Vectors in the previous session We define the space and in the space of some This vector on lines the sum of the areas, We said we'd do that for integration. Now we call them and line integrals first line in the plane We'll start with the orbit. Because our approach before all I can understand the topics in two dimensions, then to generalize to three dimensions was generally easy going. Now the line integrals In three scopes are on duty. One, in numerical functions collection are calculated. We know that the integral A collection eventually. Second, one of the vectors along the line of the total or a calculation of the vectors one side to the other side of the line transitions to calculate the amount, i.e., the first in case of projection on a tangent, In the latter case the unit vector perpendicular We want to collect on the projection. In this Article I, fully digital value, is the value of a function. The integration and the identification of generally calculation We can do three types. Arc length is important. Because an aggregate along a line If we want, for example, a mass If you want to find the arc length units We offer mass, total want. Or Based Popu- an electric charge, Suppose distributed. Units in arc length We give an electrical charge. Accumulated a total of how much electricity? You have a wire or heat-temperature heat, Multiplied by the number of times gives heat energy. What is the total accumulated heat energy? Unit to find it again spring We need the length. Thus, for a total arc length important criterion for giving unit length. But to calculate a curve at a single We can show in terms of parameters. t. x in the plane of the x and y There are sizes. The x, y and z dimensions in space shows a vector. A vector indicating the position. That is the position of this point We define the vector x. This is given in terms of b t. A second way, is available in this plane. y is equal to x in the plane for defines a curve. Not very useful in space. Because of two functions, and two surface defined by the need kesifsÃ¼m. He also is not practical. Instead, the parametric space The notation we use. Let's open a little more in Council this just what you're saying. Open trajectories between points a and b we may face as a trajectory or We may encounter a closed trajectory. While we follow this example, You have a point here begins. Are you walking along a curve, You can come back to where you started. These lines also show the following. Each integral every as we did large range infinitesimal We divide intervals. Each one infinitesimal size range account whether the collection here we go. Now let's look first arc length. Whether you get an open curve Whether you like it closed curve As the delta is a bit of s As we grew up we do. We are standing at a point x. This delta x as x are increasing. x plus delta x is happening. Then we are in b y. y plus delta y is happening. Now they, delta x and delta y we obtain a beam combining. This is slightly different from the beam spread. But we went to the limit of infinitely small When this beam spread between the there will be no difference, te beginning of the tutorial We have seen from the curves in space issues. This is a bit more simple for regulating. This of course the length of the beam immediately Can be calculated from the Pythagorean theorem. Delta s, the length of the beam, delta delta x squared plus y because here there is a right triangle the square. Here it is written that arc length. Now we said, We can show two types of curves. We can show a parametric. t as functions of x and y. With an outdoor function y equals We can show as f x. In both cases, the delta s the need to aÃ§Ä±klayabilme, We need a more open and write. If x and y t given in this statement, Divide the delta t, delta t'yl we stand. In the delta t share outside the denominator, because it is the square root For entry into the bottom of the square root, square root of the When entering the delta t goes to the square below. Right here we see that, delta t limit delta x divided by the time we go here delta t, x is the derivative with respect to t. Delta Delta T in y divided by y will be derived with respect to t. Therefore derivative of x with respect to t at point x if for the sake of easy writing, y according to the t derivative of y at the point, we s to a delta, these deltas s finite number because the more we're going to the limit. When we go to the limit deltas That's what we always encounter integral derivative in the accounts d is changing. where d s is equal delta t delta limit x x point x to the point where the frame. Wherein Y point times square d t is happening. Delta is always going to t. T I've written here explicitly. Just to emphasize that the function of t but generally mean square for point x, y point say enough frames. Because this d t t of these functions already shows that in terms but subject In order to emphasize their function t To emphasize that we wrote it. The second notation, the again using a similar approach, again this delta delta x to s Delta X, we divide stands. We hit the delta x here, 's denominator. Those in the denominator here again As in the delta t enters into with the square root. See here, the delta x divided by delta x occurs spontaneously. This would give us one. Here delta delta x divided by y. We went with the same approach limit d when is divided by x to give. Because there was such a derivative. Now that the delta of s When we take the limit In the above context again delta d is happening. d p. This differential or infinitesimal icon. d s is equal to the square root I stayed here under a. From here came the square of the base year. Gene of base x of y function but is clearly I've written here to highlight thoroughly. In general, so you do not need to write. We can now say y base frame. Delta x D x comes from. Now what you have won here? Now it s important delta. d s also important, but the delta with s not able to do much work. Because the functions of arc length usually not stated in the neck. you referred to herein as the parametric As would open or wherein We're with the function expression. D say here that in terms of s t, in terms of the parameter t or x denominated argument We're going to have found expression. Now we have three kinds. Such a closed here trajectories illustrated but clear trajectory from A to B. applies equally in everything. Only limits will change. This will make applications We will also further consolidate. Has a function. This function, for example a wire The bulk density as for You can look at or electric charge density as Or you can look at this temperature heat energy trippin ', is to see if you can physics, classical physics, each karÅÄ±lÅÄ±l anything in this field. This is the unit for spring s and the ball hit the delta, wherein the