Hello. Our main preceding session physical size of three conservation laws was removed. As you can see all three physical Although many different origin of the exactly the same mathematical method and Gauss, Or Green Gauss divergence Using the theorem is achieved. Now a vector magnitude We will achieve conservation. Before this mass of what we obtain a electric charge numeric function numeric functions, temperature, followed numerical functions of heat energy. Now, the vector functions We will see the use. Vector-borne momentum basic sizes, electric field and vector field. We remove the heat flow equation, heat flow caused by temperature all, Put the rules of the Fourier heat flow is defined and is going to appear. Now here's momentum I'll start by conservation. It has three main areas of classical physics. Mechanical strength, mass of working with such size, here the most important momentum One of the concepts. Thermodynamics of heat and temperature are interested and with the current, with transmitted. Third, the electromagnetism. As I told numerical heat transfer function can be solved by We have to finish it. Now we come to the conservation of momentum. We all know that since time immemorial, Newton's The second law is a fundamental rule of nature. Momentum is the force derivative with respect to time. We know the momentum, mass times velocity. If we say that there is a change in speed and mass of the If we work with the systems momentum means that the time derivative of the mass fixed that is the time derivative of the speed. Speed with respect to time We call derivative momentum. So Newton's momentum conservation force equals mass times acceleration law Coming at this time, since many everyone, including high school we have learned. Now we will do our continuous For media will do it. Mean continuum solid Or a liquid substance or a gas, their dynamics. Here, let's write a delta V volume before. If the density of a small delta V ro ro times the delta V is the volume of the mass. We want to find more momentum precisely where the mass times acceleration We want to find the mass times acceleration ie with a mass of these ro V'yl delta, Delta V mass an infinitely small ro volume, it is multiplied momentum. This is a delta V. You find the total mass times acceleration If we want to subject an integral collection, Why is it we're getting triple integrals because we are working in three-dimensional space and still the same that the previous We limit as a region. Ron mass in this region have because the force may be inside, forces from the border could be a such as the accumulation of water in a pressure Or you book a solid object boundaries in a taking There are forces from outside the region, We show them how to sig. This unit of area We call it stress. Multiplied by n, it gives us strength, Sigma's a force multiplied with n See gives a vector force. Although you are just a vector sigma took a number from the inner product, whereas the sigma-called tensor matrix in the nature of a thing. Two indicators are determined, Showing vectors in a single index It is stated in a single indicator. Here, perhaps, the first that can confuse but it looks like this might force To remove the border of a vector which forces the contents of a quantity of multiply a vector with n not to give the sigma vector, matrix must be it is called the tensor physical aspects. This double indexing We show called sigma i j. Of course, the strength of this was seen in the electrical engineering many of the properties of crystals, etc. Showing with such tensor. Now what do we do? We'll find strength, we have found here. Again, this will be our kind of liked. This forces from the border force therefore, an integral two-storey, but structure because this is exactly what we wanted in the components of the tangent to the boundary limits it does not produce a force of sigma they come and go tangential limit. Produces a force perpendicular to the boundary components, This will force acting on the object generates. This tangential components also have the ability to but I do not want to mix it here. Again, this divergence theorem When we use See this comes sigma divergence. So you have found the force. Our Newton's law was, mass times acceleration equals force. E mass times acceleration We found; ro times the mass dv, it also three acceleration dV dt ply because the integral The volume contains over The mass times acceleration. So as the situation, from outside forces within force, but considered I think about that right now, Let's take a simpler model many Although he can be handled in a way. From outside forces such as under You are throwing a ball from outside forces and the mass in the body would give a momentum equation this. Now both of these random volume is on in any case it for In order to validate this integration of the contents of must be equal to each other. Here ro divergence dV dt We find that even came to sig. See here for a vector There you have the size of a vector. Sigma tensor matrix vehayut physical sense by saying, Get it by an inner product here comes a vector. Now here's a vector There are a total derivative on the field. If you remember