Uploaded by UCBerkeley on 17.11.2009

Transcript:

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.

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.