Change of Variables in Multiple Integrals

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint\limits_Re^{\frac{x+y}{x-y}}dA} , where $R$ is the trapezoidal region with vertices $(1,0),(2,0),(0,-2),(0,-1)$,

it would be helpful if we can substitute $x+y$ as $u$ and $x-y$ as $v$ because $e^{\frac {u}{v}}$ 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 $f(x,y)$. We want to calculate the expression:

$\iint \limits _{R}f(x,y)dA$

However, the area $R$ is too complicated to be written out in terms of $x,y$. So, we want to change the variables so that the area $R$ 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 $u,v$ that have connections with variables $x,y$ that satisfy:

$x=x(u,v)\quad {\text{and}}\quad y=y(u,v)$

The original integral can be rewritten into:

$\iint \limits _{S}f(x(u,v),y(u,v)){\bigg |}{\frac {\partial (x,y)}{\partial (u,v)}}{\bigg |}\ du\ dv$

The purpose of this section is to have us understand the process of this transformation, excluding the ${\bigg |}{\frac {\partial (x,y)}{\partial (u,v)}}{\bigg |}$ part. We will discuss the purpose of ${\bigg |}{\frac {\partial (x,y)}{\partial (u,v)}}{\bigg |}$ in the next section.

Introduction

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 $x,y,z$ into $r,\theta ,z$ with relations:

$x=r\cos \theta \quad y=r\sin \theta \quad z=z$

As a result, the function being integrated $f(x,y)$ is transformed into $f(x(r,\theta ,z),y(r,\theta ,z),z(r,\theta ,z))$, thus giving us:

$\iiint \limits _{R}f(x,y,z)dV=\iiint \limits _{S}f(r\cos \theta ,r\sin \theta ,z)dz\ r\ dr\ d\theta$, which is the formula for polar coordinates integration.

The second example, integration in spherical coordinates, offers a similar explanation. The original variables $x,y,z$ and the transformed variables $\rho ,\theta ,\phi$ have the relations:

${\begin{cases}x=\rho \sin \phi \cos \theta \\y=\rho \sin \phi \sin \theta \\z=\rho \cos \phi \end{cases}}$

These relations can give us that

$\iiint \limits _{R}f(x,y,z)dV=\iiint \limits _{S}f(\rho \sin \phi \cos \theta ,\rho \sin \phi \sin \theta ,\rho \cos \phi )\rho ^{2}\sin \phi \ d\rho \ d\theta \ d\phi$, which is the formula for spherical coordinates integration.

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=r\cos\theta\quad y=r\sin\theta\quad z=z} , we will talk about general changes. Let's start from two variables.

We consider a change of variables that is given by a transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} from the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uv} -plane to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} -plane. In other words,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(u,v)=(x,y)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)} is the original or old variables and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u,v)} is the new ones.

In this transformation, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y} are related to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u,v} by the equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=g(u,v)\quad y=h(u,v)}

We usually just assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^1} transformation, which means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g,h} have continuous first-order partial derivatives. Now, time for some terminologies.

• If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(u_1,v_1)=(x_1,y_1)} , the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1,y_1)} is called the image of the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u_1,v_1)} .
• If no two points have the same image, like functions, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} , the transformation, is called one-to-one.
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} transforms region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} into region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is called the image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} . The transformation can be described as:

$T(S)=R$

• If $T$ is one-to-one, then, like functions, it has an inverse transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T^{-1}} from the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} -plane to the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uv} -plane, with relations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=G(x,y)\quad H=h(x,y)\quad\text{and}\quad T^{-1}(R)=S}

Regions

Recall that we have established the transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(S)=R} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is the region in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uv} -plane while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is the region in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} -plane. If we are given the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} , we are expected to calculate the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} . For example, a transformation is defined by the equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=u^2-v^2\quad y=2uv}

Find the image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , which is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\{ (u,v)|0 \le u \le 1,\ 0 \le v \le 1 \} } .

