Calculus/Change of variables

Multiple integration Vector calculus The Jacobian matrix and the change of variables are proven to be extremely useful in multivariable calculus when we want to change our variables. They are extremely useful because if we want to integrate a function such as, where is the trapezoidal region with vertices ,it would be helpful if we can substitute as and as because is easier to be integrated. However, we need to be familiar with integration, transformation, and the Jacobian, which the latter two will be discussed in this chapter. Transformation Let us start with an introduction to the process of variable transformation. Assume that we have a function . We want to calculate the expression: in which is a region in -plane. (Another notation for is . ( here is not differential.) ) However, the area of is too complicated to be written out in terms of . So, we want to change the variables so that the area of can be more easily expressed. Furthermore, the function itself is too hard to be integrated. It would be much easier if the variables can be changed to more convenient ones, Assume there are two more variables that have connections with variables that satisfy: The original integral can be rewritten into: in which is another region in -plane transformed from the region in -plane. The purpose of this section is to have us understand the process of this transformation, excluding the part. We will discuss the purpose and meaning of in the next section. Introduction In fact, we have already encountered two examples of variable transformation in . The first example is using polar coordinates in integration while the second one is using spherical coordinates in integration. Using polar coordinates in integration is a change in variable because we effectively change the variables into with relations:As a result, the function being integrated is transformed into , thus giving us:, which is the formula for polar coordinates integration. (It will be proved later) The second example, integration in spherical coordinates, offers a similar explanation. The original variables and the transformed variables have the relations:These relations can give us that, which is the formula for spherical coordinates integration. (It will be proved later) Generalization We understand the transformation from Cartesian coordinates to both polar and spherical coordinates. However, those two are specific examples of variable transformation. We should expand our scope into all kinds of transformation. Instead of specific changes, such as , we will talk about general changes. Let's start from two variables. We consider a change of variables that is given by a transformation from the -plane to the -plane. In other words,, where is the original or old variables and is the new ones.In this transformation, are related to by the equationsWe usually just assume that is a transformation, which means that have continuous 1st-order partial derivatives. Now, time for some terminologies. If , the point is called the green image of the point . If no two points have the same image, like functions, , the transformation, is called green one-to-one (or injective). transforms region into region . is called the image of . The transformation can be described as: If is green one-to-one, then, like functions, it has an green inverse transformation from the -plane to the -plane, with relation Regions Recall that we have established the transformation , where is the region in the -plane while is the region in the -plane. If we are given the region and transformation , we are expected to calculate the region . For example, a transformation is defined by the equationsFind the image of , which is defined as .In this case, we need to know the boundaries of the region , which is confined by the lines: If we can redefine the boundaries using instead of , we effectively will find the image of . As a result, the image of is We can use the same method to calculate from . The Jacobian The Jacobian matrix is one of the most important concept in this chapter. It "compromises" the change in area when we change the variables so that after changing the variables, the result of the integral does not change. Recall that at the very beginning of the last section, we reserved the explanation of from here. To actually start explaining that, we should review some basic concepts. Review "u-substitution" Recall that when we are discussing -substitution (a simple way to describe "integration by substitution for single-variable functions"), we use the following method to solve integrals.For example,If we add endpoints into the integral, the result will be: If we look carefully at the "rearrangement" and "remember" part in the solution, we find that we effectively changed our variable from to through this method:, which is what we have mentioned above. The appearance of the term not only is a mathematical product of deduction, but also serves a intuitive purpose. When we change our function from to , we also green change the region we are integrating, which can be seen by looking at the endpoints. This change of region is either green "stretched" or green "condensed" by a factor of . To counter this change, is deduced to compromise (recall that ). We can simply think this term as a compromise factor that counters the change of region due to a change of variables. Now, let us put our focus back to two variables. If we change our variables from to , we also green change the region we are integrating, as demonstrated in the previous section. So, continuing our flow of thought, there should also be a term deduced to counter the change of region. In other words:Note that the symbols used here are for green intuitive purpose and not for official use. Official terms will be introduced later in the chapter, but for now, we use these terms for better understanding. In this case, when we change the function from to , we green "stretched" or green "condensed" the area of our region, by a factor of ; therefore, we need to counter the change with a factor of . The Jacobian matrix for two variables is basically an expression for calculating in terms of , so that we are able to integrate the new integral after transformation, since the function involved in the new integral can only in terms of , but not (we need to express and in terms of ). The Jacobian Double integrals Now, it is time for us to deduce the Jacobian matrix. In the review above, we already established green informally that the Jacobian matrix for two variables is basically , with being the infinitesimally small area in the region in the -plane and being the infinitesimally small area in the region in the -plane. Since we are changing our variables from to , we should describe and in terms of over a region in -plane. Let us start with first because it is easier to calculate. We start with a small rectangle , which is a part of , in the -plane whose lower left corner is the point and whose dimensions are . Thus, the area of isThe image of , in this case let's name it , is in the -plane according to the transformation . One of its boundary points is . We can use a vector to describe the position vector of of the point . In other words, can describe the region given thatThe region now can be described in terms of . The next step is to utilize the position vector to calculate its area . The shape of the region after transformation can be approximated, which is a parallelogram. As we learnt in algebra, the area of a parallelogram is defined to be the product of its base and height. However, this definition cannot help us with our calculations. Instead, we will use the cross product to determine its area. Recall that the area of a parallelogram formed by vectors and can be calculated by taking the magnitude of the cross product of the two vectors.In this parallelogram, the two vectors and are, in terms of :It seems very similar to the definition of partial derivatives:As a result, we can approximate that:Now, we calculate , given that :We can calculate (we take absolute value to prevent negative area). You can review the cross product in Chapter Vectors. Note that the inner bar of || is for calculating the magnitude (or norm) while the outer bar of || is for taking the absolute value.Then, we can substitute our newly deduced terms. Finally, we derived the green absolute value of Jacobian. The definition of Jacobian is as follows: darkblue Definition. (The Jacobian for two variables) The Jacobian of the transformation given by and whose partial derivatives exist and are continuous, is We will then use the Jacobian in the change of variables in integrals. The absolute value is added to prevent a negative area.Here is the theorem for the change of variables in a double integral and we have green explained intuitively why and how it works, but the above explanations are green not green proof of this theorem. In particular, we make some approximation, while the statement in the following theorem is green equality, and not approximation. The actual green proof is quite complicated and advanced, and thus not included here. tomato Theorem. (Change of variables for double integration) Suppose is a transformation whose Jacobian is nonzero and that maps a region in the -plane onto a region in the -plane injectively, via the change of variables and . Suppose that is continuous on , we have rosybrown Remark. If we change some notations, we can get ( maps a region in -plane onto a region in -plane in this case.) which may be a more convenient form to be used sometimes. <quiz display=simple> { red Example. } {Choose correct expression(s) for the Jacobian in which and .} + || Since and , ||=\begin{vmatrix}\frac{1}{2}&0\\0&\frac{1}{5}\end{vmatrix}=\frac{1}{10}</math>. - || This is the expression for . - || This is the expression for . - || This is the expression for . - || This is the expression for . {Choose correct expression(s) for the integral in which is a region bounded by and . |type="[]"} - || Let and , and the bounds in a transformed region via this change of variables are and . || Then, the Jacobian is || Thus, - || Let and , and the bounds in a transformed region via this change of variables are and . || Then, the Jacobian is || Thus, - || Let and , and the bounds in a transformed region via this change of variables are and . || Then, the Jacobian is || Thus, + || Let and , and the bounds in a transformed region via this change of variables are and . || Then, the Jacobian is || Thus, + || Let and , and the bounds in a transformed region via this change of variables are and . || Then, the Jacobian is || Thus, {Choose correct statement(s) from the following statements.} + If both and are independent from both and , the Jacobian . || This condition implies . || Thus, the Jacobian equals zero. + If both and are independent from both and , the Jacobian . || Since both and are independent from both and implies that both and are independent from both and . (They have the same meaning) || This condition implies . || Thus, the Jacobian equals zero. + If is independent from both and , while is dependent from both and , the Jacobian || The condition is independent from both and implies that . || Thus, the first row of the Jacobian a zero row, and it can be seen that the Jacobian equals zero. + If is independent from both and , while is dependent from both and , the Jacobian || The condition is independent from both and implies that . || Thus, the second row of the Jacobian a zero row, and it can be seen that the Jacobian equals zero. - If in which is a real number, then the Jacobian || Since and in this case, || the Jacobian equals || So, this statement is incorrect unless , i.e., it is not always correct and thus not a correct statement. </quiz> red Example. Consider a region that is bounded by the lines , , and . Prove that Proof. Let and , and be the transformed region via these changes of variables. Solving these two equations, Therefore, the Jacobian for this transformation is Also, the bounds for and in are and . So, the bounds for and in are and . Thus, the desired integral is Triple integrals If we continue our flow of thoughts, we can also find the Jacobian for three variables. Suppose there is a function . has relations with , which are is a region in the -space, and is a region in the -space, with transformation . To calculate the Jacobian for three variables, we go through a similar process. The process of transformation will be: a rectangular prism with dimensions in the -space to a parallelepiped in the -space and a volume of . The parallelepiped can be described with the position vector:The three sides of the parallelepiped can be described by the position vector as:Since the derivatives of are defined as:The three vectors can be similarly approximated into: Since the position vector is , the partial derivatives for are:Recall that the volume of a parallelepiped determined by the vectors is the magnitude of their scalar triple product:We just need to substitute the vectors with what we have yielded.Thus, . darkblue Definition. (The Jacobian for three variables) The Jacobian of the transformation given by functions and whose partial derivatives exist and are continuous, is The absolute value is added to prevent a negative volume. Then, we have the following theorem which is analogous to the theorem for double