Hello again. This section of derivatives The last part about the process. Now here's middle name and the Jacobian we will see many of these comprehensive While many business naturally but also hardly occurring that's a concept that many plant-available form. Here, where we will see the calculation method, We'll both get some new concepts. The history of this univariate something that is also in function. But very variable When we get some richer events are faced with as always. In a univariate function component functions which can be called The function of the function There are functions. So y is a function of a work function of x. In this step function If we start to write the differential d f f u the only direct function in is divided by the d f Du Du. This d * 'If we divide by the derivative of f, u is the derivative with respect to x, this We know derivative calculation only Even in variable functions. There is already putting chain derivatives, The term comes from here. You can not directly calculate the f's this because the derivative with respect to x You do not know directly. Indirectly, you know. In a step for him to u We take the derivative with respect. Then we take the derivative of u with respect to x. Of course not, but this is their fractions u, we simplify symbolic If we consider the right-hand side of the left-sided compatible because there where f, where f there. X remains at the end. u and du are also symbolic gone but it's not as these fractions. This is a function of its own. This in itself is a function, 're getting. Now it bivariate Let us say that in the function. So once again, but the function of f u x and y are functions of both. Will behave the same way again. Since we only u z function in the first stage. f the infinitesimal, the We're now account for differential. d f d d u u times. When we divide it x take the same where x to see the divide, is divided by x, where x is divided into d. Not anything else. d x is divided into. But here is a matter to be considered the difference here and there from one variable because it is a function of only x here comes a time partial derivatives. X and y are both because of function. But it comes as a derivative of d f is flat. For u because only one variable function. Yet here too the partial derivative According to one variable is coming. Indirectly, because for both function of x and y. When we do a similar process to y'yl, As you can see you're getting. A full equivalent of formula. x instead of coming years. Here, too, just like the symbolic If we make the simplification, f and x have left. Where f and f, and x is staying or left While staying here for years and years. This is the one that we have done show consistency. Here is an example of this with respect to x we'd like to find the partial derivatives, u to say this exponent, only a function of f u. Therefore f with respect to x derivative of the derivative of f. U e to the room. Then the derivative with respect to x. Minus two x's. y is the same from partial differentiation by for here, Or when applying the formula You can also go to this simple logic, adopted the concept of a chain derivatives. derivative of f u e to u. u is the derivative of y. Here comes the minus six years. Second derivatives It is recommended that you do. There is a difficulty. Now that we promote bi step further, are faced with something like:x and u and v is, If v is a function of that X and Y, for example, x r cosine theta, y is the sine of theta, r u, If we think of theta as there is also a coordinate transformation occurs. Now this coordinate transformation chain derivatives under Let him rule what. We, us, our x's and y's open Suppose the given representation. Here we know that the tangent and the equation of the plane here is the differential, the infinitely small. Here comes the partial derivatives. The equation of the tangent plane at work wherein the first partial differentiation, second partial derivatives of x and delta y was multiplied by the delta. If you remember the first part already This course sequence, straight chained up there already We saw derivation rules. There x was on his own. y is a function of x was. It is not a coordinate transformation. Looks like these. But not the same thing as a concept. But the process completely follow one another. Take it to the divide. divided by d u. Where d x d u like what you see will be divided. It is a significant thing because u and v is a function of x. Here we divide the other, d y d u would divide. See're getting it. Issues to be considered here partial derivatives everything now. And indirectly, because for x, and both of x u and v because v is connected to the function, Y, and therefore connected to V indirectly, if not directly for as a function of u and v. So, according to the partial derivatives of u and all all of these partial derivatives are happening. Since f x and y are functions of the u and v is a function of x. These partial derivatives are achieved. This software may also show short. According to the partial derivative of f is equal to partial derivative of f with respect to x, According to the partial derivatives of x and The term can likewise. Yet here symbolically If we make simplification, and the denominator is the denominator of the x's, d y in the denominator and the denominator of the symbolic again the more simple and consistent again we see