In this case, we need to know the boundaries of the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , which is confined by the lines:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u=0\quad u=1\quad v=0\quad v=1}

If we can redefine the boundaries using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y} instead of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u,v} , we effectively will find the image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \text{When }\quad & u=0 \quad (0\le v\le 1) \\ & \begin{cases} x=-v^2 \\ y=0 \\ \end{cases} \quad \text{(substitution)}\\ \text{Thus, }\quad & y=0 \quad (-1\le x\le 0) \\ \end{align} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \text{When }\quad & u=1 \quad (0\le v\le 1) \\ & \begin{cases} x=1-v^2 \\ y=2v \\ \end{cases} \\ \text{Thus, }\quad & x=1-\frac{y^2}{4} \quad (0\le x\le 1) \\ \end{align} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \text{When }\quad & v=0 \quad (0\le 0\le 1) \\ & \begin{cases} x=u^2 \\ y=0 \\ \end{cases} \\ \text{Thus, }\quad & y=0 \quad (0\le x\le 1) \\ \end{align} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \text{When }\quad & v=1 \quad (0\le u\le 1) \\ & \begin{cases} x=u^2-1 \\ y=2u \\ \end{cases} \\ \text{Thus, }\quad & x=\frac{y^2}{4}-1 \quad (-1\le x\le 0) \\ \end{align} }

As a result, the image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R=\{(x,y)|0\le y\le 2,\ \frac{y^2}{4}-1\le x\le 1-\frac{y^2}{4}\} }

We can use the same method to calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R } .

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigg| \frac{\partial(x,y)}{\partial(u,v)}\bigg|} from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint\limits_S f(x(u,v),y(u,v))\bigg| \frac{\partial(x,y)}{\partial(u,v)}\bigg| \ du\ dv} here. To actually start explaining that, we should review some basic concepts.

Review "u-substitution"

Recall that when we are discussing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u } -substitution (a simple way to describe "integration by substitution for single-variable functions"), we use the following method to solve integrals.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^bf(x)\ dx=\int_c^df(x(u))\ \frac{dx}{du}\ du \quad\text{where }c=x(a),d=x(b) }

For example,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int \frac{\sin(\ln (x))}{x}\ dx }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \text{Let } \quad & u=\ln(x) \\ \text{Thus, } \quad & \frac{du}{dx}=\frac{1}{x} \\ \Rightarrow\ & du=\frac{1}{x}dx \\ \end{align} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int \frac{\sin(\ln (x))}{x}\ dx & = \int\sin(\ln (x))\ \Big(\frac{1}{x}\ dx\Big) & \quad\text{rearrangement}\\ & = \int\sin(u)\ du & \quad \text{remember }u=\ln(x)\text{ and }du=\frac{1}{x}\ dx \\ & = -\cos(u)+C & \quad\text{integration} \\ & = -\cos(\ln(x))+C & \quad\text{resubstitution} \\ \end{align} }

If we add endpoints into the integral, the result will be:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_e^{e^2} \frac{\sin(\ln (x))}{x}\ dx & = \int_e^{e^2}\sin(\ln (x))\ \Big(\frac{1}{x}\ dx\Big) & \quad\text{rearrangement}\\ & = \int_1^2\sin(u)\ du & \quad \text{remember }u=\ln(x)\text{ and }du=\frac{1}{x}\ dx \\ & = \Big[-\cos(u)\Big]_1^2 & \quad\text{integration} \\ & = \cos(1)-\cos(2) \\ \end{align} }

If we look carefully at the "rearrangement" and "remember" part in the solution, we find that we effectively changed our variable from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u } through this method:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_a^bf(x)dx=\int_{x=a}^{x=b}f(x(u))\ d(x(u))=\int_{u=x(a)}^{u=x(b)}f(x(u))\ \frac{dx}{du}\ du } , which is what we have mentioned above.

