Hello. About the previous session chain, We saw chain derivatives. Here you will see nothing new. However, an interpretation of the derivative chain We will reach a new clutch making. This is the second of a concept to us than the concept of skin will bring. It will be the concept of the gradient. Therefore, bi directional derivative comments The concept will reach. Here we will find the gradient of the gradient Supremely very common derivative processor and an important means now we'll see soon. Now we know:is equal to f x y z When is a surface in the x y z space. B are also given as a parametric curve c x and y plane. This c curve as g * y is equal to As can be given as a parametric an S parameter in the x, y a s We can give in terms of the parameter. T have used before. But what we may call a parameter name. But here we choose a special parameter we want. Whether it is the arc length of this p. In the third section we saw the space curves in the plane and wherein arc length of the curve that 've seen. This arc length parameter we choose. Will be a convenience that it brings. We know it. After t d f, the total derivative of f with respect to t and x the parameters of the partial derivatives and again by the second partial derivative of y According to these parameters is composed of derivatives. This parameter is the parametric curve defined as the size. It takes both of them before we We have seen built upon this plane curve c in cylinder that z is equal to f x and y from the surface segments We find that the curves. X is zero in any plane of the curve y If we choose the zero point With this binary cross-section of this surface We find the slope of the curve on. We know that. So why t'yl s always trying We have now? So what's the difference arc length is he doing? Now let's see it. Here we repeat the same formula. When we think of the arc length, see There where d x d s. There are d y d s. Combine them in a vector If we build it the third section we saw in the unit tangent vector see that. Unit tangent vector in the plane. Wherein a is the length We show. But this is a very interesting product because the product The first component of this vector with the d f d x multiplied. The second component is multiplied by the d f d y. Already at seeing them together We're used. Because they form a pair of them, this they form a team. Let us define a vector of them. Let's say you have this vector. See where the size of this first the second vector, the first, second The first component of the vector product of plus the product of the second component. But I also know what it is. This bi inner product. With a t where d x, d s, d y, consisting of d s t and that our creativity Let's say using positive to say that it naturally vector v is obtained by selecting the internal product. The inner product of vectors. Here are faced with an interesting concept. This d f, d x, d f d d d y if x Disconnect the f's, d d y- Or as i j unit vector If we write vectors with Such a vector qualified derivative processor emerges. That sandstones, it is called gradient. Gradient call processor. Affects on the form for the gradient We call the gradient vector of f. This vector qualified in terms of algebraic CPU. Derivative processor. This is the one variable that we already know d a multivariate function of x with d generalization. How is it d d x a f on the If etkilet derivative, we obtain b. Wherein a gradient for these four, two variable If a gene for a function etkilet we obtain the derivatives team. So in one variable, such as transactions but overall its generalization to two dimensions of the size of the b happens when we get a vector. So a qualified derivative vector We're going to have information about the processor. Now repeat it here. This d f d s was:z is equal to f with the surface c c s, with the curve defined by Built on the cylinder The curve of intersection on the curve The derivatives along. We give it a new name so. This is what we call directional orientation. Why? Because we discovered the previous step b d f d s an aspect of and that the gradient vector of a vector We are obtained by multiplying. So it works fine just here on. However, no information from f in t. F there is no information in the gradient of t. Therefore, this information has been divided into two We're going. This is because the two vectors derived qualified shown here with one component As with size, but according to what the cross from the inside inner product? And a directional unit vector internal obtained by multiplying the value would have. So, in the direction of this key chain derivatives is the value of derivatives. We call this directional derivative work. Sometimes directly, direction, direct given the right to a u'yl. For example, x is a global rather than a curve we take an accurate, yet You will recall from the second part of the bi x is zero an accurate given direction about the point We define the vector. a parameter t. If we take the derivative of x with respect to t it, that t According to the parameter If we take the derivative of a vector, but that the interests vector per unit length is not. However, where the unit vectors was needed. This is the unit vector u to get the very easy. So, if u take its longitudinal split Remove the unit vector. In this way, when defining t when positioned to take of the product of the gradient of f u'yl Remove the split lengthwise. Paint division is important because u bi if you receive a vector-mile, bi kilometers a vector or a neck ALSA if you receive a vector in centimeters the same direction. The same direction. Thus, this directional derivatives unit come vector math in terms of natural as a means to arrive there. Using this meaning here as it is already become independent of the length