So far, we bivariate We've worked with function.

Here chained derivatives, directional derivatives I have seen the concept of gradient.

Now that our knowledge of three variables We will generalize functions.

But you'll see that three variables 20 to generalize

to generalize the concept of variable no difference.

An additional term of three variables to generalize not to put anything else.

Now, bi, big, say f, this function of x of y and z get.

We use it only for small x and from the function y

To emphasize that each individual or We could say something as a name.

We know that two variables, this formula We saw a given parametric.

t is a function of x and y is fully derived by to account

This means that the partial derivatives and partial x and y derivatives

According derivative of y with respect to t We were obtained by multiplying.

For this reason, already chained derivatives said.

Therefore, these partial derivatives are already is the sum total derivative is called for.

If a third variable x, and y

We do the same process of this third We also use variable.

Thus in this way a chain derivatives is achieved.

Three variable gradient is still the same the two variables with this little fun

and we show only the x's and y's

partial function from the function We would obtain derivatives.

If the derivative at z are putting this z'l.

First, get, two variables previously two gradient while achieving dimensional vector

wherein for three-variable, three-component we get a vector.

As of this writing the equivalent of can be:where

i j vector with components which we k we can show with.

I still only in two variables and to j'l where x and y'l when there is k'yınc

ie z-direction and z components of the components There are partial derivatives based.

F wherein a common gene

thinking that the term outside the brackets can get.

Behind this time consisting of partial derivatives remains a processor.

So how d d x single

If you are using variable functions In bivariate d d

in three variables x and y d d d of z'li 're getting as well.

As we can this summer with i j k We can also write ingredients.

This is no longer a function of a number

to take derivatives, derivatives of vector qualified processor is going to take.

A size that a process is going on.

As I said this in a single variable d money

d X, then it was a digital processor.

We saw the directional derivative.

This arc length as a parameter was obtained when select derivatives.

We showed that with d.

According to one aspect of this it

been turned by the unit vector multiplied gradient was happening.

This is completely the same as in two dimensions structure, only in the

There are three components, the three components of the gradient f is there.

The largest values still the same.

The maximum value of the length of the gradient is going to increase.

Minus the minimum value of that going on.

Gradient direction in which the maximum value occurs direction

The minimum value occurs at the direction of the gradient in the direction of the minus sign.

Zero slope in the direction of the gene in the direction t is given by the direction of the derivative is zero.

This gradient direction vector whereby gives, say t, the

This t u were saying that u is zero divided by perpendicular to the direction of gradient.

Now he turns a difference.

A function of two variables in the plane I have a vector.

Zero directional derivative that gives direction perpendicular to the direction of gradient.

These two kinds.

One of it in the opposite direction in this direction.

Therefore, one has straightened.

However, when we came to the plane in space the gradient vector.

There are infinitely many vector perpendicular to f.

It also infinitely many random vector

, not from the vector in a plane will occur.

This small difference.

20 20-D would even be sized plane.

Of course, I can not draw, but the conceptual to as

no obstacles in the process to understand the does not exist.

Here, business, visual

As discussed in the plane is equal to f

shows a line of x and y equals zero.

On this line, this line perpendicular to the gradient that direction.

So the directional derivative zero

so that the direction perpendicular to the gradient In the interest of tangents.

If the gradient at a point in three dimensions When the account will be a vector.

This vector is perpendicular to the surface is.

This is a tangent plane to the surface of the perpendicular vector does the opposite.

Here there were tangential direction.

Tangent plane is going on here.

They'll also see a little more detail.

Now let's do an example.

x, y and W as a function of z get.

Of course, it needs the four-dimensional space is there.

You can not draw.

This surface, two, one, minus three points Let's take.

At this point again as the others, the negative

one, minus one vector in this random we choose a direction vector.

We want to find the derivative in this direction.

Of course, because you do not draw four dimensions

There is a need to account exceptional simple.

Again, this direction of the gradient vector and the internal product.

Yet the largest and the smallest values and We want to find their direction.

I also find that aspect of the derivative is zero We want also want to find direction.

Now here first big job function f to determine.

One of x, y and z, as a function w is equal to this great effort, great f.

Here we need gradient.

Mean gradient of partial derivatives with respect to x, y fractional

find derivatives and partial derivatives of z by a To create a vector.

But as it functions as a general We do not want to leave

because the two of us, one, minus three points given.

When you calculate the value at this point, two for x, y, a, z also already here

does not appear, but if there were minus three would put it.

So here four, six, a vector We find the gradient.

This prepared for this one, minus one, minus

the derivative with the inner product of a specific direction is going on.

Multiplied by the length of the gradient of u dividing the product.

Here is a given vector, minus one, minus one.

But one plus the square of the square of height in the one plus

including the square root of one of its We find the three.

This means that the inner product, the numbers one by one

As you can see each other, multiply by simple account.

Minus six minus four minus three of us giving.

There are also three in the denominator a root, the root mean three derivatives involved in this direction.

The largest derivative thereof in the same two dimensions such as

The length of the gradient, gradient, four, six, We found a vector.

Its length, the first of the frame 16 plus 36 52 plus 53.

The square root of 53.

This value is also the direction in which it occurs gradient

itself but to find a unit vector If we want will divide the length of the gradient.

So four, six, this is a vector we divide the length of the stem 53.

It is the smallest value of the opposite sign,

and also largely the opposite direction direction.

Therefore, it immediately without account can write.

He can do a checksum:We say The largest value, the root 53.

Minus the minimum value stem 53.

We have calculated a value of u, if Our accounts

If true, the value of u between these two It is necessary to remain.

Indeed, roughly seven in 53 bit roots large, root

around seven minus three minus a comma one thing.

Indeed, from 53 to seven with a comma is small.

It also minus minus 53, stem from 53 is greater.

So we provide that too.

So we choose a random derivative in accordance with the largest and most

which claims to be small, thus

indeed, we calculate values remains between.

This is also a nice supply.

Zero to find directions inclined we We need to find steep gradient vector.

But the infinite in space perpendicular to a vector There are vectors.

These are not random plane in the vectors.

We do not know this for vector x, y, z let's components.

If we multiply this with the gradient for zero We want it to be.

Gradient four, six, one.

We have calculated this on the previous page.

We do not know the vector components x, y, z said,

As you can see that's a plane forms.

Four of the six inner product of x plus y plus z, x, y, z, all

this because it comes with the first force implicit function occurs here.

This equation of a plane.

Or, if you want such as separating z You can write.

The difference in this work plane plane

one a direction perpendicular to curves There are straightened.

you would be Artillerie but you would be minus direction is the same, but we think in space

When the concrete as a plane that exit We see here.

As a second example I give homework.

This gene provided in a plane curve,

the center and a radius of zero In a circle with two.

We take this point on this circle.

I really testable, it really on the circle?

This is on the circle easily We'll find.

And at the other, the difference in the previous example direction was given.

If you are here to go on a curve We are having difficulty.

This way the surface on the curve

find out what happened on the slope we want.

This account by following the previous we can do.

This is a problem because it is already easier two free-variable problem.

This helps us to understand the meaning of the gradient will be problems.

Now a little more clear gradient meaning as

I want to give it exceptional gradient an important process.

To do this, take a break.

Our next meeting this gradient,

We will see sense.