The appearance of the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{du} } not only is a mathematical product of deduction, but also serves a intuitive purpose. When we change our function from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x(u)) } , 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{du}{dx} } . To counter this change, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{du} } 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy } to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uv } , 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint\limits_Rf(x,y)\ dA_1=\iint\limits_Sf(x(u,v),y(u,v))\ \frac{dA_1}{dA_2}\ dA_2 }

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 $f(x,y)$ to $f(x(u),y(u))$, we "stretched" or "condensed" our region, which is an area, by a factor of ${\frac {dA_{2}}{dA_{1}}}$; therefore, we need to counter the change with a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dA_1}{dA_2} } . The Jacobian matrix for two variables is basically the process of calculating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dA_1}{dA_2} } in terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y,u,v } instead of arbitrary areas.

The Jacobian

Double integrals

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dA_1}{dA_2} } , with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dA_1} being the infinitesimally small area in the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} -plane and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dA_2} being the infinitesimally small area in the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uv} -plane. Since we are changing our variables from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u,v} , we should describe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dA_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dA_2} in terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u,v} .

Let us start with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dA_2} first because it is easier to calculate. We start with a small rectangle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0} , which is a part of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} , in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uv} -plane whose lower left corner is the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u_0,v_0)} and whose dimensions are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta u,\Delta v} . Thus, the area of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta A_2=\Delta u\Delta v}

The image of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0} , in this case let's name it Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_0} , is in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} -plane according to the transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(S_0)=R_0} . One of its boundary points is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0,y_0)=T(u_0,v_0) } . We can use a vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}} to describe the position vector of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_0} of the point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (u,v)} . In other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}} can describe the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_0} given that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}(u,v)=x(u,v)\mathbf{i}+y(u,v)\mathbf{j}\quad\text{where }u_0\le u\le u_0+\Delta u,\ v_0\le v\le v_0+\Delta v}

The region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_0} now can be described in terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u,v} . The next step is to utilize the position vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}(u,v)} to calculate its area Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dA_1} .

The shape of the region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_0} after transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_0=T(S_0)} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{b}} can be calculated by taking the magnitude of the cross product of the two vectors.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta A_1=|\mathbf{a}\times \mathbf{b}|}

In this parallelogram, the two vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{b}} are, in terms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u,v} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}=\mathbf{r}(u_0+\Delta u,v_0)-\mathbf{r}(u_0,v_0) \quad \text{ and }\quad \mathbf{b}=\mathbf{r}(u_0,v_0+\Delta v)-\mathbf{r}(u_0,v_0)}

It seems very similar to the definition of partial derivatives:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}_u=\lim_{\Delta u\rightarrow 0}\frac{\mathbf{r}(u_0+\Delta u,v_0)-\mathbf{r}(u_0,v_0)}{\Delta u}\quad\text{ and }\quad \mathbf{r}_v\lim_{\Delta v\rightarrow 0}\frac{\mathbf{r}(u_0,v_0+\Delta v)-\mathbf{r}(u_0,v_0)}{\Delta v}}

As a result, we can approximate that:

${\begin{aligned}&\mathbf {a} =\mathbf {r} (u_{0}+\Delta u,v_{0})-\mathbf {r} (u_{0},v_{0})\approx \Delta u\mathbf {r} _{u}\quad {\text{ and }}\quad \\&\mathbf {b} =\mathbf {r} (u_{0},v_{0}+\Delta v)-\mathbf {r} (u_{0},v_{0})\approx \Delta v\mathbf {r} _{v}\\\end{aligned}}$

Now, we calculate $\mathbf {r} _{u},\mathbf {r} _{v}$, given that $\mathbf {r} (u,v)=x(u,v)\mathbf {i} +y(u,v)\mathbf {j}$:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \mathbf{r}_u=x_u(u,v)\ \mathbf{i}+y_u(u,v)\ \mathbf{j}=\frac{\partial x}{\partial u}\ \mathbf{i}+\frac{\partial y}{\partial u}\ \mathbf{j} \quad\text{ and } \\ & \mathbf{r}_v=x_v(u,v)\ \mathbf{i}+y_v(u,v)\ \mathbf{j}=\frac{\partial x}{\partial v}\ \mathbf{i}+\frac{\partial y}{\partial v}\ \mathbf{j} \\ \end{align}}

