Hello. This is a new part of our session here we go. B alone before starting here Let's review what we have done. In this section, derivative and integral calculation methods and practices to be seen. Last issue we handle derivatives and They were the basic concepts integral. So we have defined the integral derivative and very variable functions and achieve an infrastructure here We have. As we reconsider what we do Before the single variable functions We remind derivative. Wherein the derivative is the slope of the tangent 've seen. It multivariate functions We generalize. Partial derivatives of functions of several variables see that. Because both x and y with respect to x because that derivatives sub-sub-indicator to indicator for x or f As years have shown. Slope of two turns. So when there is a dip in one variable here comes two slopes. E two slope from the two tangent vectors consists of a plane. So instead of tangent in one variable here we obtain the tangent plane. We will see in the tangent plane is extremely favorable The key concept and many things, since. Yet if we consider the integral univariate function is the integral of the area We know. It's a collection. In multivariate functions of two We have integral floors. Shown in this bivariate with the function The volume beneath the surface shown in We have seen that. Now that the basic infrastructure on here on the concept of first derivative calculation methods We'll see. The main chain derivatives of words. It also functions of one variable in has its equivalents. Already univariate functions I'll start by reminding. There the concept of directional derivatives that a single variable functions No equivalent. Because a single variable functions There are one-way. You can advance or back along the curve I'll go. However, on a surface here forever able to advance in the direction. So that's going to change in the slope we see. Yet in a single variable functions We will see the equivalent of a non-concept. This concept gradient. We know the gradient derivative transactions and derivative transactions in the form of vector extend, generalize and multivariate A concept of generalized functions. Having seen the derivative calculation methods After it My two main main field of application We'll see. These Taylor series. In a univariate function in You will recall it was of paramount importance ie, the minimum and the end value problems maximum problems. Ekstrim will see problems. These derivatives is not the end of the application but The most basic, must be learned first apps. You know the second part of the course There are further examining the issues. I've shown that of multivariate functions in two parts. Here, the fundamental gradient and nature We'll see what the equations and gradient concrete is suitable as a processor We will examine. A fifth section will now then. How in the derivative calculation methods and derivatives I have seen more in future applications We will see other applications. The application of integrals and integral There are methods of calculation. We will see them. Now let's go to our subject. Here again, we always do in of functions of one variable, such as a Let's start with remembered. We know of functions of one variable geometric meaning of derivative at a tangent. And this, we in this tangent, tangent on the while moving from a point i.e. deviations did farther main deviations from the function is increasing. But this is the point we calculate the tangent If we stay around on the tangent Our calculation of the function on becomes a good approximation of the calculations. As you can see here we see an x When it comes to the point of this curve is the actual value while on the calculation of the tangent If we do a little different but rather a value close we find. If a bit away from the actual value curve more over here on the tangent incorrectly, a different but still significant We find an approximate value. Multivariate its equivalent We'll start seeing in functions. Yet if we continue to remind x zero right at the point of this tangent rather we find the following equation:y from the y is zero the difference ie the right of the equation x minus x is zero b This gives the slope. This is no zero at a point x The slope of the data. This is the slope of the tangent line to the should be equal. Here tangent equation y times equal derivatives x minus x is zero x variable x is zero, this value given in point. y is zero at this point, the calculated Y value including the value of the function. This more compact tangent equation can be expressed in a collective manner. The difference delta symbol in mathematics show. That's what they and the rest of the word differential in ancient Greek initials. The difference in delta y. This year, we bring zero to the left y minus y becomes zero. The difference in this year. Derivatives had here. The differences in the x is x minus x is zero. As you can see this same equation more gives a neat structure. It's a start. Let's say a key. Here deltas that we put in place d's When d is equal to wherein the derivative d times x, x is zero one in that well, we are taking a zero. We leave it as x. This is a relationship in terms of infinitesimal is going on. This differential symbol from Western languages call or infinitely small icon We can say. The main difference here a number of years the delta. a number of times that the value of y you will find. When you give the value of x, for example x If you go to a x is a number with a minus x is zero. It's a numbers show. However, d's symbol, not a number. So you can ask the following question, and rightly