Hi there. Now you can sense your way. We started from Cartesian. There are infinitesimal volume We calculate the volume. Delta x, delta y, delta z. Then we move on to the cylinder coordinates. There is also very easy. We know infinitely small area of the apartment. r r delta delta theta. We stood him z'yl delta. In spherical coordinates in the same We will proceed with the concept. So, as we did in Ã¶bÃ¼rkÃ¼ E, We'll find a geometric ways. BI through the Jacobian. Essentially this. Our understanding of some geometry I think it will be useful. In spherical coordinates of a point in space We give the position with three arguments. From the center is great. Ie from the northern axis z the angle from the axis of fi. For those North said:These sailors use the z-direction is the north pole From there to measure angles latitude in which they find themselves. This second angle. Again, this point p in the space of x and y 're getting onto the plane. Also in the case of the cylinder coordinate distance to center as it is small. But that is a small global not of coordinates. One assistant. This is how small the cylinder r at coordinates x and y axes taking the projection on the x and y we find the same thing here. R b is the size of a small help. Here we consider the right-angled triangle, from the p, z oh that is perpendicular to the axis it is in length, r is smaller with higher size sinus is who is possessed. We show here this helpful information. Small r r times larger sinus fi. Once there is a small x small r times cosine theta. E is a small sinus was fi'yl large r. Cosine theta. y is still small, r, the times are great times for sinus sine of theta. As for the cylinder z has coordinates z z. There is a small three coordinates r, theta and z has. Here we compute the z in terms of the others have. See here again, other than p z, Let's go to the other. Wherein the edge of this length z Because the projection of the side will be determined by the cosine for. Cosine fi. Means that x, y, z is trusted, fi, and we have identified in terms of theta. Now here again We need to find sizes. We napÄ±yon? t even Cartesian one variable keep hard, we are increasing slightly. On a sphere Let's keep hard before r. After that, let's keep a constant theta Keep the theta time that he the z-axis in such a line We came circling the south pole of the sphere. Here we have drawn bi meridian. Increase a little more of this theta We have found the second bi meridian. When it comes to latitude, latitudes variable radii. In the same meridian radius. Has always been. Where e, how do we find this latitude? We know geography. Your position at one point, Although this example from Greenwich Istanbul came over from the lat. E are changing the way theta. Changes to these theta and its radius is small. The radius of the meridian is great. Small r is the radius of latitude. Now let's write them. Here from A to B. ie latitude We start by finding the spring on. This is a small delta theta times. Of course, from a small spherical coordinates r but we know it. Small r r times larger sinus fi. But our work in geometry that works. The length of the radius of the meridian is a great and we get back up to the delta for the length of time we have achieved. This is shown here with a length d. It is also great for times of the delta. BI is also finally got the delta. Delta is in this sphere rather than the radius r r plus we take the delta delta r We will be gone. For example, I want to give as an example b. Consider Bi orange peel. Has thickness of orange peel. Let's say it is the delta. Let's draw on the meridian. Two of the meridian line. Two of the latitude lines. See the idealized orange here, assuming that the global This small rectangle but curved edges We find this quadrilateral formed. Orange peel is the delta. Because the inner radius r. R is the radius of the outside plus the delta. That's the point. Now if you multiply these three size here still bi bi six-sided prisms have achieved. Curved edges. But not all sides of equal wherein with respect to infinitesimal In your second order error will be this is approximately the volume of data. So this publication length of a circular b'yl in length, radius neck meridian delta r gets hit in the neck length we obtain the volume. Here is a small place is great sinus Insert the plug is large sinuses have fun. There are many BI here. R squared was for him. Then run the delta, for the delta, delta theta. As you can see this in simple geometry we can get. It's important to understand something. Because all the three bi size of thinking about thinking You need to improve your skills. Geometrisel it a bit more, To understand the geometry of different We can try to understand. The same as we did on the surface of the cylinder. See, I have given this just in Or of orange peel meridians over the world, from the intersection of longitude and latitude The edges of these four We call it the delta water to the field. This delta p as follows:This along meridians on a d for latitude with the r multiplied by the delta theta. Here we see this product. Longitude are traverse length on times because the delta for longitude radius r on large hard. However, there is little on latitude. R times smaller delta theta. But at small r r times We know that the sine fu. Here, have this. The thickness thereof still ka, On the obtained orange peel If he thought a small volume of the delta r'yl We find the same size gets hit again. However, these three issues in which steep, If you hit the hit will be the same thing. But here it gets bi container, thickness of the bump, the horizontal area where this apartment have latitude here as horizontal space. Latitude is the length of time the small delta theta, r times smaller delta theta. Delta is the other aspect. Small r r times larger We know that the sine fu. Delta delta r teta'yl product. This gives the floor area. Delta gives. Times the height. Height rather than the latitude longitude traverse length on. He is also the time for the delta. We find the same thing again. Finally, these three vectors were finding the product. Algebra, vector algebra from the we know that u, v, w, three vectors Although all three of these multiplication mixed gives the volume of the vector form. On a sphere e is the vector from the center. Meridians of longitude on the vector e f. Or on latitude and vector E parallel on theta. Their length is the delta, the delta for the teta'yl those obtained by multiplying the delta. This vector also certain to find our way. As we saw in the former space of T The only parameter curve is determined. He received the derivative with respect to the parameter space of We find on the curve tangent. means the partial derivative of x by r-theta Or do not and kept constant for. So the only variable by r The vector data for derivatives of r. It hit the delta r'yl that this vector edge with the vector length We're getting hit. Similarly in the direction F The following derivatives of this gene has changed so lon vector on giving. Longitude also when passing on the delta for We're going to have to give this size. Similarly, on the latitude theta is changing only. From the derivative with respect to theta Because the partial derivatives are and f is kept constant here. Delta gets hit in teta'yl We find that edge. When we multiply these three, triple product here. Triple multiplication here Let's write clearly. X is the position vector of the space. r, fi, in terms of theta. When we get its derivatives, by r partial derivatives, According to the plug, wherein by theta There's no use telling my bi. These are very simple derivative. These sizes are achieved. They're going to get the triple product. So it stands for bi previous page. Means taking triple product The first line of the first vector, The second line of the second vector, the third line of the third vector writing said three by three of the thus obtained means to find the determinant of a matrix. This u, v, w product. Delta r for the delta, delta theta common multiplier Because they are coming out. See here quite a few such as mixed Even though the terms that appear in mathematics the beauty of these terms of I can easily see each other only at the end of simplifying r squared sine fi is staying here. So it is zero line or When you can see by the column these terms square cosine theta plus A square sine of theta is going up once. Sine-squared for the gives reunite for a cosine square. Back just a sinus fi is staying at. As you can see that they still others give the same result. But this blind also you can do The geometry also thinking something you can do. Because the Jacobian determinant here. We have already seen two bets, two episodes ago. In case of three variables jakobiyan with changes in parameters You can find the volume gets hit. In two dimensions, when the Jacobian gave an infinitesimal area. So it can be done blindly. So the infinite sphere Find a small volume would have. Now moving on rotating objects I want to call again before b. Bye for now. Surface on the rotary body We have to calculate the area. Wherein one surface area the volume occupied by packed assuming that the volume Finding ways to be seen. Bye for now. After completing this approximately infrastructure we have created and then We will see examples of various applications. We choose examples from everyday life Examples of technology will be encountered. Goodbye.