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Hi there.

Multi-variable functions We start the second part of our lesson.

After taking this course with some of the two

weeks until the first We worked together on some.

Have received the first part of this course There are those who do.

Of course, requires a prior knowledge.

I'll show a little more detail later.

But those who take the first course in the

little effort prior knowledge can complete the open.

Perhaps you already own before Universities in other environments that

infrastructure have provided.

First, the location of this issue I'm starting to recall.

Higher mathematics at the university infrastructure, engineering,

in areas such as science consists of the following four courses:this

Of course the way to do four courses may notice from school to school,

philosophy of education and training from but philosophy is outlined.

Derivatives of functions of one variable here, integral and an infinite series can be seen.

This multivariate functions we did the first part of the course.

We wrote it in bold letters.

We learned our lesson.

Linear algebra and differential equations are observed.

These first and second courses It is the purpose of their own.

However, linear algebra and for differential equations

creates an infrastructure.

Arranged as a series of these two classes

Our lesson multivariate functions Me first person, put it here with the

I wrote to highlight, rather than We focused on basic concepts.

That is part of functions of several variables Derivative and integral multilayer and some

We have seen methods, among them The most important rule in a chain derivatives.

I wrote it a little lighter mainly because this weight

arose from the development of basic concepts.

Multi variables functions was our aim to recognize the important.

This previous lesson.

About this course even more There are some advanced concepts.

I wrote knowing obscure concepts.

Least some of the concepts we'll see.

But with this course puts weight as it is written and new methods

see new applications.

Here multivariate functions

b we briefly review qualifications are as follows:y is equal to f x.

y is a number of x being a number for functions of one variable.

y wherein y as one, two years, Y is composed of as many

As we saw earlier, a vector x in which x is a gene,

x is two, consisting of x a vector of a function to.

Therefore, the number of vectors are changing.

Of course, for this very generality of some We are interested in special structures.

x as the dependent variable,

so instead of x when y instead of the numeric variable x

When we receive a variable which t vector qualified function is going on.

Their geometric meaning show the curve in space.

If u go by LDs, ie independent is a vector of variables,

the minimum number of two-component but it is a vector

If the dependent variable y where a number.

This is a second end.

This is because the function of two variables

and in the space that is called anti-surface geometry.

There are also vector fields I have yet to see it.

Typically a physical the size of nature,

such an electric field, for example, a fluid velocity field.

For example, a heat conduction heat transfer vector.

This space x, y,

z coordinates at the point specified by the and at time t can be defined.

As you can see this is a multivariate function of the size but endless in number

Or a very large number There is also a variable.

Both the dependent variable, it is usually geometric size and time is going.

As well as the dependent variable.

Here we are approaching in this structure.

Of course, it is a very general dependent variable,

very generally independent possible to get variable.

But it would not be very efficient way.

We are derivatives of functions of several variables There are two steps and integration.

The first course,

preceding lessons and basic concepts As I said a little earlier methods.

Here, the same content, the same content but in more advanced concepts,

some of the more advanced calculation methods We will see and applications.

This new lesson is this.

Here we briefly first course content, I sift heads.

This consisted of nine sections.

General topics, namely mathematics from nedir Starting from very basic to the increasingly

and preparation of single variable functions a skim information.

Vectors and then confirmed using vectors and

We saw the plane.

Then a variable vector

t depending on the saw, We have examined the curve in space.

Then you know the basic functions.

So the paraboloid, hyperboloid the parabolic hyperboloids,

such as elliptical hyperboloid.

Pits, bumps as we have seen surface.

If the curve, if you have seen the surface.

They were about to create an infrastructure.

Our most basic issues multivariate derivatives of functions in the

partial derivatives were saying it.

Because each of the two variables is it may be a derivative according to the partial

derivatives were saying, and the only variable is different from with integration as a two-story

We started and their multivariate In case that more than two variables

In case of multi-storey integrally We saw generalizations.

Some basic derivative calculation methods we have seen.

They are at the beginning of this chain The concept sounded derivatives.

This is also part of our new lesson derivative calculation methods

Further applications and methods to be seen.

This short, with partial functions We see two main applications.

Taylor series expansions and extreme values.

I mean, minimum, maximum values.

Yet the basic two-storey integral I have seen methods of calculation.

That in Cartesian coordinates substantially saw.

But the bi-circular coordinates We have also made an entry.

About this course than in circular coordinates by many applications of these concepts,

We want to reinforce these issues.

And when working with partial derivatives and

suddenly while chained derivative We are faced with a very interesting concept.

