 Dear student, today we are going to learn the Jacobian. So, what is the Jacobian? Basically, the Jacobian is a matrix of partial derivatives that describe the rate of change of a set of variables. So, basically, what do you have Jacobian? Transformed variable's rate of change. This is called the Jacobian. So, here is the introduction of the Jacobian in mathematics. The Jacobian is a matrix of partial derivatives that describe the rate of change of a set of variables with respect to the another set of the variables. You have the Jacobian as a set of variables with respect to the another set of the variables. The Jacobian matrix provides a way to compute the gradient. So, gradient is the inclination of a line or you can say that the slope of a line or the vector of the partial derivative or the vector valued function. So, this is the function. Function f x1, x2 up to so on xn. This is the function that maps n variable to m variables. The Jacobian d of f is an m into n matrix. Here is the m into n matrix. Whose elements are the partial derivative of the output variable with respect to the input variable? That is, journal term consider f x1, x2 up to so on xn. Now, look at this. This is the function. Partial derivative we have taken function 1 with respect to x1. Partial derivative f1 with respect to x2. Partial derivative f1 with respect to xn. This is the matrix made by you. Whose m into n? Second, we have function and that we have taken partial derivative with respect to x1, f2 with respect to x2, f2 with respect to xn. Up to so on, how much we have made? m into n matrix. Partial derivative of fm over partial derivative of xn. So, look at this. Is the each row of the Jacobian represent the gradient of one of the output variable with respect to the input variable? Slope of a line of the output variable with respect to the input variable. Overall, Jacobian matrix is a powerful tool for analyzing and understanding the behavior of multivariable functions. Now, here we do not have one variable. We have a multivariable, i.e. x1, x2, up to so on, xn variables. Here is the example 1. A simple numerical example to illustrate the Jacobian in multivariable calculus. Suppose we have a system of two equations. We have the equation number 1, x2 plus y2 equals to 4 and the equation number 2, x plus y equal to 3. Here, we can define a vector valued function. This is the function f, f1, f1 and this is the f2. That represents the system of equation. Look at this. Here is the function f1, f2. In f1, we have the first equation and in f2, we have the second equation. To find the Jacobian matrix of f, we need to compute the partial derivative of each component of f with respect to x and y. Journal form, Jacobian f of 1 and f of 2. Curly f1, i.e. partial derivative of function 1 with respect to x. Partial derivative, we have f1, i.e. the first equation with respect to y. So, look at this. Here is the curly f1 over curly x. This is the f1 and this is the f2. Now, we have taken the derivative of f1, partial derivative with respect to x. If we take the derivative of f with respect to x, what will happen? 2x. Here is the 2x. Then, we have taken the derivative of f1, i.e. partial derivative of the first equation with respect to y. What did we get? 2y. Similarly, the second derivative of f2, we have taken the derivative of f2 with respect to x. When we take the derivative of partial derivative with respect to x, you get 1. And partial derivative, we have taken f2 with respect to y. Here, we have taken f2 with respect to y. So, you get 1 partial derivative. Now, put these values in this matrix. So, first, we have the here is the 2x. Then, 2y, 1 and 1. We have the Jacobian of the f1 and the f2. To evaluate the Jacobian matrix as a specific point, for example, Suppose 1 and 2, 1 for x and 2 for y. This is the Jacobian of the function of f1, f2, 2x, 2y. And y square, we have replaced it with 2. Now, how do we evaluate the Jacobian further? We have to take the mod of the determinant of the Jacobian. If we say mod, we are taking its determinant, i.e. we take its positive value. We have to find its determinant. For determinant, you know that we have the diagonal term multiplied by the off diagonal term with the negative sign. So, we have the determinant minus 2. This is the mod of the Jacobian, modellized. So, modellized means negative value is positive. Here is the mod of the Jacobian. We have the Jacobian of 2. Similarly, here is the another example. Example number 2. Now, how many variables do you have? With two variables. Now, what are we doing next here? A simple example to demonstrate the Jacobian in multivariate calculus. Suppose we have the system of equation. Now, here we have three equations and we have three variables x, y and z. Now, f1, x plus y plus z, this is the second equation, y minus z and here is the third equation, x plus y plus 2z. We want to calculate the Jacobian matrix at a specific point. Previous, we have a specific point, which according to which we found the Jacobian matrix, was 1 and 2. In a particular example, we check if we have any value or just we have x, y, z, or they have 1, 1, 1, 0, 1, 2, any value. On that, we have to check the Jacobian matrix. The Jacobian matrix is a matrix of partial derivative. You know that this is the matrix of the partial derivatives where each entry i and j represent the partial derivative of the i-th equation with respect to the j-th variable. This is the partial derivative of the i-th equation with respect to the j-th variable. To calculate the Jacobian matrix, we need to find the partial derivative of each equation with respect to x, y and z. Now, we have to take its journal term, partial derivative with respect to x, y and z. This is the journal form, curly f with respect to x, curly f2 with respect to x, curly f3 with respect to x. We have differentiated function 1 with respect to x, x differentiated function 2 with respect to x, 3 with respect to x. Now, function 1 with respect to y, function 2 with respect to y, function 3 with respect to y. Similarly, this is the 3 cross, 3 with respect to Jacobian. Now, when we are taking its derivative, you know that the first equation we have is x plus y plus z. Here is the x plus y plus z, y minus z, y minus z and the third is the x plus 4 y plus 2 z, x plus 4 y plus 2 z. First, we have f1 with respect to x differentiated, you have 1. Then f1 with respect to y differentiated, again 1. I am doing it in the first column, f1 with respect to z, 1. Okay? In the second equation, you are seeing that x is not there. That is, what we have here is f2 with respect to x, x is not there, 0. f2 with respect to y minus 1, 1 and with respect to z minus 1. So, we have taken the partial derivative of these three equations. After that, we have found its mod and the value of the mod has come to us, 4. So, this is the mod of the Jacobian. The Jacobian matrix basically provides the important information about the local behavior of the system of the equation. You have the system of the equation. Students, in this particular example, we have checked the transformed variable's rate of change find. The variable we have transformed, we have taken its derivative, transformed variable, we have found Jacobian, we have found its rate of change find. So, you have the mod of the Jacobian, the mod is with us of the Jacobian. Now, this Jacobian will be used in further multivariate normal. So, the basic concept we have determined here, then we will apply it further in multivariate normal distribution.