length of a delta h infinite between two lines Delta water springs were too small. So this small size The data in the weight. And when we take the sum of them, we find the total weight. If the mass is weight mass Or, if the electric charge density, If that's what comes to your mind, we find their sum. And when we go to the limit. Where k on it, here, where all these on endless tracks will give us a total integration. We already know more I have seen many times before the integral symbol of German sum or in English, s letter in the word summa work Ã§ekiÅtirilerek be made a symbol. Leibniz found it. Newton did something a little different, but At the end he was pleased with it. As you can see in this integral k because the ends Coming on collecting them, d s is transformed into an integral over. The second thing we on the trajectory of a Although vector, such as a gas stream comes into an area, a heat flux, an electric current comes into an area. During this wire that these orbits for the component, can deal with. To find this trajectory has a tangent at each point. This tangent to the projection of u on t happen if we take the time, kth small, If you are on this end of the infinitesimal arc The calculated values ??of x and y, where we have chosen points x k, where y is the value of k. Tangent at this point and When we receive the projection, Because the unit length tangent here Remove the direct projection. It would be a numerical function that the inner product is an end as f consists of numeric functions. When we went to open it in the limit u function, but this time space of vector functions x the value at that point of the year in the the sum of the tangent is possessed. Yet again, and you Please assume that the gas through the conduit currents, currents, water, oil flow there. You want to find the sum of Vertical component of course that this curve projection on this does not move along the curve. Therefore, we take this t y projection. But as a matter ilgileÅsek, Please assume that there is an article here. Here you have a membrane. Osmosis incident wherein the material on this side or there will be a heat here, There are temperature, inside temperature difference here you will pass through a heat flow. Let's say it again. This time, we are concerned that the wire Or for that for this limit not perpendicular vector components on the component that is important. Both of these different issues responding to the requirements. But both the inner product t because of u'yl gets hit with a You can find numerical functions. u'yl of n'y numerical gets hit again You can find a function. It's the same when you go to the limit As above this u'yl time of the product of n p means the integral over. We knew t. t x is the derivative with respect to s. In contrast, the vector n is perpendicular to t. Now when we get a vector, See other vectors out of this t it is towards the other, There are going inwardly a perpendicular vector. Thus there are two options. To eliminate these options with uncertainty because in mathematics not possible to proceed. To make certain single n so that x-y plane perpendicular to k unit vector tangent vector We define the vector product. Immediately following this page The expansion will see it. When we do that, but this You can see the stage. Finger perpendicular to the plane Bring on the location of t. Go to your finger that n When an inward be the vector that intuitive, but as you'll see here I want to show as an account now. t, perpendicular to n. t is the vector product of the k'yl. t was derived based on the definition of x s and here was making unit vector. I hope that you remember it. If you do not remember the first of these courses In the first part has space curves. There is clear and they has long presented in detail. Also at the beginning of this part of the first section in the summary of all that we know. n is the multiplication. This vector multiplication. Vector is multiplied We would like this account. i, j, k of the first row We are writing all the time. The components of the first vector We are writing in the second row. k vector is perpendicular to the plane zero for x and y components. 1in the z-direction component. To do this the vector product a triple expansion needs. Here we are in the plane where t We gave two component, but We can easily generalize it to three dimensions. Because the third component is zero. You write three component We are opening this determinant. To open Determinants ii Let's open by the first line. In the column and the line temporarily discard the remaining two binary matrix We are determinant account. Here comes the zero-zero reset. Then on the diagonal dualistic multiplied by minus're getting. As you can see the merger d y, is the axis of s, but comes with the product. We're starting with a spike in ji'li. You'll recall plus signs, minus, plus going. We start with the ex. the row j, i.e., the first line and the second We take the column temporarily. Here again, the multiplication Reset the product of zero. Birla d x, d s is multiplied. Second because it comes with a minus sign but at the beginning of the diagonal j There are a minus. Therefore, a positive value is coming. k you can see when we look at be reset as the product of the k No component. Leaving no space here to hunch account would remove as easily. Now we can control. If you are here t d x, d s, d y, d p. If this is also a product of these two in the X, as you can see with the terms y'li product comes in both terms, but In the first coming as missing, As the notion of residual second coming, inner product of these two is to collect. Therefore, the zero that the inner product We're going to have control, are making. Now we need to u'yl t is the product of S and d. Think of times before t DS. t, d x d s respectively. So d * t happens only once ds. d x d x and d y are the components also. there are times where n d s. d s is n times the product of the where we multiply by n d s. Then here minus d y and d x is coming. Now get this product means The components u and v, respectively. This d x, d y and will multiply. So the first vector u, d x second there is going to be the year where the term component. You can see in the second integral u will like minus multiplied by d h whether he is suitable to sort y'l I put the term in the second row. u d y minus. Here you have the d y times We will multiply by d *. with v, the first term of d * We put here. Thus in this integral bit We are writing a different structure. Here we are freed from d s. in terms of x and y we are able to write it. Now here I am taking a break. They try to keep in remembrance, Let's try to learn but where we are Identify the end of this integral. Now the next session integrally related by examples how to apply We will see these forms. Bye for now.