This concept gradient.

Gradient univariate we know of functions d d x,

that is, a function is to take the derivative to multivariate generalization.

A vector processor and its importance, basic nature

denotes the equation be the most appropriate approach.

and wherein in a partial equations We have also made a presentation.

Of course we diphenyl partial, differential We are not in a position to solve the equation.

I say this just before the four The last link in classrooms directory.

But what this partial derivative works I always ask when he was a student.

At this stage and this bi deserve to be seen.

Because we are making transactions with derivatives, and their partial derivatives are account

by itself, but also of importance thereof Using the most important places of nature

equations, or even just the nature for example, not statistical sciences,

such as probability theory, which are produced by the human mind

science in the media equations in the expression works.

As you can see these nine episode was a lesson.

Our lesson is that the current nine episodes.

This section names Starting give him.

Because these two coupled constitute an important unit.

Here we will see topics two-storey circular coordinates

integration had started, but larger We want to see applications.

We have seen applications to derivatives, but it We will see more of a variety of applications.

I have seen the concept of gradient.

Nature equations, but we see genelletÅ to them further,

generalizing the more fundamental We will see applications.

We rather to a dependent variable

in the relationship between two independent variables We saw a chain derivatives rules.

For this, two and three dependent variables We will see that he could use.

In this coordinate transformation is happening.

As the Jacobian of a sweat, concepts we'll see.

Infinitely small fields, the infinitesimal volume and complex functions at bi

an expansion structure tells us, generalization.

In the first part, but the essence is there too.

In the first part there.

We are in a two-storey integral plane We saw in the two-storey integral.

As in Cartesian coordinates.

So there in the plane b with a surge on

but we know they were a region There are also bi surfaces in the space.

They are not straight.

This space on the surface integrals of paramount importance.

This is done in a two-storey integrals but some of the issues we have learned.

We have two floors with integral Cartesian We were limiting ourselves.

Also integral two-storey circular saw.

But many of triple integrals In case of using Cartesian

but also cylindrical coordinates br In spherical coordinates, and this

operations extraordinarily common which finds application concepts.

Now, after seeing them a tool that gives us quite a few.

The last three sections of this vector field.

I told vector fields.

To them, rather, physical quantities,

in space and time defined physical quantities.

We call these vector fields.

These are the above There are going to do the process.

At the end, in nature conservation laws and We will see briefly the basic field equations.

So in mechanics of continuous media We also both mechanical and solid objects

liquid bodies, gas equation gas environments.

The basic equation of heat conduction and the basic electromagnetism

equation that we learned information quickly and with exceptional

In a short time, and all compact possible to remove the mass in one place.

16:09

You can not learn or topics you can not consolidate the topics thoroughly.

We will remind them.

These bi recall and learning opportunities will occur.

Now this, this part of the extraordinary I want to go fast.

This is a very variable functions A detailed summary of the first lesson.

I would recommend this to areas first lesson.

I once go out.

When I remember to do anything more You will feel better yourself.

If you know you are will be forced to remember a little bit.

They will pass each page but I will not dwell on.

Our first topic was general topics and preparation.

Mathematics starting from nedir a fundamental cultural dimension

at the location of the course has a section that describes.

Thereafter itself a sense that,

value which, where applicable, We have examined the vector.

These vectors were fixed vectors.

Lines and using them We have obtained the equations of the plane.

We found some applications.

This vector operations with , in particular collecting secure bi multiplication,

We saw the inner product and vector product.

First, we start from two dimensions.

Then they have expanded into three dimensions.

Triple products in three dimensions We have seen that it is.

In the following sections of this vector variable

with a numeric argument as a function,

vector as a function of We have identified and examined.

This is a curve in space We have seen that coming the opposite.

But before curves in space We recall curves in the plane.

Especially the curvature of a curve and

radius of curvature We have seen where that came from.

Because suddenly comes out in three dimensions I do not know if this concept

My experience so far driving students this coming.

However, when you see it three too much generalization in size

not something that requires imagination.

T curves in space function in the x component,

y component and z component being We approached two types.

This publication describes and an arc length in terms of the length of the space curve

the unit tangent vector, the unit upright vector and the second vector of unit

We have identified and are parallel to and the curvature radius of curvature

Or not in the anti-plane curve We have introduced the concept of zero torsion.

This in essence as a bending curve E, showing deflection

size and arc length of in terms of how beautiful,

elegant and useful and memorable which is very easy to remain in a

structure, namely the Frenet formulas It's called, can be expressed saw.