that software. On the left side, and have fun. Denominators for the denominator u. Here the denominator for the denominator u. Denominator f, u staying in the denominator. This shows consistency. And also keep in mind This is a convenience to. I wonder if I've done wrong and for understanding the formulas, useful to keep in mind in terms. Example, as a function function of x and y is given u and v in the x and y but As though in terms of a transformation. This is actually a spherical surface the x and y the angle from the north pole and the equator opened on them as values??. But I do not know the geometry of There's no reason I can not account for. We are currently doing:take, e, We can do this in two ways. Since we know that x I'll take place here. Sine cosine cube cube would. I put it here, since the cube again sinus, sinus u and v in terms of the cube consists of terms. I take it directly derivatives. You can do it here because There is quite a simple function. And in this way we obtain. So here we are doing before conversion, then take derivatives. But we're doing the opposite chain in derivatives. Before we take derivatives. Then we put transformations. Coming Transformation of the derivatives. Our goal is to learn the chain rule, chain derivatives learn the rules. In the majority of cases because it simple functions, but more complicated functions evolved and the There is no other way to account factors also. Will be put in place, very strange, terms will be mixed. Also make the correct account almost impossible. In some cases, You may even be the possibility of placing even. Then this chain derivatives, remains the only way. We function of x and y from the UE and we will take the derivative with respect to v. That chain derivatives Insert rule step by step when positioned to take derivatives 're getting. See here before we take derivatives, and x's here, and y's, u's, there's a temporary but very this account with a simpler account are able to able to achieve. In a second step that the coordinate we put the conversion. Similarly to the real. Let's take a function as little more difficult. Gene x plus y cube cube, had in the previous example. But let's find the cube root of it. Still get the same conversion. See here for a little more placement Do we would not get much easier. Terms involved, e, obtained in terms because it is rather complicated calculations would not be able to do this directly. Its derivative chain rule for remains the only reliable way. To do this, for example, We'll do it together here. Once out more I would advise you to switch. Alone derivative thereof, derivative of f with respect to x is easy. one-third will fall forward, three x from x cube will be square, three will take each other, and it will remain just x squared term When you receive a negative first derivative We will add strength, so minus two divided by the third power remains. Similarly, in this year. x and y certain thereof will take derivatives. See it here first term. According to u then x We will take the derivative of this place. We will take the derivative of y. Here's a little long as they but looks them directly we would not place the account Do you suffer much more. This is an exercise again. But it can be done in two ways chain rule, at least your mistakes, You can reduce your chances of making a mistake and Because in the event of a sudden go step by step Do karÅılaÅmass embroiled terms. One example given in the answer to that. z minus x squared minus y squared, so that the surface of a sphere, the northern hemisphere. x and y, they wonder geometry For the spherical coordinates. u, the angle from the north pole, v, Turn on the equator, ie, latitude and longitude angles, variables. You derivative chain I'm waiting for you to make the rules. Here also selected a simple function again. There are more complicated function. Now, the following question comes to mind. We have made a coordinate transformation. I.e., f, x, and y are has given function. Concatenated using derivative rules We calculate the derivative with respect to u and v. We can write this in matrix form. See here has coefficient matrix. d x, d, u d y, d D x, D v, D H d v. They are writing here. In column d f, d x and d f, d y. Matrix multiplication that you need to know the front of the policy, one of pretreatment. We take these vectors, we invest, the coincident terms We find multiplication and collect. Already you can reproduce it. We're going to the second line. We turn again to the right vector. We collect hit mutual terms. We find it. One important thing we do here this d f, d u, d f, d have to spend the right. Now the question we We can ask:What is it In contrast we go, Did you find the only solution? Is that exactly this transformation? Here d f, d u, and we to them you a single d f, d x min would come against? Before that first what we see here, conveniently this matrix we see that structure. See, x and y, i.e., x, y vektö, components x, According to the derivative of the vector y. Gene same vector, components x, The derivative of the V y. Gene preparation of information We know that two, single binary equation In order to be solution two binary matrix determinant must be nonzero. These determinants at work in many areas A key task will be done. This matrix determinant see here, When the corners of the matrix, the determinant when it comes to straight, This is