Previously, we dealt with one equation, defining just one implicit function. So, we either considered a function which had x and y two variables, and we were looking for a function y as a function of x, or in the latter case, we dealt with one equation, which defined a function y. This time a function of many variables, but quite often in economics, we have to deal with the systems of equations. We're looking for several implicit functions or sometimes we can call it vector valued functions, but I will try to abstain from this term. So, let's consider m functions, they are functions of x and y. So, the arguments for all of these functions are written in the form of, I'm using x letters and also y letters. So, this is equation number one and I continue until I reach the final equation. So, the question is whether it's possible to find m functions denoted by y letters, all of them being the function of x alone. But firstly I need to provide a definition of a very important and quite useful concept in the calculus of several variables, which is called Jacobian or Jacobian matrix of a system of functions. To begin with, we have written within the braces, the system, each of them depends on n variables and within the system we have m functions. Now, let us introduce a matrix, each row of which consists of a gradient of all these functions, starting from the first. So, in order to fill in the entries in this matrix, we take the very first function, f1, and we differentiate with respect to x and fill in the entries. So, it becomes. Now, in order to fill in the second row, we need to take the second function, f2, and differentiate it with respect to these variables, that's how we get and we continue until we reach the nth function at the very bottom. So, here I put dots indicating that there are more rows until we reach the nth row. That's how we get this matrix, which has the name a Jacobian matrix or simply Jacobian for short, and this is m by n matrix. As I said earlier, it's made of the gradients of all these functions drawn from the system. Now, going back to this implicit function theorem number three this time. Let's suppose that all these functions are defined in some ball, so they're continuously differentiable, there is a ball. Ball centered at point x0, y0, it belongs to the n plus m dimensional vector space, R n plus one, n plus m. We are considering a system, let me unite all these m equations with a brace and I'll write a number for this system of equations. We assume that this particular point satisfies all m equations being after substitution. Now, we need to formulate a condition under which the system of implicit functions exists. This condition is formulated in terms of the non-singularity of the Jacobian matrix. This Jacobian matrix is based on these m functions but we differentiate only with respect to y variables, that's how we get a square m by m matrix. So, the condition which we need to check in order to be sure that m implicit function will exist, takes the form the determinant of the Jacobian matrix, J, is not zero at a given point. Where J is a matrix, as I said, a square matrix where we differentiate respectively, all these functions with respect to y variables. So, we have here, I'll fill in the final row. So, once again let me repeat, we deal with a system of equations based on a system of functions. All these functions are continuous and differentiable on some ball. The ball has a center at a given point, and this particular point satisfies all these equations. Now, we check at a given point the non-singularity of the Jacobian matrix which is a square matrix. We have information, this is a non-zero number. Now, the claim of the theorem is as follows, there exists a ball, B tilda, in n-dimensional space centered at x0. There exist a system of implicit functions. I will be using a shorthand notations in the vector form to make it shorter. This theorem claims that there exists a ball, B tilda, in n-dimensional space centered at exactly x0 point. There exists a system of implicit functions from y1 through y nth, and the following conditions hold.