so can be. So why unnecessarily a differential I can bring something more to say. Because a lot like that she immediately. Because it has a lot of benefits with this icon the ability to operate as algebraic fractions gives us. This is achieved by taking the limit of the delta The results will be much more beeline for this piercing, d's, differentials, this infinitesimal icons such as algebraic operations useful to be able to use. That shortens the job. Now this is a right bend, on a curve I find the tangent tangent line on a surface rather than try to find the equation of the plane. Shorter, in the definition of the partial derivative with respect to x ErUzdU holding y constant. Similarly, the partial derivatives at x to y is kept constant. Now let's start with the first. y the surface when we fix the function is equal to z When we say f x and y the y constant You can see this single-valued function of two variables structure came into function. Now what is the geometric meaning of this? z for x and y has a surface. y is equal to zero, y is zero or y hard is the set of points. E x and z axis to zero this year the plane parallel to the equation. Whether you have a combination of both when we say Two geometric shape equation intersection is obtained. So for z is equal to x, but y When secures this on the surface, surface scratch this year The intersection of the vertical plane. As this will yield a line. We are at x is zero y is zero. We are at the point p is zero this. This e, we can draw a tangent to the curve that at some point. Let's call this vector. Similar process x is equal to zero, we call x So when we see where x is zero, with the vertical plane passing through z is equal to f x y the intersection of the surface. If he thought there such a surface. For example, consider a watermelon. And axis, parallel to the vertical s planes As you take a knife dropped off so bi Let's think. This line will appear on the surface b. If we divide these two lines, for example following one bi curve in the XZ plane projection of x we get y plane will be such a curve. You also know the slope. Because this is no longer one-size, variable we have reduced function. Already the basic methods of multivariate any functions wherein by fixing the variable We manage to do this. Reduced to a single variable functions. Why are you doing this? Because what functions of one variable with Because we know we will do better. Means that x z plane so that one bi The following curve which is tangent, the y z plane of öbürkü tangent to the curve that you have. This shape to reflect approximately 're getting them. Now we have two vectors u and if there is Request vectors thereof, t in a plane can be initial Under vectors in space planes We have seen in section. This means that we proceed. This is a vector in the x direction When we move in the direction of y No changes. Because we fix the y y is zero. When x is zero in the z direction for the changes. This defines a vector. x is a similar vector v as an No changes. Because x L x zero said. When a move in the y direction z Our progress towards this, the slope of the amount is happening already. That the definition of the slope. What is a vertical scroll horizontally when You can change up? If E is a plane now two vectors From these two vectors to write it take to obtain orthogonal vectors financing as a servant of the plane perpendicular vector, We have to use. These two vectors when administered us again we see the beginning of orthogonal vectors Under the vector obtained by multiplying vector Thank you. Vector multiplication of the following determinants came out. i j k. First E, vector're writing. The second vector are writing. These determinants are calculated. As you can see i see i component reset Multiplying this f x y gives zero. Less f x is zero. The i component. Similarly, the j component axis We start anyway. f y becomes zero. k where k is the component of this column is We're closing. We're shutting down the row k. As you can see these two binary backwards matrix remains. This is also a determinant. I hope that you forget that these vectors the relevant part. If y'all forget your little look back will help you remember. If we write this without using i j k u and v vectors obtained so steep that the tangent vector plane of the surface at that point, and z is equal to f x with y Find a vector perpendicular to the surface has been We're going. How did buluyo a plane? Bi plane to better Need point. BI also need at that point perpendicular vector is there. Orthogonal vectors, all vectors in the plane other means that are provided in the following way. a floating point x on the plane. Our basic point that x is zero. So x minus x is zero, this all vectors in the plane, will show by changing x. This has to be perpendicular to the vector. You're getting a job in this way. They saw TA vectors initially section we examine the issues. Thus, this means that the inner product of n are writing over here. We are writing zero components of x with x older. Where the three components. Where the three components. Each hit opposing terms We collect. As you can see there that bi minus x is zero for There are negative for the year multiplied by x to y is zero multiplied. Bi z negative z is zero multiplied by the merger. As you can see in terms of x, y and z first I have found an equation that includes forces. This equation of a plane. Which plane What's this? This e, the plane tangential to the curved surface equation. Edit this equation we see that z z as a zero here. already function x y z is zero zero at zero value. Here, let f zero. When they go to the right