Let's learn a little bit about the gradient, and we'll use the
same function we've been using since we're pretty familiar
with it's graph and it's partial derivatives.
So let's say that f of x y is equal to x squared plus x
times y plus y squared.
Now we're going to take the gradient of this, and then I'll
give you a little bit of intuition on what
a gradient is.
We're going to take the gradient in two-dimensional
space, and you'll see what that means in a second.
You can extend this to any dimension.
So the gradient that we use this vector differential
operator-- this upside down triangle-- the gradient of f is
equal to the partial of f in the x direction times the unit
vector i-- which is the unit vector in the x direction so it
gives the magnitude of the slope and the x direction, and
then it points the vector in that direction-- plus the
partial with respect to y in the j direction.
And obviously if there's a function of more variables you
could keep going and you'd multiply the magnitude of the
derivative in each of those dimension's directions times
the unit vector in that dimension.
Well what does this mean?
And another way to view it, you could have also written this as
the gradient of f is equal to the partial with respect to x
of f of x y and then all of that in the i direction.
This is just different notation.
Plus the partial with respect to y times f of x y.
Not times, operated on f of x y.
So the partial of f of x y with respect to y times
the j unit vector.
And because of this notation, a lot of people view this delta
operator-- and actually it's pretty consistent when we learn
about the divergence and the curl-- a lot of people view
this delta operator, they actually define it.
They say that is equal to the partial with respect to x in
the i direction plus the partial with respect to
y in the j direction.
And then if you wanted to do it into three space-- that's not
going to apply to our problem because we only care about
two dimensions right now.
And you'll see what I mean.
Because even though we're applying it to a
three-dimensional surface, but then you can do the partial
with respect to z times the k vector.
And then you could extend it to n dimensions, but it becomes
very hard to visualize.
Actually it becomes hard to visualize beyond what
we're doing right now.
So what does this mean?
Well before I go into what it means, let's actually calculate
it for this and then I'll show you what it means.
Let's actually calculate the gradient of f.
So the gradient of our particular function, it's a
partial with respect to x.
So we figure that out.
That's 2x plus y, and then that doesn't matter anymore.
So it's 2x plus y.
That's the partial of this function with respect to x.
And we multiply that in the i direction, or
in the x direction.
Plus the partial with respect to y, and this is 2y plus x.
And we calculated this in the two videos on the
partial derivatives.
And we multiply that in the j direction.
So what does that look like?
Well what are each of the components of this vector?
What is 2x plus y in the i direction?
What does that vector look like?
What is 2y plus x in the j direction look like?
And this I have graphed.
So this is the same surface we've been working with, but
now I've plotted these points right here.
These are just points that the software has picked to actually
display these vectors.
And if you look at it, this is the x-axis.
And I can rotate this.
I can pull it down a little bit.
And then I can spin it around.
I think that's pretty neat.
But anyway.
This parallel to this line is the vector in
the x-axis, right?
So at this point I've actually calculated the gradient.
This vector says, what is the magnitude of this vector is the
partial derivative of the function, or the partial
derivative of the surface, or the partial derivative
of z with respect to x.
And it's direction, it just goes in the x direction,
because it's that times the i unit vector.
So this vector is the partial derivative of z with respect
to x times the i unit vector at that point.
So we just calculated the partial derivative
at that point.
It gave this length.
And then the direction is just the i unit vector, or in the
direction of increasing x.
Similarly, this vector right here-- I hope you can see it.
Let me see if I can actually zoom in a little bit.
That might be useful.
Zoom in, there you go.
And I wanted to see the axes, which you can't see now that
I'm zoomed in, but you took my word this was the x direction.
Now this is the y direction, or the same direction
as the j unit vector.
The j unit vector goes in the same direction as y.
And the magnitude is determined by the partial derivative of z
with respect to y at that point.
And that's the magnitude.
And we saw there was some symmetry, so the magnitude of
this vector is the same as the magnitude of this vector.
And then when you add the two vectors together,
you get this vector.
And notice that none of these vectors have any dimensions
in the z dimension.
They're all kind of giving you directions in the x y plane.
And why is that interesting?
Well the gradient-- and this is the intuition-- the gradient
tells you the direction in the x y plane you should travel
in order to get the maximum slope in the z dimension.
Or another way to view it.
Remember the partial derivative with respect to x said what is
the slope in the x direction.
The partial derivative with respect to y said what is the
slope in the y direction.
But you could take the partial derivative with respect to any
direction, and the gradient gives you the direction in
which the slope is the largest.
So let me zoom out a little bit because I want you to see the
actual axes There's the axes So all this says if I were to
go in this direction, I get the maximum slope.
So if I go in this direction, my z goes up like that.
Let me see if I can rotate this a little bit.
I don't want to scale it anymore.
Let me do the rotated.
See if I can show that to you.
So if I go in that direction in the x y plane, I get maximum z.
If I go in that direction I get a maximum upward slope.
That's what the gradient tells you, how do you get the
maximum upward slope.
And if you were to take any closed line where the z is
constant-- well actually I don't want to get into that too
much, but this gradient will actually be normal to any, or
it'll be perpendicular to any curve where z is constant.
I don't want to get too involved with that right
now, so let's go back here.
Maybe it's more clear if we look from below the graph.
So if you go to this point right here, this is the
magnitude of this vector, shows what is the slope
in the x direction.
The magnitude of this vector is what is the slope
in the y direction.
And then when you add them together, you get the gradient.
And the gradient says, well, if I travel in this direction in
the x y plane-- notice that none of these have any z's.
The plane defined by all of these vectors is all
flat in the z dimension.
But if I would have traveled this direction in the x y
plane, then I will get the maximum increase in z
per unit that I travel.
And let's actually show you what it looks like
in the x y plane.
So if I just go head on-- so I'm above the graph looking
straight down at the graph, and then the colors just show kind
of you know where we are-- if I travel in this direction in
the x y plane, I get my maximum increase in z.
If I'm here-- notice here the x component of the gradient is
much larger than the y component.
So I need to travel a little bit more in the x direction,
and I'll get the maximum change in z if I travel there.
Another way to think about it.
If I'm on a hill, the gradient of that surface will tell you
at any point what direction you need to travel in to go
up the hill fastest.
Or the direction which the steepness of the
hill is maximum.
This is more of a bowl as opposed to a hill, but anyway.
I'll just rotate it around.
Hopefully that makes some sense.
And I want you to think about it a little bit more.
And we'll do a few more problems where we just
calculate gradients, just because I think it's useful
to get the mechanics.
But in my opinion at least the intuition is a little bit
harder to get your head around than the actual mechanics,
but once you get it, it makes a lot of sense.
I will see you in the next video.