Here with arc length transactions.

See derivatives wherein all by arc length.

But usually spread curves in terms of length is given.

Is given in terms of a parameter.

This parameter can be made in more favorable to the account.

They have seen how to do.

This function, This function is a vector components x, y,

girls first of the vector function, calculate the second and third derivative work

with the process of the curve I have seen can be understood.

The subsequent section

basic level this time, ie z

function of x and y equals

The multivariate functions exception.

These geometrically shows a surface in space.

There are three main classes of these surfaces.

A pit, As we say, if a peak at a

Or combine the two valleys We can think of it as a surface.

As their algebraic and geometric the nature and without integral

Our know what we will try to There he saw the benefits of this surface.

They saw several main types.

How do you see the truth in the plane, immediately after you see a parabola,

hiperpol you see, you can see the ellipse.

Such as these.

These are some of the foundations of There are also benefits to recognize faces.

Because the derivative will receive You will receive an integral,

need to understand what you're trying to.

Here we see the basic types of these surfaces.

I'm going through them.

Gene globe,

Or, hyperboloid of revolution, such as cone

one-piece, two-part, which We saw hyperboloid surface.

And in the functions of a complex variable the fact that two bivariate

function, la phrase I have seen can be.

The aim of this lesson complex variables but also complex variables

of functions of several variables a very special case.

Very useful at this stage to see them.

Because you do not learn something new an application other what you have learned.

The fifth chapter the main department.

Derivatives and integrals of the single storey generalized to multi-storey.

A multivariate generalization.

We know that the slope of the derivative.

The concept of univariate Locate it in a multivariate

We saw how generalized partial derivatives.

Your views on this page I'm leaving for.

How in the single variable functions If you have a derivative at a point

If the count here at some point There are the partial derivatives.

But they also point to the number of When you make a single variable

derivative function also becomes variable.

How do you take the derivative of the sine, cosine involved.

He is also a function.

In multivariate functions There are derivative functions.

We saw the meaning of derivatives.

We know that the slope of one variable.

In multivariate functions slope again but that we are on a surface

in two major directions to each other is substantially perpendicular to the slope.

So therefore univariate The main concept of the slope of the function is available.

However, when one in two variables It is understood that there will be more incline.

We saw higher order derivatives.

How in the single variable functions There are first-order derivative,

If there is a second order derivatives here There are second order derivatives.

Of course, a little more rich.

Because the two variables is able to produce more possibilities.

Integral is a collection of We knew it was.

Area under the curve dividing it into small pieces

The presence of this field is the sum thereof.

In multivariate functions The same thing also.

But because the two variables There are two variables on the collection.

As a result, two-storey we arrive at the integral.

How much of it in a natural way We saw that in the first lesson.

This is related to the integral We have seen some important theorems.

E.g. of the order of integration change does not change the result.

Integral over a region Instead of making two regions

Even if you do integrally on their 're getting on the total integration.

Although three rather than two variables 're getting triple integrals.

Not even three, though the variables integral to the story we're getting.

I have seen them.

Derivative calculation methods te,

How the univariate tangent function is important in

If doing a task, where the tangent plane is doing an important task.

This is the equation of the tangent plane were found.

Using this tangent plane We have the concept of differential.

This already exists in one variable.

And here the chain We have derived the concept.

We have a full concept of derivatives.

Here, the exact derivative calculation method

chain rule derivatives obtained derivatives was going full size.

This means that we have seen have studied.

Conveyor derivative meaning.

We saw it in a different application.

If you are a real line g y is equal to x not given as a parametric

x x t y t if given as a parametric chain derivatives of these rules

have achieved and how full of derivatives Accounts can be found.

Making them the gradient We've got the concept and

We have the concept of directional derivatives.

They are also extremely important We will see applications of this

Our second lesson.

This is hardly exceptional directional derivative concept The concept is a concept not a miracle.

In our daily life a concept that we encountered.

We have reviewed them.

These first two

The variable that we have done We generalize to three variables.

This supremely easy.

If there were two variables x and y, If a third-z also the same x

z and y are as task doing the same task.

We have generalized gradient of the three variables and saw the meaning of the gradient.

A digital function gradient gradient in these two variables

f x y is y is that is, a zero off the

If this function is a curve that We have seen that the curve is perpendicular to the vector.

Or is equal to f x y z equals zero but it still functions as closed

time between three variables off function.

This gradient perpendicular to the surface We have seen that vector.

That is the main concept here is: Gradient, no matter how many dimensions it

geometric object, going perpendicular to the surface.