side of the plus signs will be. This is the equation we have achieved. This is the same single variable functions inscribed in the structure of the equation. See, because this is the first year the terms had not been When we get out of this structure is the same. Reset x where x derivative of course the old one There are many derivatives of a variable. There are two variants here. As you can see the same structure but of course becomes more genellenmiş. As I said at the beginning we have always functions of one variable in one We will make reference to concepts. We will remember them. Thus, two birds with one stone in the sense that b blast. Both univariate function better 'll understand. If you will remember that you forgot. In both single variable functions in many cases the results immediately two variables, and hence three and to the variable functions generalizations will understand. Now we've turned the functions function We can deliver f x y z equals the We can deliver or off function. Off function always in off the function from an open function We can now. See this for the left and let x to y. As you can see here, y and z of x we obtain a function. If it is zero at all Find the equivalent would have. Here f by f large part by Taking derivative of x, z with respect to x and y hard to be seen, to be held temporarily for becomes smaller as the partial derivative with respect to f. Have a negative sign because of the negative. Great partial derivatives of f to y When we receive these z again b will do little to hard task to f y would be based derivatives. This is again due to the negative here There are a minus. a derivative with respect to z happens. As you can see us here that we find 're getting exactly the same Great fun with a minus sign because the small fx x. Small f with a minus sign, then the a great big f by f the z partial derivative. Therefore, the function off If it is still too easily According to x to y and z partial derivatives We find the perpendicular vector. Outdoor functions of x and y in the open independent variables, the dependent variable z is considered. However, in closed functions x, y, z equals are traded. Its perfectly symmetrical for a phrase here we find. Watching it the equation of the tangent plane to be found immediately. Because the vector perpendicular to the x minus x is zero vector multiplication. Gene surfaces We know that a third representation. given by the parameters u and v, x vector. We are writing here again. When you write on here as components an x component u and in terms of v, y, and z is the same way. Surface again with two free variables found that expressed the know. As you can see here, the variables and going there. We have two free variables. That means that we show a surface. This surface again earlier logic When we secure that within the framework of V. the function of x only happens. This is a right. E, sorry curve. A curve in space. We have already seen this in the space curves section of the vector function. Similarly, a constant value of If we choose this The vector x is again a univariate function is happening. Let's say you have this vector. As a function of v is small. This again dependent on the single variable vector function. In the previous section we saw more third again I know that a curve in space, we find. So, given these parametric surface equations obtained in the two curves We are. U and V curves and the tangents thereof wherein the derivative of the vector. Already mean partial derivatives of V fixed we're holding. Similarly, in partial derivatives by V. We keep in constant mean. So the partial derivatives of the vector x If we achieve this and vectors We are. After obtaining these vectors wherein We are writing again partial vector, with the partial derivatives of this vectors. These multiplication of vectors is happening. Let's say a b c components of this vector. This vector, characterized by plane equation x minus x reset again n the product. x minus x n is reset from here If we get zero times Since there is zero these three complex we obtain the vector multiplication. The second part will look at it again. Vectors section. Triple product as determinants of bi used to calculate. First vector to the first row, second line the second vector, the third line of the third vector. As you can see it when we open the xy z appear only the first force. All of the other terms of constant values. x z y is zero zero zero constants. The reset values specified in this The components of this vector. the vector constant vector. The only variable where x minus x is zero, this the sliding surface over, so here on the plane, x on the tangent plane. Here again the linear terms of x y z seen that the first equation forces. This is a surface. This is because the surface of a plane x y z only with the first force seen. It's three in the second part we see vectors, lines and planes in the section 've seen. If y'all forget now look back bi will remind you. Now it resolved in the next section problems are going to do. Let's take a little break before making them. If you get the opportunity to consider them. I once you get the opportunity to review. These three concepts that we have seen concepts We have seen here. Tangent plane as a function of the open Giving off or parametric function is given as given. The equation of the tangent plane with them we find. Only in the plane tangent plane equation variable functions of the tangents a generalization. I can look at these issues as follows. There's nothing too complicated. But of course, like learning any new topic also needed. Your opinion until one more goodbye.