some full on derivatives for example, a full-derivatives with respect to t the partial derivative of f with respect to t full derivative plus the derivative of f with respect to x that the derivative of x with respect to t gradient had already come here. If we put in place for any one in this place We will obtain, but the x position, the time derivative of the position vek, speed velocity vector. This also means that we have here where the total derivative dV dt stands for the partial derivative Once these have pros have twice because it looks exactly the derivative of v is We take this for coming there instead. In this way, we are found. This is the general condition of the momentum equation. Where will the momentum, I'm here, I hit a solid object would be in When a wave would produce a I change the shape of solids When I'm doing an event. That's it, but I'm doing this through strength outside forces affecting the body This sig is defined mAlArlA. This sigma studies, this time taking the divergence everywhere sigma value is reached. Now continuum in each of four Is the basic conservation laws. A mass will be protected and will be protected momentum, angular momentum will be maintained and energy will be preserved. Now with some idealization we can simplify it. No angular momentum now Let's not confuse it be shallow so that We also stress the stress tensor on the tangential component. For now, ever confuse him. We saw mass conservation. Momentum conservation equation was removed. If the heating temperature does not interfere it is self-balancing and self-providing drops from the equation. Thus, conservation of mass, with back conservation of momentum remains. In the first session the conservation of mass We have issued the law. Now we have the momentum interests. Or so the room environment We think that the temperature is constant, We do not need the heat equation. Electromagnetic properties Suppose that, not subject to an electromagnetic effect. Thus, the sheer mechanics a particularly trying. Now there are solids, liquids there. Fluid is also divided into two parts. Them apart We need to distinguish. They can be distinguished With this happening and sigma With ro happening. Now with solid objects The main difference of fluidized solid If so how much change in body materially change the environment Near points equilibrium position remains, so they do not drift too far away. However, in fluid full the opposite can move too far away. A lot of intensity in solids associated with the previous one because it does not change of particles is very does not always go away close vicinity of each other A remaining deformations, If there are deformations ron can be assumed to remain constant. Therefore, the conservation of mass self- need not be provided, and equations. There are two types of fluid in the fluid We said:a fluid in a liquid. Sıvılarda- 'excuse me from the gas in the fluid There is a liquid with a gas. The most important of liquids with gases if the difference of the intensities. Incompressible fluid in the environment i.e., density change, may vary slightly, perhaps, Even if you do change in water pressure. But ten thousand of meters from the sea Are you going down to the bottom, There are great pressures on him nevertheless does not change its intensity. Most of the liquid The alteration in feature density nozzle. And with the overall strength of a pressure determination. Fluids can be divided into two:a There are excellent fluids, therein parç- of water molecules with each other friction and do not lose energy, If there is no friction at the macro level We call them the perfect fluid. These equations that take Let's say iii equation. But in many liquids of various There are in the order of friction. Both go through the wall of the tube with There are energy losses due to friction, both among themselves momentum exchange are doing and therefore this friction energy loss, the loss of momentum going. These are not perfect, Or frictional viscosity liquids. These are the Navier-Stokes equation he is taking an equation. We will see them. Gas in full, the situation is different. Here incompressible media and he does not change the density, on the contrary, the density in this pressure is changing, and indeed we know that. When we blow air into a balloon at As the pressure increases the volume is changing. Therefore, the pressure-density there a relationship between. Yet these gases in the line at the same In some dynamic agent perfectly acceptable, friction is important in some cases. We call this the viscosity, As in the same liquid. Important gases with liquids similarities in the material points can go too far, can replace large. But differences in the volume of liquid de- density does not change, can not be compressed. On the contrary in gas be compacted at important features. Momentum loss may or may not, perfect fluids, excellent fluid or gases can be perfect. If this is a gas in the density and pressure Small changes speeds If it shows them sound wave form and acoustic call They constitute the subject of science. Now we first floor Let's start from objects. Though even if the solid liquid the momentum equation