 Hello, and welcome to this screencast on section 11.9, Change of Variables. When we switch between Cartesian coordinates and polar coordinates for double integrals, or between Cartesian, cylindrical, and or spherical coordinates for triple integrals, we're using a change of variables with a specific set of substitutions. In this section, we'll look at how to apply a change of variables for double and triple integrals that will work no matter what we substitute. Let's start with double integrals. Consider the double integral of the function f over the region d. Suppose this integral is difficult, or even not possible, to evaluate in terms of x and y. We wish to simplify the integral by making a change of variables, where we substitute x and y for new variables, s and t. To complete the substitution within the integral, we need to understand how the area element, dA, changes with the substitution. To understand this, we use a familiar procedure. We proceed with the basic idea here. You can see your textbook for additional details. Starting with our new variables, s and t, we partition the corresponding intervals of s and t into subintervals. These subintervals partition the region we are integrating over into subrectangles. Let t be one of these subrectangles, pictured on the left. T lives in the st plane. This means that there is some region of x and y that maps to t under our given substitution. Let this region be called t prime, pictured on the right. To approximate the area of the region t prime, we use a parallelogram formed by the vectors v and w. It turns out that v and w can be approximated with the following two vectors, which involve partial derivatives of x and y with respect to s or t. The area of this parallelogram, formed by v and w, can be found using the magnitude of the cross product of these two vectors. As the number of subrectangles we use increases without bound, this expression turns into the following, which tells us how the area element changes with our given substitution. This quantity is called the Jacobian, where we denote it with the following shorthand notation. Note that the Jacobian can be written as the determinant of a following two by two matrix made up of these partial derivatives. We summarize our work with double integrals here, showing how to convert a double integral from variables x and y to new variables s and t. Switching now to triple integrals, the general process for change of variables is the same. However, the argument used is complicated, so we won't go into the details here. Instead, we'll state the Jacobian and summarize how to use change of variables for triple integrals. First, it turns out the Jacobian for triple integrals, given by this notation, can be computed using the following determinant of this three by three matrix of partial derivatives. Recall from earlier in the course, we can compute a determinant of a three by three matrix as follows. Lastly, we summarize our work here, showing how to convert a triple integral from variables x, y, and z to a triple integral with variables s, t, and u.