Welcome to a video
on the applications of the gradient.
We first mentioned the gradient when discussing
the directional derivative in the previous video.
In this video we'll take a look at specifically
what the gradient tells us given a function
in terms of two variables.
Remember the gradient of f is equal to the vector
in which the x and y components are formed
by the partial derivatives of f with respect to x
as we see here for the x component,
and then partial with respect to y
for the y component of the vector.
We refer to this in many applications,
we would like to know in what direction to move
so that f of x, y increases or decreases the most.
This direction is called the direction
of steepest ascent or steepest descent.
The gradient actually gives us this information.
So here are some properties of the gradient.
If the gradient of f of x, y is equal to the zero vector,
then the directional derivative of f of x, y
will also equal zero for all unit vectors.
For this video, we'll focus on the next two properties.
The direction of maximum increase of f is given
by the gradient of f of x, y.
And the value of the directional derivative at x, y
or the rate of increase is given by the magnitude
of the gradient of f of x, y.
Connected to this, the direction of maximum decrease
of f is given by the opposite of the gradient of f of x.
And it also follows the maximum decrease
of the directional derivative of f of x, y
or rate of decrease is given by the opposite
of the magnitude of the gradient of f of x, y.
So what this means is for a snowboarder,
the opposite of the gradient of f of x, y
would indicate the direction of steepest descent
or steepest decline and on a hot metal plate,
the gradient of f of x, y gives us the direction
of greatest temperature increase.
We do need to be aware, though,
that the gradient will change as soon as we move
from a given point.
Let's look at a couple of problems.
Here we want to determine the gradient of f of x, y,
and then determine the direction of steepest ascent
from the point one, one, zero,
and then determine this rate of increase.
So the direction of steepest ascent will come
from the gradient of f evaluated at the point one, one.
So, let's go ahead and find the gradient of f of x, y.
Remember this implies that f of x, y
is equal to natural log xy squared.
So the gradient will be equal to the vector,
or the x component is equal
to the partial of f with respect to x.
So we'll have one over xy squared
times the derivative of xy squared with respects to x,
that'll give us y squared.
And the y component will be the partial derivative of f
with respects to y.
So we'll have one over xy squared
times, and then u prime would be two xy.
Let's go ahead and simplify this.
We have
one over x, with the x component,
and this would be two over y for the y component.
So to determine the gradient at the point one, one, zero,
we need to evaluate this when x is one and y is one.
This will give us the direction of steepest ascent
from the given point.
So we'll have one over one that's one, two.
So this tells us the direction
of steepest descent,
and then the actual rate of increase is equal
to the magnitude of the gradient at the point one, one.
That's going to give us the square root of one squared,
that's one, plus two squared, that's four
or the square root of five, which is approximately 2.24.
Before we look at this graphically,
remember the opposite direction of this gradient
would tell us the direction of steepest descent,
and the opposite of this magnitude would tell us the rate
of decrease as well.
And we'll show both graphically.
So what we're seeing here is the surface in gray
and the point one, one, zero is this purple point.
And I plotted the gradient
as the red vector here.
And again, remember it's pointing in the direction
of maximum ascent or largest increase.
So from this point, if we move in this direction,
this would be the direction of maximum ascent
and the slope of this line is equal to the magnitude
of the gradient which was the square root of five.
Now, if we go in the opposite direction,
I plotted the tangent line in blue
to represent the direction of maximum descent.
So moving in this direction,
the slope would be negative square root of five.
Let's take a look at another example.
The temperature of a circular plate is given
by this function.
At the point two, one, determine the direction
of maximum temperature increase
and the maximum temperature decrease.
And then what is the rate of increase and decrease?
Again, all this information can be determined
by the gradient of T.
So let's start by determining gradient T.
To determine the x component,
we'll find the partial derivative of T with respects to x.
I'm going to go ahead and apply the quotient rule,
so we'll have our denominator squared,
and our numerator is going to be our denominator
times the derivative of the numerator, that's zero,
minus the numerator, so negative four
times the derivative of the denominator with respects to x,
so that'll be two x.
The y component would be the partial derivative
with respects to y.
So we'll have x squared plus y squared plus one squared,
and then our denominator,
times the derivative of the numerator which is zero
minus the numerator
times the derivative of the denominator with respects to y,
and that'll be two y.
Now we'll go ahead and evaluate this at the point two, one.
So if x is two, our numerator is going to be negative 16,
the denominator is going to be four plus one plus one,
six squared, that'll be 36.
And for the y component, we're going to have negative four
times two times one is going to be negative eight
over the same denominator.
And these do simplify.
Here we'd have negative 4/9
and here we'd have negative 2/9.
So negative 4/9, negative 2/9
tells us the direction of maximum temperature increase,
which means the opposite of this vector would give us
the direction of maximum temperature decrease.
So 4/9, 2/9 can be the direction
of maximum temperature decrease.
And now to determine the maximum rate
of increase and decrease, we need to find the magnitude
of the gradient vector at the point two, one.
Let's go ahead and do that on the next slide.
So we have the square root of negative 4/9 squared,
that'll be 16 over 81 plus negative 2/9 squared,
that'll be four over 81.
It's going to give us the square root of 20
over the square root or 81.
It's going to be two square root of five all over nine.
Let's go ahead and get a decimal approximation for that,
it's approximately .497.
So .497,
maximum rate of temperature increase
at the point two, one.
And that means the opposite of this, or negative .497
is the maximum rate of temperature decrease
at the point two, one.
Let's take a look at this graphically.
In gray we see the surface.
In purple we see the point two, one.
Here we see the gradient vector in red,
again, pointing in the direction
of maximum increase or maximum ascent.
So the slope of that red tangent line, as we see here,
represents the maximum rate of increase
which was approximately .497.
And then moving in the opposite direction,
this should be the direction
of maximum rate of decrease in temperature.
I want to mention one more thing about the gradient.
The gradient and level curves have a special relationship.
So here's the same surface that we just saw
and here we see the graph of the level curves.
The gradient of f of x will always be normal
to the level curve.
So, for example, if we want to know the direction
of maximum temperature increase or decrease,
from, let's say, this point here,
the gradient will be normal to this level curve.
So the direction of maximum increase would be
in this direction, and the maximum rate of decrease
would be in the opposite direction.
That's going to have to do it for this video,
I hope you found this helpful, thank you for watching.