is the same. Of particles in solids They do not change much space. We displacement is based. If you say displacement, v is the time derivative of displacement. DT square frame is going to say here, that is. The second derivative is coming. We know this already from classical mechanics x is equal to the force at two points. X is equal to mass times the two points he knew force. Here comes something similar. As less of these substituted neglecting the nonlinear size We can here because There is a small value. of the derivative, the derivative is also a small space. This small gradient that. This non-linear, nonlinear term, we are taking it. This approximation can do. Rode hard call. ro is staying here. See our equation solid comes to the following structure for the objects. But I still have to report to sig. On the nature of these substances sigma given by something. Now we equation Let's look at the simplest structure. u is a vector Although the three constituent components can numerically that Although only one from, and Although only the x and T-dependent one-dimensional wave will lead us to the equation. Already wave equation We have seen the beginning. u dt d square frame looks here. Or he and iii law as the coefficient of elasticity Or elastic modulus as equivalent to each other terim- Progress has a known e. It says on an object Add a tensile which he did with coefficients değiştirmesi- way deformation of the u AnIdIr gradient substantially. In one variable x to It is based on partial derivatives. Such given he says. Them so that also in physics courses Nuzi anyway. Now here, in fact sigma a matrix and a tensor. But one sigma, were reduced to a size where simplification because when the other one-dimensional sigma It was the only x-dependent one-dimensional. A numerical value divergence sigma DX will give you square it is will lead us to the square. Now take it when positioned here Our problem is solved now. Here is e times the square will be the DX frame. See here once rode on the left DT had square frames d. There are other times square to square DX. See also this same physics we know mass times acceleration. See that this is a mass momentum. Displacement with respect to time the second derivative is equal to the force. Understanding that such a force of the force was attempted. Now take this rode divided by right-hand side, we say that c square, Because here because squares, This shows the speed of the outgoing wave. That Lack the wave equation di. That is the nature of the three basic equation We have removed, with the most simple cases. Now the three-dimensional elastic body If we look at the situation in tau followed If you can have a little difficulty, but the main The idea that you kavr I hope and believe. We have the momentum equation was as follows: is the time frame of the vector ie the mass of the second derivative these times acceleration equals force. Now this object from another This definition distinguishes sigma. As in solids is happening to this sig It's called stress, stress English It's called, in our daily lives entered in the psychological stress situation. This displacement by sigma see between a There was a law in size very easy. The strain of the associated with a gradient measure. This structure also in three dimensions as follows:iii and iii twice in numbers as The simplest case is determined. Iron and steel work for them More concrete another other times for plastics numbers. Thereof is determined by experiments but all this is in the structure. where u is the divergence times a unit vector here because I We want to create a tensor, We want to create a matrix. gradient of a matrix, it also MATS We're taking the transpose, it is also a matrix. Now we'll take this divergence. That's when you get the divergence of these so you have to follow step by step but not essential, as is an equation, This equation is called the elastodinamig. How in the Gauss' law This vector basically plus a rotational gradient useful as a reserve by We have seen things here places and this equation diverjan- If we take s and rotational two kinds of visible waves. See them again, wave equation, hyperbolic equations. iii equals variable numeric the second derivative with respect to time variables or According to the second derivative of the fourth variable. This is a wave equation. These terms are also used a wave equation. See where the coefficients When we divide the speed we were finding. Wave is moving at a breed specific, A second wave is going at another rate. It earthquakes in our daily lives We see in the waves. They squeezed in the direction of progress shows the outgoing waves. Their speed is greater. First, these waves are seen Or in seismology at home If you saw sitting earthquake shaking. behind are s is more destructive than the shear wave The building shook with such cutting-called or bottom of such waves waving come, they come in different speeds. They are coming back, Unfortunately, the waves are becoming more effective. Now we see solid objects. Now the same momentum equation We will apply fluid objects. Now here I am taking a break. But they internalized thinking lest time occurs. Goodbye for now.