Calculus/Change of variables

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Change of variables

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:

 

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.

IntroductionEdit

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)

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 1st-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 (or injective).
  •   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 relation

 

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 (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 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" 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 JacobianEdit

Double integralsEdit

Now, it is time for us to deduce the Jacobian matrix. In the review above, we already established 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   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   (we take absolute value to prevent negative area). You can review the cross product in Chapter 7.1. 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 absolute value of Jacobian. The definition of Jacobian is as follows:

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 explained intuitively why and how it works, but the above explanations are not proof of this theorem. In particular, we make some approximation, while the statement in the following theorem is equality, and not approximation. The actual proof is quite complicated and advanced, and thus not included here.

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

 
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.

Example.

1 Choose correct expression(s) for the Jacobian   in which   and  .

 
 
 
 
 

2 Choose correct expression(s) for the integral   in which   is a region bounded by   and  .

 
 
 
 
 

3 Choose correct statement(s) from the following statements.

If both   and   are independent from both   and  , the Jacobian  .
If both   and   are independent from both   and  , the Jacobian  .
If   is independent from both   and  , while   is dependent from both   and  , the Jacobian  
If   is independent from both   and  , while   is dependent from both   and  , the Jacobian  
If   in which   is a real number, then the Jacobian  

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 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 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,  .

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 not proof of this theorem.

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

 
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.

Example.

1 Choose correct statement(s) from the following statements.

 
  gives the 4-dimensional volume under the graph of   over the region   in  -space.
  gives the 4-dimensional volume under the graph of   over the region   in  -space.
  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.

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 systemEdit

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.

Proposition. (Changing Cartesian coordinate system to polar coordinate system for double integration) Let   be a continuous function defined using Cartesian coordinates, and let   be the same function expressed using polar coordinates. Suppose the region   in the polar coordinates is mapped injectively to the region   in the Cartesian coordinates. Then,

 

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,
 
 

Proposition. (Changing Cartesian coordinate system to cylindrical coordinate system for triple integration) Let   be a continuous function defined using Cartesian coordinates, and let   be the same function expressed using cylindrical coordinates. Suppose the region   in the cylindrical coordinates is mapped injectively to the region   in the Cartesian coordinates. Then,

 

Proposition. (Changing Cartesian coordinate system to spherical coordinate system for triple integration) Let   be a continuous function defined using Cartesian coordinates, and let   be the same function expressed using spherical coordinates. Suppose the region   in the spherical coordinates is mapped injectively to the region   in the Cartesian coordinates. Then,

 

Proof.

 
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,
 
 
Proof.

If we change from Cartesian coordinate system to cylindrical coordinate system, we have the relationships

 
Thus, the Jacobian is
 
By the theorem about change of variables for triple integration,
 
 

Example. (Volume of the cone) Prove that the volume of a cone with radius   and height   is   by triple integration. (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 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
 
 

Example. (Volume of the sphere) Prove that the volume of a sphere of radius   is   by triple integration. (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 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
 
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Change of variables