Of course, do not draw later than three dimensions but as a concept it again

minus one order of n-dimensional space dimensional vectors are perpendicular to the surface.

They are in physics, geometry, which is always useful and concepts that are very broad applications.

After seeing them in their methods of calculation were.

Some applications have made.

They saw two main applications.

Taylor series and the local end i.e., minimum and maximum values

problems at a point near Calculation of the largest and smallest values.

Review them here I would advise you to spend.

Finding the best with extreme values There are applications of optimization.

We are only in the first lesson We saw the local extreme values??.

But the local end of this lesson, we and also to values

and using part of the idea here using the absolute extreme values??,

restrictions under the extreme values and the integration of the largest and most

in the presence of small value it is called calculus of variations.

We will see them.

Here, briefly introduced the absolute value.

This endpoints presence and quality.

Also in univariate How as the

y is equal to x is a necessity for the condition adequacy of the derivative is zero

the smallest value for the condition f of two base plus is valuable.

These two variable function

We have seen that all have a similar structure and they brought out the criteria.

Single in three dimensions, ie two independent variable functions

as well as a minimum and maximum value it was making the point geometry.

The only variable No anti-in functions.

Here are three basic surfaces in a variety of

By studying of these criteria We have seen applications.

After this two-storey integral We passed on the calculation method.

From here in the simplest Starting more complex regions

and integrally on the integral calculation which

the order of performing I have seen would be appropriate.

Our basic approach, in derivatives, two

variable function of a variable the other is held constant change.

Integral also like that.

From the simplest of jobs, We started from the rectangular region.

Where x is held constant on the first take the integral, where y is the constant

began to take hold x on the integral We have seen applications calculations.

And in these two different account We understand how to give the same result.

Regions a bit more complicated,

I do not want to say more difficult, but in mathematics I do not want to use the term at all difficult.

All he could find a way to already successful in mathematics.

Everything is simple to because it can reduce successful.

Both concepts and calculations.

Curved boundaries such that refer to cases

in a direction of constants but in the other direction curvilinear boundaries.

Here, conversely,

y direction where fixed values while in the other direction with curved boundaries

how accounts will be defined when and we have seen many examples of them.

And by the way the two curves

The region defined by the integral We have seen how to do.

Sometimes in the case of an ellipse, a may limit even in the circle.

How it can be done, where the curves are cut artificially.

Then turning to a previous problem.

Two limit defined curved,

but the other two boundary fixed with the values ??determined.

Where x is constant on the here including constant values ??over the year.

I have seen these types of problems.

Here it is important

Even if the same result will be in order to facilitate the calculation

We have seen many examples of the importance of and tried to understand the overall structure.

Moment of our many applications made on account.

Moments and why it is important calculating a set of random integration

Instead of torque calculations and calculation skills

both benefit from the increase in The results obtained in

useful results from the fact that little We pay attention to the starting torque.

The first and second torque We have seen the importance of moments.

Then in circular coordinates we went to the integral double-decker.

Because many forms in nature Thank him for a circle,

you are a hoops yaklaÅtÄ±rÄ±lÄ±y, Even even though a bit exaggerated civilization

proposal began with the discovery of the wheel, you've heard of the proposition.

For this kind of event, and in recurring events

circular coordinates more can best be expressed.

We calculate integrals related.

Ora, how in the Cartesian coordinates and delta x, delta y is

circular coordinates is also infinitely small area,

d r d theta, we saw that coming.

And some related We saw the main application.

And we have seen some moments.

We've made a very simple geometry.

This new lesson we more advanced applications will do.

Both the concept and the calculation of We hope that the consolidation method.

Last in our department, In a presentation as an introduction

gradient of the partial nature of the applications and We introduce differential equations separated.

About this course on this subject 'll come back and

more broadly and more We will see wide application.

Our first course consisted of them.

I'm going through it.

Then we'll get a new lesson.

I can revise them b.

And of course a little air attributes can be entered into.

Why did he take this course in two parts If we think of the whole experience is showing

starting with the derivative of the derivative too far When you go from one place to practice

An important part of the student we can say that the end of the rope is missing.

Because the first sitting of the basic concepts, maturation,

being absorbed too far We do not see useful applications.

This also applies to derivatives, applies to integrally.

Its our lesson for this We have structured way.

We describe the main concepts in the first course.

The second lesson that the main concepts in briefly review and more

We will see further applications.

We will see more advanced calculation methods.

I'm here now adjourn.

To meet again.

Goodbye.