We can calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta A_1=||\mathbf{a}\times\mathbf{b}||} . Note that the inner pair of || is for calculating the magnitude while the outer pair of || is for taking the absolute value.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta A_1 & = ||\mathbf{a}\times\mathbf{b}|| \\ & = ||(\Delta u\ \mathbf{r}_u)\times(\Delta v\ \mathbf{r}_v)|| & \quad \text{approximation} \\ & = ||\mathbf{r}_u\times\mathbf{r}_v||\Delta u\Delta v \\ & = \begin{vmatrix} \begin{vmatrix} \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & 0 \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & 0 \\ \end{vmatrix} \\ \end{vmatrix} \end{vmatrix}\Delta u\Delta v & \quad\text{cross product} \\ & = \begin{vmatrix} \begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{vmatrix} \mathbf{k} \end{vmatrix} \end{vmatrix} \Delta u\Delta v & \quad\text{evaluation} \\ & = \begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{vmatrix} \end{vmatrix} \Delta u\Delta v \\ \end{align}}

Then, we can substitute our newly deduced terms.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\Delta A_1}{\Delta A_2}=\frac{\begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{vmatrix} \end{vmatrix} \Delta u\Delta v}{\Delta u\Delta v}=\begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{vmatrix} \end{vmatrix} = \Bigg| \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial u} \Bigg|}

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

The Jacobian of the transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=g(u,v)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=h(u,v)} is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\ \end{vmatrix} =\frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}} }}

We will then use the Jacobian in the change of variables in integrals. The absolute value is added to prevent a negative area.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \iint\limits_Rf(x,y)dA & \approx \sum_{i=1}^m \sum_{j=1}^n f(x_i,y_j)\Delta A \\ & \approx \sum_{i=1}^m \sum_{j=1}^n f(x(u_i,v_j),y(u_i,v_j))\Delta A_2 \\ \text{Since } & \Delta A_2 \approx \Bigg| \frac{\partial (x,y)}{\partial (u,v)} \Bigg|\Delta u\Delta v \\ \sum_{i=1}^m \sum_{j=1}^n f(x(u_i,v_j),y(u_i,v_j))\Delta A_2 & \approx \sum_{i=1}^m \sum_{j=1}^n f(x(u_i,v_j),y(u_i,v_j))\ \Bigg| \frac{\partial (x,y)}{\partial (u,v)} \Bigg|\Delta u\Delta v \\ & \approx\iint\limits_Sf(x(u,v),y(u,v))\ \Bigg| \frac{\partial (x,y)}{\partial (u,v)} \Bigg|\ du\ dv \\ \end{align}}

Here is the theorem for the change of variables in a double integral.

Suppose the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^1} transformation whose Jacobian is nonzero and that maps a region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uv} -plane onto a region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xy} -plane. Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint\limits_Rf(x,y)dA=\iint\limits_Sf(x(u,v),y(u,v))\begin{vmatrix} \frac{\partial(x,y)}{\partial(u,v)} \\ \end{vmatrix}\ du\ dv}

Triple integrals

If we continue our flow of thoughts, we can also find the Jacobian for three variables. Suppose there is a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y,z)} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y,z} has relations with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u,v,w} , which are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x(u,v,w),\quad y=y(u,v,w),\quad \text{and}\quad z=z(u,v,w)}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} is a region in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xyz} -space, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a region in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uvw} -space, with transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(S)=R} .

