Calculus/Multivariable Calculus/Change of Variables and The Jacobian

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.

TransformationEdit

Let us start with an introduction to the process of variable transformation. Assume that we have a function  . We want to calculate the expression:

 

However, the area   is too complicated to be written out in terms of  . So, we want to change the variables so that the area   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:

 

The purpose of this section is to have us understand the process of this transformation, excluding the   part. We will discuss the purpose of   in the next section.

IntroductionEdit

In fact, we have already encountered two examples of variable transformation. 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.

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.

GeneralizationEdit

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 equations

 

We usually just assume that   is a   transformation, which means that   have continuous first-order partial derivatives. Now, time for some terminologies.

  • If  , the point   is called the image of the point  .
  • If no two points have the same image, like functions,  , the transformation, is called one-to-one.
  •   transforms region   into region  .   is called the image of  . The transformation can be described as:

 

  • If   is one-to-one, then, like functions, it has an inverse transformation   from the  -plane to the  -plane, with relations

 

RegionsEdit

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 equations

 

Find 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 JacobianEdit

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"Edit

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 change the region we are integrating, which can be seen by looking at the endpoints. This change of region is either "stretched" or "condensed" by a factor of  . To counter this change,   is deduced to compromise. 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 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 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 "stretched" or "condensed" our region, which is an area, by a factor of  ; therefore, we need to counter the change with a factor of  . The Jacobian matrix for two variables is basically the process of calculating   in terms of   instead of arbitrary areas.

The JacobianEdit

Double integralsEdit

Now, it is time for us to deduce the Jacobian matrix. In the review above, we already established (unofficially) 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  .


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   is

 

The 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 that

 

The 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  . You can review the cross product in Chapter 7.1. Note that the inner pair of || is for calculating the magnitude while the outer pair of || is for taking the absolute value.

 

Then, we can substitute our newly deduced terms.

 

Finally, we derived the Jacobian. The definition is as follows:

The Jacobian for two variables

The Jacobian of the transformation   given by   and   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.

Change of variables in a double integral

Suppose the   is a   transformation whose Jacobian is nonzero and that maps a region   in the  -plane onto a region   in the  -plane. Suppose that   is continuous on  .

 

Triple integralsEdit

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 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,  .

The Jacobian for three variables

The Jacobian of the transformation   given by   and   is:

 

The absolute value is added to prevent a negative volume.

 

Change of variables in a triple integral

Suppose the   is a   transformation whose Jacobian is nonzero and that maps a region   in the  -space onto a region   in the  -space. Suppose that   is continuous on  .

 

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 polar coordinate system and the change of Cartesian to spherical coordinates.

ExamplesEdit

Let us start with the change of coordinates from the Cartesian coordinate system into the polar coordinate system. In previous chapters, we have found out the relations between the two coordinate systems. If we want to transform a polar coordinate   into a Cartesian one, the transformation is as follows:

 

Now, assume there is a function   and an integral:

 

We want to change the variables in the function from   to  . According to what we have learned so far in this chapter, the integral after the change should look like this:

 

Then we evaluate the Jacobian.

 

So, we yield:

 

which is the integral for polar coordinates.


In the case for spherical coordinates, recall that the relations between   and   are

 

Thus the integral   can be changed into

 

Then we evaluate the Jacobian.

 

As a result,

 

We can yield:

 

which is the integral for spherical coordinates.