of the vector We bring the directional derivative. Because even if bi bi centimeters kilometers though it yo, always the same in the direction derivatives. Therefore, regardless of size should be. For him, this unit according to the paint Fetching. In many applications of this vector directly administered. You do not have to be true, even with such. Directly administered. Now I wonder if this artificially a Does the team have produced a new concept called consider. So according to the orientation direction we ourselves We produce? No. According to the orientation direction already-day our lives there. According to this aspect of mathematics in everyday life that pointed us to the concept of functions opportunity to express the very significant gives the opportunity to express. Let's consider a few examples. As you get bi hill. You probably have of riding a bike or Embark on foot. Such that you can not get too steep topping. You can be making such a zigzag. What's going on at the other so if I'm going been a steep uphill going. When you follow a zigzag finger If such a slope corpuscles, reducing we are come. Let's say that will exceed the Taurus Mountains You want to find the orbit. M, you can not switch from the other. No te car can not go. People also can not go walk. Such twisted with appropriate slope Are you going. It turns out the following:on a surface Locate the slope to the direction we look is changing. Let's take a concrete example of gene b. Let us behold in childhood BI sphere hope we play with the sphere. A world map can be drawn. Bi longitude while running on a very steep ways. Bi slope at latitude zero on the way. The slope between the two when Meanwhile, we see that the value of b. So again we see the same thing. We were at a point E, the slope of this depends on where we are heading. That revealed that directional derivatives involved. This function f geometry favorites here often give. Because the geometry is easy to revive. This is also a potential for energy can function. When you draw it again as a surface we draw but this is not a surface in the geometric sense. Or b of a corresponding electrical field and may be potential or a temperature distribution indicates the distribution of a substance may be a function. Heat how to proceed with this temperature distribution When? Again, this will be determined with a gradient. They are also a bit more detail We'll see. So it only in geometry a concept which is not. Recently we have seen in the examples. Concretely longitude on the globe when there is a very steep slope. We go in the opposite direction the same as the absolute value but going down a slope. Will have a negative value at Makiki Or at this latitude tilt in the direction we should go will also be zero. Now how these numerical In mathematics, the function expressed by will be able to? They are the largest, the smallest slope and zero What is the direction of the slope? Now let's get our formula again. As the slope relative to the direction oluşuyo:One direction a gradient vector b with we choose the vector x is zero y is zero the value at. We know the formula for the inner product. The length of the first vector inner product e, t has a unit vector. I've written here clearly. Otherwise you have to already be a unit vector If not anything specific. Length a. Do not ever wrote for him. The length of the second vector of this double line We show in the neck. Multiplied by the cosine of the angle. We show the cosine of tetayl him. Now we're seeing right now. Wherein the gradient direction independent of f. Partial with respect to x at the point where this is located derivatives, y by forming the partial derivatives The length of the vector. Nothing 'blah blah has nothing to do with the direction. Therefore, the only variable here this cosine theta. What is the cosine of theta? we proceed in the direction that the vector t and the angle between the direction of the gradient. Now here on out:The biggest value single variable because it is simply the cosine of theta This is where a theta. Because the maximum value of the cosine of theta. Minus the minimum value. So the largest and smallest values that we find. When a cosine of theta-theta zero. This means that the t and the gradient angle, The angle between zero. So t in the same direction and gradient. Conversely, when there is a negative cosine theta t with a gradient but a direction vector identical they went a different direction. Here, too, that cos theta minus one value of theta is pi. This minimum value of the smallest value we say. Values minus the birisinin plus other is going on. Because the cosine of theta one or minus one when. When it is zero cosine theta E theta pi divided two means that. So t is the gradient perpendicular to each other. This directional derivative that is zero show. For example, in the case of the sphere along latitude If you scroll directional derivative is zero. Because no te place your yükselmezs. Would you like stops rotating. However, up to now little aberration You can go right or down. This is the direction perpendicular to the gradient show. Now as you can see, this new concept of bi but it's interesting to us both bi and bi gradient reviews brought the concept to both, depending on the direction their derivatives between the largest and smallest values showed that selection. Or we know the nature of the problems at the bi thing in technology problems and even in the economy at large problems We're looking for the biggest or smallest mostly. This is the opportunity to address such issues to give. Now instead of me to move a little If you give it time to think again I want. Now we see then that related to the concept of digital examples, we will see the problem resolved. Until next time, goodbye.