integrals. Again, we should aware that the above explanations are green not proof of this theorem. tomato Theorem. (Change of variables for triple integration)Suppose is a transformation whose Jacobian is nonzero and that maps a region in the -space onto a region in the -space injectively, via the change of variables and . Suppose that is continuous on , we have rosybrown Remark. has the same meaning as . If we change some notations, we can get ( maps a region in -space onto a region in -space in this case.) which may be a more convenient form to be used sometimes. { red Example. } {Choose correct statement(s) from the following statements. |type="[]"} - || ||This is not equal to in general. - gives the 4-dimensional volume under the graph of over the region in -space. || The absolute value of the Jacobian is an extra term. Without it, this is true. + gives the 4-dimensional volume under the graph of over the region in -space. || By the proposition about 4-dimensional volume given by triple integration, this is true. + gives the volume of the region in -space, that is mapped from the region in -space by a transformation satisfying the conditions mentioned in the theorem about change of variables in a triple integral. || This integral equals which is the volume of the region in space, || since the function involved in the above integral is (constant function), the value of function still equals one after changing variables and to and respectively by setting and , || and we need to insert an extra term to the integral, namely absolute value of the Jacobian. Now, we understand the purpose and the derivation of the Jacobian. It is time to apply this new knowledge to some examples. The first two examples consist of the change of coordinates from the Cartesian coordinate system into the polar coordinate system and the change of Cartesian to spherical coordinates. Change of coordinate system Sometimes, we may change the region we are integrating over to another region in other coordinate system. This can simplify the computation of integrals, especially when the region in Cartesian coordinate system is related to circle, e.g. sphere, cone, circle, etc. Let us start with the change of coordinates from the Cartesian coordinate system into the polar coordinate system. prp:Changing Cartesian coordinate system to polar coordinate system for double integration purple Proposition. (Changing Cartesian coordinate system to polar coordinate system for double integration) Let be a continuous function defined using green Cartesian coordinates, and let be the green same function expressed using green polar coordinates. Suppose the region in the polar coordinates is mapped injectively to the region in the Cartesian coordinates. Then, violet Proof. If we change from Cartesian coordinate system to polar coordinate system, we have the relationships Thus, the Jacobian is By the theorem about change of variables for double integration, prp:Changing Cartesian coordinate system to cylindrical coordinate system for triple integration purple Proposition. (Changing Cartesian coordinate system to cylindrical coordinate system for triple integration) Let be a continuous function defined using green Cartesian coordinates, and let be the green same function expressed using green cylindrical coordinates. Suppose the region in the cylindrical coordinates is mapped injectively to the region in the Cartesian coordinates. Then, prp:Changing Cartesian coordinate system to spherical coordinate system for triple integration purple Proposition. (Changing Cartesian coordinate system to spherical coordinate system for triple integration) Let be a continuous function defined using green Cartesian coordinates, and let be the green same function expressed using green spherical coordinates. Suppose the region in the spherical coordinates is mapped injectively to the region in the Cartesian coordinates. Then, violet Proof. thumb|Illustration of Rule of Sarrus. Red arrows correspond to the positive terms, and blue arrows correspond to the negative terms. If we change from Cartesian coordinates to spherical coordinates, we have the relationships Thus, the Jacobian is By the theorem about change of variables for triple integration, red Example. Prove the proposition about changing Cartesian coordinate system to cylindrical coordinate system for triple integration. (green Hint: the proof is quite similar to that for the proposition about changing Cartesian coordinate system to polar coordinate system for double integration, and you may find some results in the steps in the proof useful) Proof. If we change from Cartesian coordinate system to green cylindrical coordinate system, we have the relationships Thus, the Jacobian is By the theorem about change of variables for green triple integration, red Example. (Volume of the cone) Prove that the volume of a cone with radius and height is by triple integration. (green Hint: You may put the base of the cone on the -plane with centre , point the cone to the direction of positive -axis, and use cylindrical coordinates) Proof. First, put the cone as instructed in the hint. Let the region bounded by the cone in Cartesian coordinate system and cylindrical coordinate system be and respectively. Then, using cylindrical coordinates, by the proposition about triple integration using cylindrical coordinates and the proposition about volume given by triple integration, the desired volume is Next, we need to find the bounds for and in the region . First, the bounds for is (by the definition of cylindrical coordinates). Then, given a fixed , we consider the corresponding -plane to see whether we can obtain any relationship between and . Since the region in the -plane (it is -plane in Cartesian coordinate system when ) over which the integral is taken is the green triangle with vertices and , for which the equation of the region is Therefore, given a fixed (this shows that is actually independent from .) and given fixed , (this shows that is actually independent from .) Therefore, the desired volume is red Example. (Volume of the sphere) Prove that the volume of a sphere of radius is by triple integration. (green Hint: You may put the centre of the sphere to the origin, i.e., .) Proof. First, put the centre of the sphere to the origin. Let and be the region bounded by the sphere in Cartesian coordinate system and spherical coordinate system respectively. Using green spherical coordinates, by the proposition about triple integration using spherical coordinates, the desired volume is Since the bounds for are and (by the definition of spherical coordinates) in the region , the desired volume is Multiple integration Vector calculus