- [Voiceover] So here I'd like to talk
- [Voiceover] So here I'd like to talk

about what the gradient means
about what the gradient means

in the context of the graph of a function.
in the context of the graph of a function.

So in the last video, I defined the gradient,
So in the last video, I defined the gradient,

but let me just take a function here.
but let me just take a function here.

And the one that I had graphed is x-squared plus y-squared,
And the one that I had graphed is x-squared plus y-squared,

f of x, y, equals x-squared plus y-squared.
f of x, y, equals x-squared plus y-squared.

So two-dimensional input, which we think about
So two-dimensional input, which we think about

as being kind of the xy-plane,
as being kind of the xy-plane,

and then a one-dimensional output that's just the height
and then a one-dimensional output that's just the height

of the graph above that plane.
of the graph above that plane.

And I defined in the last video, the gradient,
And I defined in the last video, the gradient,

to be a certain operator.
to be a certain operator.

An operator just means you've taken a function
An operator just means you've taken a function

and you output another function,
and you output another function,

and we use this upside down triangle.
and we use this upside down triangle.

So it gives you another function that's also of x and y,
So it gives you another function that's also of x and y,

but this time it has a vector valued output.
but this time it has a vector valued output.

And the two components of its output
And the two components of its output

are the partial derivatives, partial of f with respect to x,
are the partial derivatives, partial of f with respect to x,

and the partial of f with respect to y.
and the partial of f with respect to y.

So for a function like this, we actually evaluated it.
So for a function like this, we actually evaluated it.

Let's take a look.
Let's take a look.

The first one is taking the derivative
The first one is taking the derivative

with respect to x, so it looks at x and says,
with respect to x, so it looks at x and says,

"You look like a variable to me.
"You look like a variable to me.

"I'm gonna take your derivative, your 2x."
"I'm gonna take your derivative, your 2x."

2x, but the y component just looks like a constant
2x, but the y component just looks like a constant

as far as the partial x is concerned.
as far as the partial x is concerned.

And the derivative of a constant is zero.
And the derivative of a constant is zero.

But when you take the partial derivative
But when you take the partial derivative

with respect to y, things reverse.
with respect to y, things reverse.

It looks at the x component and says,
It looks at the x component and says,

"You look like a constant.
"You look like a constant.

"Your derivative is zero."
"Your derivative is zero."

But it looks at the y component and says,
But it looks at the y component and says,

"Ah, you look like a variable.
"Ah, you look like a variable.

"Your derivative is 2y."
"Your derivative is 2y."

So this ultimate function we get, the gradient,
So this ultimate function we get, the gradient,

which takes in a two variable input, xy,
which takes in a two variable input, xy,

some point on this plane,
some point on this plane,

but outputs a vector,
but outputs a vector,

can nicely be visualized with a vector field.
can nicely be visualized with a vector field.

And I have another video on vector fields
And I have another video on vector fields

if you're feeling unsure.
if you're feeling unsure.

But I want you to just take a moment, pause if you need to,
But I want you to just take a moment, pause if you need to,

and guess, or try to think about what vector field
and guess, or try to think about what vector field

this will look like.
this will look like.

I'm gonna show you in a moment,
I'm gonna show you in a moment,

but what's it gonna look like,
but what's it gonna look like,

the one that takes in xy and outputs 2x, 2y?
the one that takes in xy and outputs 2x, 2y?

Alright, have you done it,
Alright, have you done it,

have you thought about what it's gonna look like?
have you thought about what it's gonna look like?

Here's what we get.
Here's what we get.

It's a bunch of vectors pointing away from the origin.
It's a bunch of vectors pointing away from the origin.

And the basic reason for that
And the basic reason for that

is that if you have any given input point,
is that if you have any given input point,

and say it's got coordinates x, y,
and say it's got coordinates x, y,

then the vector that that input point represents
then the vector that that input point represents

would, you know, if it went from the origin here,
would, you know, if it went from the origin here,

that's what that vector looks like,
that's what that vector looks like,

but the output is two times that vector.
but the output is two times that vector.

So when we attach that output to the original point,
So when we attach that output to the original point,

we get something that's two times that original vector
we get something that's two times that original vector

but pointing in the same direction,
but pointing in the same direction,

which is away from the origin.
which is away from the origin.

We kind of drew it poorly here.
We kind of drew it poorly here.

And of course, when we draw vector fields,
And of course, when we draw vector fields,

we don't usually draw them to scale.
we don't usually draw them to scale.

You scale them down
You scale them down

just so that things don't look as cluttered.
just so that things don't look as cluttered.

That's why everything here, they all look the same length,
That's why everything here, they all look the same length,

but color indicates length.
but color indicates length.

So you should think of these red guys as being really long,
So you should think of these red guys as being really long,

the blue ones as being really short.
the blue ones as being really short.

So what does this have to do with the graph of the function?
So what does this have to do with the graph of the function?

There's actually a really cool interpretation.
There's actually a really cool interpretation.

So imagine that you are just walking along this graph,
So imagine that you are just walking along this graph,

you know, you're a hiker and this is a mountain.
you know, you're a hiker and this is a mountain.