called Jacobian determinant. Even in German, In pronouncing that he says Yakob, because a scientist named Yakob Germans have anticipated the importance of this work for the first time. So the question we We have asked our hands d f, d and d f, when D r, we have found them, I wonder which one team did, d f, d x, d f, d y is coming from If this conversion is not exact? This is an important job. Because we have done all of this, but exactly If you do not know which one came from. Mathematics does not like uncertainty. So you do not like it means: would not be able to make progress. And we show it with this J and we call Jacobian, We call Jacobi term terms we find the answer. If this term is zero Jakobi If the conversion is not exact. Because these two equations with two unknowns. So you just put it in three to five numbers, A two-variable equations. Here's a function but the result does not change and The only solution to this equation with two unknowns In order to be Jakobi term, to be the determinant is zero. We for example, circular coordinates we have worked with. Our first B curvilinear coordinates, These were our first coordinate transformation. Here we are looking to work more generally. x, u, v, y, one of u, v, when some I will show examples of this circular Beyond the curvilinear coordinates. Now we can ask a question like this:We We just deal with a fundamental problem. X and y are us given function. But X, y coordinate undergone transformation, Under this transformation the partial derivatives of f Is one to one relationship with each other? I have found the answer to that. We find in Jakobiyan'l. Now a second question We can ask it looks like. X and y, such a wonder transformation of an individual under If the answer comes a point Please assume at a point x, y there, you are doing such a transformation The point of this is going to ten, or vice versa, this is not an appropriate thing. Wherein a point x, Opponents of the year comes a point in m, several m, or even in some cases Does infinite point coming the opposite? These points are called tekinlik. This relationship will also be important, and these two, With this question in a previous issue We will see that the response is the same. We answer it this way. Let d * account. That is an infinitesimal near x of x in the region, the change in x. We find that with partial derivatives. The change in this year. We also again these two equations matrix can be written using. See the matrix coefficients, d x, E u, d x, d v. So u and v-derivative of the vector x, and v by the derivatives of the x component. Wherein Y variable u and v based derivatives. BI have noticed a little more than before. You will remember where x by u, There were derivative of x to v. If you look here and see x According to the first derivatives of y x and the second in line again Derivatives according to v y. If one of x and u on here, both V-derivative, a single y both ua, the V-derivative. If we think of these two as a matrix matrix transpose of each other. So, row, column, or if you change on the diagonal to replace the term If you change here d y, revenue is down, d x, d v income. We get the same matrix. Yet here is the question:is us x d have them when given Yet here is the question:is us x d h when given these Do u an area from a single Is it a literal d u, d v, If you have more than me is coming from? The answer to this is also the determinant of gene is defined by the absence of zero. This is also a determinant of is the same as the previous. Because the purchase of a matrix transpose When trying to transpose in Turkish, When that transfer of this matrix, column lines, When you make the columns line obtained determinants is the same. That means that we are Jakobiyan. Also a bit more fancy representation, such as we are writing. x, y, u, I have to u and v by the x derivative, and v by y ua As derivatives are showing. We call this the work Jacobian. Let's write a combination of the two results. See, there's a tiny difference, but determinants sound the same. See, I have just said. derivative of y by x, and, V-derivative of x and y. In the denominator the same. The opposite here, share the same ones. u and v based on the derivative of x, and v by the derivative of y. But the result does not change, We want to know because the determinant. Determinant on these two diagonal The term of the product, as seen this. Whether you received it, or taken it. d x, d y and d, d v. See here, here, see if you can account also, Y'all account of here, the same thing is going. Naturally, it turns out Jacobian of this transformation concept and where the partial derivatives of the transformation, where the coordinate transformation ensure that exactly the Jacobian size. I would like to draw your attention to one more thing. Here, for although there are for a given function outside. This means that if and only if the coefficients, In both cases only related to the coordinate transformation. It should also be noted, note was that, I need to observe, to be aware of Not much is already clear. Now here I need respite. I'm a bit to digest them. Please pass over. Then we meet again Jacobian of this appeared again in an unexpected place We will see that out and still we did not expect it, we can expect a part, in three dimensions generalizations We will see that interesting. Now you can take a break. To meet again. Goodbye.
