jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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5DUQ3-Y_gX4	So we'll call it a vector space as well.
5DUQ3-Y_gX4	STEVEN G. JOHNSON: Yeah.
5DUQ3-Y_gX4	But it's incredibly useful, though.
5DUQ3-Y_gX4	Because very often, especially in physical sciences,
5DUQ3-Y_gX4	you have something where conceptually you're
5DUQ3-Y_gX4	solving for a functions.
5DUQ3-Y_gX4	So you're solving for the fluid flow or something
5DUQ3-Y_gX4	around an airplane wing.
5DUQ3-Y_gX4	And what you want is that then take that fluid flow
5DUQ3-Y_gX4	and compute the drag on the airplane wing.
5DUQ3-Y_gX4	And then in order to optimize it,
5DUQ3-Y_gX4	you want the derivative of the drag with respect
5DUQ3-Y_gX4	to that flow field, with respect to the function,
5DUQ3-Y_gX4	or with respect to the shape of the airplane, which
5DUQ3-Y_gX4	is a function.
5DUQ3-Y_gX4	So it's very, very nice to be able to take derivatives
5DUQ3-Y_gX4	connected to functions and work with vector functions
5DUQ3-Y_gX4	as vector spaces.
5DUQ3-Y_gX4	And very soon we're going to be able to do that
5DUQ3-Y_gX4	with this notion of a derivative.
5DUQ3-Y_gX4	Because we're going to be able to define linear operators,
5DUQ3-Y_gX4	functions that act on functions.
5DUQ3-Y_gX4	And linear operators are functions.
5DUQ3-Y_gX4	But that's getting a bit too far ahead of ourselves.
5DUQ3-Y_gX4	OK.
5DUQ3-Y_gX4	So the point is that the 18.01-- so far, we
5DUQ3-Y_gX4	haven't done any derivatives more than 18.01, at least
5DUQ3-Y_gX4	in my half.
5DUQ3-Y_gX4	Alan went a bit further.
5DUQ3-Y_gX4	But already we can start to see, hopefully, how
5DUQ3-Y_gX4	this is going to generalize.
5DUQ3-Y_gX4	So if you have a function f of x and you
5DUQ3-Y_gX4	make a small change in the input, delta x,
5DUQ3-Y_gX4	and you ask for the small change in the output
5DUQ3-Y_gX4	to first order, which we can denote with this d notation,
5DUQ3-Y_gX4	the derivative is the linear operator
5DUQ3-Y_gX4	that gives us that, the linearization
5DUQ3-Y_gX4	of that function for a small change in the input.
5DUQ3-Y_gX4	And that is exactly equivalent to what you learned in 18.01.
5DUQ3-Y_gX4	But it's going to be easier now to generalize this
5DUQ3-Y_gX4	to other kinds of inputs and other kinds of outputs,
5DUQ3-Y_gX4	where, in 18.01, we move this to the side where we take df, dx.
5DUQ3-Y_gX4	We divide them.
5DUQ3-Y_gX4	For numbers, that's fine.
5DUQ3-Y_gX4	For other kinds of things, that becomes a little bit weirder
5DUQ3-Y_gX4	to talk about.
5DUQ3-Y_gX4	Of course, you could define it as notation.
5DUQ3-Y_gX4	But I think it's a lot clearer if you
5DUQ3-Y_gX4	think of it in this sense once you start generalizing
5DUQ3-Y_gX4	to other kinds of objects.
5DUQ3-Y_gX4	So with that said, let's do that.
5DUQ3-Y_gX4	Now, let's revisit 18.02.
5DUQ3-Y_gX4	And let me do it in two parts.
5DUQ3-Y_gX4	So part one is going to be functions--
5DUQ3-Y_gX4	the first thing you usually do in 18.02,
5DUQ3-Y_gX4	which is functions that take a vector in or multiple variables
5DUQ3-Y_gX4	in.
5DUQ3-Y_gX4	But we'll think of it as a vector in and a scalar out.
5DUQ3-Y_gX4	So we're going to have a scalar.
5DUQ3-Y_gX4	My output is blue, right?
5DUQ3-Y_gX4	Yes.
5DUQ3-Y_gX4	We'll keep the same color scheme, good.
5DUQ3-Y_gX4	So we're going to have a scalar function f of a vector input x.
5DUQ3-Y_gX4	And I'll put a little vector sign above it.
5DUQ3-Y_gX4	I won't always do that, but it's nice to be clear sometimes,
5DUQ3-Y_gX4	which is a vector, which is a scalar.
5DUQ3-Y_gX4	So x is going to live in our m.
5DUQ3-Y_gX4	So this is going to be an m component column vector.
5DUQ3-Y_gX4	OK.
5DUQ3-Y_gX4	And what we want to do is imagine
5DUQ3-Y_gX4	what happens to the output when you change
5DUQ3-Y_gX4	the input by a little bit.
5DUQ3-Y_gX4	So we're going to take f of x plus dx.
5DUQ3-Y_gX4	Think of this as a really small change.
5DUQ3-Y_gX4	It's infinitesimal.
5DUQ3-Y_gX4	We're going to drop anything that
5DUQ3-Y_gX4	goes like dx squared or anything like that, any higher terms.
5DUQ3-Y_gX4	Let me just move it to [INAUDIBLE]..
5DUQ3-Y_gX4	It's black-- minus f of x.
5DUQ3-Y_gX4	And we're going to define this as f prime of x dx.
5DUQ3-Y_gX4	So we wanted to note we have an arbitrary change in the inputs.
5DUQ3-Y_gX4	dx is an arbitrary, very, very small vector.
5DUQ3-Y_gX4	And we want to ask, what's the change in the output
5DUQ3-Y_gX4	differential, df?
5DUQ3-Y_gX4	And the answer is going to be that this
5DUQ3-Y_gX4	is going to be, for a very small dx,
5DUQ3-Y_gX4	we can approximate this by a linear operator on dx.
5DUQ3-Y_gX4	So this is going to be a linear operator, always
5DUQ3-Y_gX4	going to be a linear operator.
5DUQ3-Y_gX4	And what is that linear operator do?
5DUQ3-Y_gX4	This one takes a vector in and gives you a scalar.
5DUQ3-Y_gX4	So this has to equal a--
5DUQ3-Y_gX4	df is a scalar, but dx is a vector.
5DUQ3-Y_gX4	So what this has to be is it has to be kind of a row vector.
5DUQ3-Y_gX4	You can think of it more as a one-row matrix.
5DUQ3-Y_gX4	Or there's fancier names for this, like covector
5DUQ3-Y_gX4	or dual vector.
5DUQ3-Y_gX4	We won't really use that.
5DUQ3-Y_gX4	I just want to throw them out there.
5DUQ3-Y_gX4	So if you want to take a vector in and take a vector out,