And you picture yourself at any old point on this graph,
And you picture yourself at any old point on this graph,

let's say, what color should I use?
let's say, what color should I use?

Let's say you're sitting at a point like this.
Let's say you're sitting at a point like this.

And you say, "What direction should I walk
And you say, "What direction should I walk

"to increase my altitude the fastest?"
"to increase my altitude the fastest?"

You want to get uphill as quickly as possible.
You want to get uphill as quickly as possible.

And from that point,
And from that point,

you might walk what looks like straight up there.
you might walk what looks like straight up there.

You certainly wouldn't go around,
You certainly wouldn't go around,

and this way you wouldn't go down.
and this way you wouldn't go down.

So you might go straight up there.
So you might go straight up there.

And if you project your point down onto the input space,
And if you project your point down onto the input space,

so this is the point above which you are,
so this is the point above which you are,

that vector, the one that's gonna get you going
that vector, the one that's gonna get you going

uphill the fastest, the direction you should walk.
uphill the fastest, the direction you should walk.

For this graph, it should kind of make sense,
For this graph, it should kind of make sense,

is directly away from the origin,
is directly away from the origin,

'cause here, I'll erase this
'cause here, I'll erase this

'cause once I start moving things, that won't stick.
'cause once I start moving things, that won't stick.

If you were to look at things from the very bottom,
If you were to look at things from the very bottom,

any point that you are on the mountain on the graph here
any point that you are on the mountain on the graph here

and when you want to increase the fastest,
and when you want to increase the fastest,

you should just go directly away from the origin
you should just go directly away from the origin

'cause that's when it's the steepest.
'cause that's when it's the steepest.

And all of these vectors are also pointing
And all of these vectors are also pointing

directly away from the origin.
directly away from the origin.

So people will say the gradient points in the direction
So people will say the gradient points in the direction

of steepest ascent, that might even be worth writing down.
of steepest ascent, that might even be worth writing down.

Direction of steepest ascent.
Direction of steepest ascent.

And let's just see what that looks like
And let's just see what that looks like

in the context of another example.
in the context of another example.

So I'll pull up another graph here,
So I'll pull up another graph here,

pull up another graph and its vector field.
pull up another graph and its vector field.

So this graph, it's all negative values,
So this graph, it's all negative values,

it's all below the xy-plane,
it's all below the xy-plane,

and it's got these two different peaks.
and it's got these two different peaks.

And I've also drawn the gradient field,
And I've also drawn the gradient field,

which is the word for the vector field
which is the word for the vector field

representing the gradient on top.
representing the gradient on top.

And you'll notice near the peak
And you'll notice near the peak

all of the vectors are pointing
all of the vectors are pointing

kind of in the uphill direction,
kind of in the uphill direction,

sort of telling you to go towards that peak in some way.
sort of telling you to go towards that peak in some way.

And as you get a feel around, you can see here,
And as you get a feel around, you can see here,

this very top one, like the point that it's stemming from
this very top one, like the point that it's stemming from

corresponds with something just a little bit shy
corresponds with something just a little bit shy

of the peak there.
of the peak there.

And everybody's telling you to go uphill.
And everybody's telling you to go uphill.

Each vector is telling you which way to walk
Each vector is telling you which way to walk

to increase the altitude on the graph the fastest.
to increase the altitude on the graph the fastest.

It's the direction of steepest ascent.
It's the direction of steepest ascent.

And that's what the direction means,
And that's what the direction means,

but what does the length mean?
but what does the length mean?

Well, if you take a look,
Well, if you take a look,

take a look at these red vectors here.
take a look at these red vectors here.

So red means that they should be considered very, very long.
So red means that they should be considered very, very long.

And the graph itself,
And the graph itself,

the point they correspond to on the graph
the point they correspond to on the graph

is just way off screen for us
is just way off screen for us

because this graph gets really steep
because this graph gets really steep

and really negative very fast.
and really negative very fast.

So the points these correspond to
So the points these correspond to

have really, really steep slopes
have really, really steep slopes

whereas these blue ones over here,
whereas these blue ones over here,

you know, it's kind of a relatively shallow slope.
you know, it's kind of a relatively shallow slope.

By the time you're getting to the peak,
By the time you're getting to the peak,

things start leveling off.
things start leveling off.

So the length of the gradient vector
So the length of the gradient vector

actually tells you the steepness
actually tells you the steepness

of that direction of steepest ascent.
of that direction of steepest ascent.

But one thing I want to point out here,
But one thing I want to point out here,

it doesn't really make sense immediately looking at it,
it doesn't really make sense immediately looking at it,

why just throwing the partial derivatives into a vector
why just throwing the partial derivatives into a vector

is gonna give you this direction of steepest ascent.
is gonna give you this direction of steepest ascent.

Ultimately it will.
Ultimately it will.

We're gonna talk through that
We're gonna talk through that

and I hope to make that connection pretty clear,
and I hope to make that connection pretty clear,

but unless you're some kind of intuitive genius,
but unless you're some kind of intuitive genius,

I don't think that connection is at all obvious at first.
I don't think that connection is at all obvious at first.

But you will see it in due time.
But you will see it in due time.

It's gonna require something
It's gonna require something

called the directional derivative.
called the directional derivative.

See you next video.
See you next video.