I've found is this vector in two space on the plane, but
if I want the corresponding vector in three space I have to
also remember the y-coordinate and the y-coordinate
will be zero.
So it will be in between these two.
So the true vector r 1.
The true vector r in and this formula is just 1, zero, and
f sub x at zero, y zero.

Why like this?
Because I found these guys 1 and f sub x, but to have a true
vector in three space I also have to give it a third
coordinate which corresponds to y.
This is a coordinate which corresponds to x and this
is a coordinate which corresponds to z.
I need one more which corresponds to y, but it is
zero because everything is happening on the plane which is
perpendicular to the y-axis.
So that's r1.
Now you need to calculate r2.
Of course, you can guess what the answer is.
The point is that now zero will migrate here.
Because now everything will be happening on a different
plane, which is the yz-plane.
So the x-coordinate will have no role at all.
The y-coordinate will now play the role of x, so this will
be 1 and this will f sub y.
It's exactly the same calculation except I should
draw it on the plane yz and I should call it r2 and I should
do the same parameterization and the same calculation.
So I'm not doing it just to save time.
So that's the answer I get.
So we are almost there because now all we need to do is to
find the normal vector to our plane, which as we already
discussed is the cross product.
So finally we get the use the cross product for something
really important.
So this is our application of the cross product.
So r1, r2 is going to be as we discussed and no doubt some of
you may have wondered why the hell are we doing
all this stuff?
But now you can appreciate that it is actually important.
For shorthand I'm not writing-- I'm dropping x zero, y zero,
but I will restore them later.
So what is this?
This is i times this which is minus fx times i.
Then minus j times fy times j and then k I just get this
matrix, so it's 1 plus 2.
Does everybody agree?
We have found a normal vector to our plane.
That's the one.
Normal vector is perpendicular to this plane.
Or if you want, like this.
So I'll say this is r2, this is r1 and this is
the perpendicular vector.
The normal vector.
That's it.
We're done because now we can write down the formula
for the tangent plane.
So the formula for the tangent plane becomes minus f sub x,
then we get minus f sub y plus z minus.

And now I just want to rewrite this in a nicer way.
z minus z zero is equal to f subx of x zero, y zero times x
minus x zero plus f sub y x zero, y zero times
y minus y zero.
That's the equation of the tangent plane.
And of course you should compare it to the equation of--
I have erased it, but I want to write it again-- in the
one-dimensional case the equation was f prime of x
zero times x minus x zero.
So it's totally analogous.
z minus z zero here was the derivative time x minus x zero.
There was only one variable therefore only one derivative.
There was no choice.
Now there are two different derivatives and both
of them show up.
This is the derivative with respect to x or partial
derivative with respect to x times the increment in x and
this is a partial derivative with respect to y times
the increment in y.
So this is a linear function.
The right-hand side is a linear function whose graph gives
you that tangent plane.
And this function is called the differential of
f at this point.

Yes?
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: This is abc.

This is what I got.
Then I wrote the equation of the tangent plane over there by
using these three coefficients.
I mean, if you want I put one here.
[UNINTELLIGIBLE]
We have two notations, two different
notations for vectors.
One is with i, j, k and the other one with
three components.
When I have a vector I call this a,b,c.
If this is a normal vector then the equation of the plane is a
times x minus x zero plus b times y minus y zero plus
c times z minus z zero.
This is what I wrote, second formula from the bottom.
STUDENT: [UNINTELLIGIBLE]
PROFESSOR: You see that now?
Second formula from the bottom is the equation for the plane,
which is perpendicular to this vector.
OK, any other questions?
All right.
So now we're almost there.

So the right-hand side now is called a differential.

The differential of f x, y at that point x zero, y zero.
It is a linear function.
It is a linear function, which approximates well our function
provided that our function is differentiable.
In fact, f of x,y is called differentiable
at x zero, y zero.
Compare it to the one-dimensional case.

If delta f, which is-- well let's write the formula.
If f of x, y minus f of x zero, y zero is equal to this linear
function up to a small correction term.
So fx of x zero, y zero times x minus x zero plus f sub y at x
zero, y zero times y minus y zero.
The correction term I will now write as sum of two terms.
Before I had a correction term which was something small times
x minus x zero, but now I will write it something small times
x minus x zero plus something small times y minus y zero.
So if you like to think in terms of Taylor series you can
imagine that I'm writing a Taylor series expansion for
my function where I first write the constant term.
Then I write the linear term, which now has two sums.
One coming from the first partial derivative.
The other one coming from the second partial derivative and
then I will have quadratic terms and cubic terms which
will involve all the mixed partial derivatives
of higher order.
But then actually I don't want to specify them other than to
say that they altogether combined have this shape.
Something which is negligible compared to these two terms.
Something that can be viewed as an error, as a
negligible error term.
So epsilon 1 and epsilon 2 have to go to zero as x, y
converges to x zero, y zero.
So in fact, I could stop here, but I want to explain the
notation again because I think all of this sounds great until
you encounter the notation and then it could become a
little bit confusing.
But now we can very easily unravel the notation as well
because we now have a very good example of that in
the one-dimensional case.
So here's how the notation works for this.

Just as in the one-dimensional case I will have dx but to
really do justice to dx I have to keep track of the reference
point, which now is x zero, y zero.
And so dx is x minus x zero.
That results to dy, which is y minus y zero.
And there is also dz or df, which is z minus z zero.
So this formula which I have framed can be written as df
equals f sub x dx plus f sub y dy.
More precisely you have to put everywhere x zero, y zero.
Here x zero, y zero.
Here x zero, y zero, here x zero, y zero and
here x zero, y zero.
And if yuo do that and you remember what this means
this will be identical.
This will become identical to this.
Nothing more, nothing less.
But in other words, this formula makes sense for a given
reference point x zero, y zero.
And it's nothing but the equation for the tangent plane.
But after this just to simplify the formulas you drop
all the indices.
You say, OK, let's just forget about this.
Let's not forget, let's remember it, but let's not
write it on the board or let's not write it
on a piece of paper.
So the formula really becomes just df equals fxdx plus fydy.
So if on the homework you are asked to compute the
differential of a function that's what you're going do.
You're going to take the first partial derivative times dx
plus the second partial derivative times dy.
So nothing could be easier than that.
You just take two partial derivatives, right?
What time is it?
Oh, it's 5:00.

But now I have explained to you what the meaning of this
formula is, so we'll talk about it more during the
review on Tuesday.