5DUQ3-Y_gX4

So we'll call it a vector space as well.

5DUQ3-Y_gX4

STEVEN G. JOHNSON: Yeah.

5DUQ3-Y_gX4

But it's incredibly useful, though.

5DUQ3-Y_gX4

Because very often, especially in physical sciences,

5DUQ3-Y_gX4

you have something where conceptually you're

5DUQ3-Y_gX4

solving for a functions.

5DUQ3-Y_gX4

So you're solving for the fluid flow or something

5DUQ3-Y_gX4

around an airplane wing.

5DUQ3-Y_gX4

And what you want is that then take that fluid flow

5DUQ3-Y_gX4

and compute the drag on the airplane wing.

5DUQ3-Y_gX4

And then in order to optimize it,

5DUQ3-Y_gX4

you want the derivative of the drag with respect

5DUQ3-Y_gX4

to that flow field, with respect to the function,

5DUQ3-Y_gX4

or with respect to the shape of the airplane, which

5DUQ3-Y_gX4

is a function.

5DUQ3-Y_gX4

So it's very, very nice to be able to take derivatives

5DUQ3-Y_gX4

connected to functions and work with vector functions

5DUQ3-Y_gX4

as vector spaces.

5DUQ3-Y_gX4

And very soon we're going to be able to do that

5DUQ3-Y_gX4

with this notion of a derivative.

5DUQ3-Y_gX4

Because we're going to be able to define linear operators,

5DUQ3-Y_gX4

functions that act on functions.

5DUQ3-Y_gX4

And linear operators are functions.

5DUQ3-Y_gX4

But that's getting a bit too far ahead of ourselves.

5DUQ3-Y_gX4

OK.

5DUQ3-Y_gX4

So the point is that the 18.01-- so far, we

5DUQ3-Y_gX4

haven't done any derivatives more than 18.01, at least

5DUQ3-Y_gX4

in my half.

5DUQ3-Y_gX4

Alan went a bit further.

5DUQ3-Y_gX4

But already we can start to see, hopefully, how

5DUQ3-Y_gX4

this is going to generalize.

