So, compared to the finite difference or the pseudospectral method, that was a very long derivation, which is why I'd like to recall just the steps so that in the end, you remember where we actually came from. So, let's start writing again the wave equation that we started, and I put the term containing the space derivatives on the left hand side with a minus sign. We integrated over the entire domain. We multiplied with a test function that we call initially Phi i or Phi j, and then we move to a local coordinate system that we called the domain De, the domain De for each element, that we said this because of the interpolation with Lagrange polynomials, that are defined from minus one to one. The equation that we obtained was the following, and you can see it here. Now, let's remember again and highlight the vectors and matrices. So, we have u the unknown vector of displacements. We have the mass matrix m, which are integrals over density multiplying the basis functions, and we have the stiffness matrix, where we have multiplication of the shear modulus, mu and the derivatives of the Lagrange polynomials. Finally, we also have the force terms where we basically can inject an external force. Now, what actually happened then to these matrices and vectors through the specific and numerical integration scheme and interpolation scheme that we introduced? Let's start with the first term. The multiplication of the second time derivative of the displacement, multiplying the integrals over density and the basis functions became a multiplication of the wave field u and the mass matrix. After putting everything together, the entries of the mass matrix M ji, actually simply the weight, the integration weights, W, multiplying the density at the collocation points Psi j, times the Jacobian, times the Kronecker delta, Delta ij. Actually, of course, that means that the mass matrix is diagonal, and that's one of as mentioned before, maybe the most powerful aspect of this particular scheme using Lagrange polynomials, and at the same time the Gauss Lobatto Legendre integration using the same integration points. The mass matrix is diagonal and we will later see also in the code, this leads to an explicit scheme just like the finite difference schemes. So, no complicated and expensive matrix inversion has to be carried out, because the matrix inversion of a diagonal matrix is of course trivial. So, the second term contains the wave field u multiplying the stiffness matrix, Kji. Kji turned through the numerical integration scheme and the Lagrange interpolation into a sum over the integration weights w. The shear modulus mu evaluated at the collocation points, and the first derivatives of the Lagrange polynomials, which you now know how to calculate. Remember, that could be done making use of the matrix dij and Legendre polynomials. So, that's straightforward. Finally, we have the force terms on the right hand side of the original equation, which is an integral over the force terms evaluated also with original integral of the force terms, but now turning into a sum also as containing the weights, multiplying the force values at the collocation points. Basically, now we know how to calculate each of those matrices. Will lead of course, again to come up with the final algorithm to apply the finite difference scheme to the time dependent vector of unknown displacement field u, and now we're done. We can develop the final algorithm with which we solve the equation system. We can now write down this system of equations that came in matrix vector form, and we have the mass matrix multiplying the second time derivative of the displacement vector. We have the stiffness matrix multiplying the displacement vector. On the right hand side, we have the force term, but this is still at the elementary level, which is indicated by the superscript e. Now, one thing remains to be done, is we need to basically put all these elemental things together to come up with a global system of equations. That process is called Global Assembly, and this will be the final step before solving the system.