5DUQ3-Y_gX4

So if you have a function f of x and you

5DUQ3-Y_gX4

make a small change in the input, delta x,

5DUQ3-Y_gX4

and you ask for the small change in the output

5DUQ3-Y_gX4

to first order, which we can denote with this d notation,

5DUQ3-Y_gX4

the derivative is the linear operator

5DUQ3-Y_gX4

that gives us that, the linearization

5DUQ3-Y_gX4

of that function for a small change in the input.

5DUQ3-Y_gX4

And that is exactly equivalent to what you learned in 18.01.

5DUQ3-Y_gX4

But it's going to be easier now to generalize this

5DUQ3-Y_gX4

to other kinds of inputs and other kinds of outputs,

5DUQ3-Y_gX4

where, in 18.01, we move this to the side where we take df, dx.

5DUQ3-Y_gX4

We divide them.

5DUQ3-Y_gX4

For numbers, that's fine.

5DUQ3-Y_gX4

For other kinds of things, that becomes a little bit weirder

5DUQ3-Y_gX4

to talk about.

5DUQ3-Y_gX4

Of course, you could define it as notation.

5DUQ3-Y_gX4

But I think it's a lot clearer if you

5DUQ3-Y_gX4

think of it in this sense once you start generalizing

5DUQ3-Y_gX4

to other kinds of objects.

5DUQ3-Y_gX4

So with that said, let's do that.

5DUQ3-Y_gX4

Now, let's revisit 18.02.

5DUQ3-Y_gX4

And let me do it in two parts.

5DUQ3-Y_gX4

So part one is going to be functions--

5DUQ3-Y_gX4

the first thing you usually do in 18.02,

5DUQ3-Y_gX4

which is functions that take a vector in or multiple variables

5DUQ3-Y_gX4

in.

5DUQ3-Y_gX4

But we'll think of it as a vector in and a scalar out.

5DUQ3-Y_gX4

So we're going to have a scalar.

5DUQ3-Y_gX4

My output is blue, right?

5DUQ3-Y_gX4

Yes.

5DUQ3-Y_gX4

We'll keep the same color scheme, good.

5DUQ3-Y_gX4

So we're going to have a scalar function f of a vector input x.

5DUQ3-Y_gX4

And I'll put a little vector sign above it.

5DUQ3-Y_gX4

I won't always do that, but it's nice to be clear sometimes,

5DUQ3-Y_gX4

which is a vector, which is a scalar.

5DUQ3-Y_gX4

So x is going to live in our m.

5DUQ3-Y_gX4

So this is going to be an m component column vector.

5DUQ3-Y_gX4

OK.

5DUQ3-Y_gX4

And what we want to do is imagine

5DUQ3-Y_gX4

what happens to the output when you change

5DUQ3-Y_gX4

the input by a little bit.

5DUQ3-Y_gX4

So we're going to take f of x plus dx.

5DUQ3-Y_gX4

Think of this as a really small change.

5DUQ3-Y_gX4

It's infinitesimal.

5DUQ3-Y_gX4

We're going to drop anything that

5DUQ3-Y_gX4

goes like dx squared or anything like that, any higher terms.

5DUQ3-Y_gX4

Let me just move it to [INAUDIBLE]..

5DUQ3-Y_gX4

It's black-- minus f of x.

5DUQ3-Y_gX4

And we're going to define this as f prime of x dx.

5DUQ3-Y_gX4

So we wanted to note we have an arbitrary change in the inputs.

5DUQ3-Y_gX4

dx is an arbitrary, very, very small vector.

5DUQ3-Y_gX4

And we want to ask, what's the change in the output

5DUQ3-Y_gX4

differential, df?

5DUQ3-Y_gX4

And the answer is going to be that this

5DUQ3-Y_gX4

is going to be, for a very small dx,

5DUQ3-Y_gX4

we can approximate this by a linear operator on dx.

5DUQ3-Y_gX4

So this is going to be a linear operator, always

5DUQ3-Y_gX4

going to be a linear operator.

5DUQ3-Y_gX4

And what is that linear operator do?

5DUQ3-Y_gX4

This one takes a vector in and gives you a scalar.

5DUQ3-Y_gX4

So this has to equal a--

5DUQ3-Y_gX4

df is a scalar, but dx is a vector.

5DUQ3-Y_gX4

So what this has to be is it has to be kind of a row vector.

5DUQ3-Y_gX4

You can think of it more as a one-row matrix.

5DUQ3-Y_gX4

Or there's fancier names for this, like covector

5DUQ3-Y_gX4

or dual vector.

5DUQ3-Y_gX4

We won't really use that.

5DUQ3-Y_gX4

I just want to throw them out there.

5DUQ3-Y_gX4

So if you want to take a vector in and take a vector out,