To calculate the Jacobian for three variables, we go through a similar process. The process of transformation will be: a rectangular prism with dimensions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta u,\Delta v,\Delta w} in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uvw} -space to a parallelepiped in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xyz} -space and a volume of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta V_2=\Delta u\Delta v\Delta w} . The parallelepiped can be described with the position vector:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}(u,v,w)=x(u,v,w)\ \mathbf{i}+y(u,v,w)\ \mathbf{j}+z(u,v,w)\ \mathbf{k}}

The three sides of the parallelepiped can be described by the position vector as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \mathbf{a}=\mathbf{r}(u+\Delta u,v,w)-\mathbf{r}(u,v,w), \\ & \mathbf{b}=\mathbf{r}(u,v+\Delta v,w)-\mathbf{r}(u,v,w),\quad\text{and} \\ & \mathbf{c}=\mathbf{r}(u,v,w+\Delta w)-\mathbf{r}(u,v,w). \\ \end{align}}

Since the derivatives of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}} are defined as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \mathbf{r}_u = \lim_{\Delta u\rightarrow 0}\frac{\mathbf{r}(u+\Delta u,v,w)-\mathbf{r}(u,v,w)}{\Delta u}, \\ & \mathbf{r}_v = \lim_{\Delta v\rightarrow 0}\frac{\mathbf{r}(u,v+\Delta v,w)-\mathbf{r}(u,v,w)}{\Delta v},\quad\text{and} \\ & \mathbf{r}_w = \lim_{\Delta w\rightarrow 0}\frac{\mathbf{r}(u,v,w+\Delta w)-\mathbf{r}(u,v,w)}{\Delta w}. \\ \end{align}}

The three vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a},\mathbf{b},\mathbf{c}} can be approximated into:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a}=\Delta u\ \mathbf{r}_u,\quad\mathbf{b}=\Delta v\ \mathbf{r}_v,\quad\text{and}\quad\mathbf{c}=\Delta w\ \mathbf{r}_w}

Since the position vector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}(u,v,w)=x(u,v,w)\ \mathbf{i}+y(u,v,w)\ \mathbf{j}+z(u,v,w)\ \mathbf{k}} , the partial derivatives for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{r}} are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} & \mathbf{r}_u=\frac{\partial x}{\partial u}\ \mathbf{i}+\frac{\partial y}{\partial u}\ \mathbf{j}+\frac{\partial z}{\partial u}\ \mathbf{k}, \\ & \mathbf{r}_v=\frac{\partial x}{\partial v}\ \mathbf{i}+\frac{\partial y}{\partial v}\ \mathbf{j}+\frac{\partial z}{\partial v}\ \mathbf{k},\quad\text{and} \\ & \mathbf{r}_v=\frac{\partial x}{\partial w}\ \mathbf{i}+\frac{\partial y}{\partial w}\ \mathbf{j}+\frac{\partial z}{\partial w}\ \mathbf{k} \\ \end{align}}

Recall that the volume of a parallelepiped determined by the vectors Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{a},\mathbf{b},\mathbf{c}} is the magnitude of their scalar triple product:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=|(\mathbf{a}\times\mathbf{b})\ \cdot\ \mathbf{c}|}

We just need to substitute the vectors with what we have yielded.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta V_1=|(\mathbf{a}\times\mathbf{b})\ \cdot\ \mathbf{c}| & = |(\Delta u\ \mathbf{r}_u)\times(\Delta v\ \mathbf{r}_v)\ \cdot\ \Delta w\ \mathbf{r}_w| \\ & = |\mathbf{r}_u\times\mathbf{r}_v\ \cdot\ \mathbf{r}_w|\Delta u\Delta v\Delta w \\ & = \begin{vmatrix} \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial x}{\partial u} & \frac{\partial y}{\partial u} & \frac{\partial z}{\partial u} \\ \frac{\partial x}{\partial v} & \frac{\partial y}{\partial v} & \frac{\partial z}{\partial v} \\ \end{vmatrix} \ \cdot\ \begin{pmatrix} \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial w} \\ \end{pmatrix}\end{vmatrix} \Delta u\Delta v\Delta w & \quad\text{cross product}\\ & = \begin{vmatrix} \begin{pmatrix} \frac{\partial y}{\partial u}\frac{\partial z}{\partial v}-\frac{\partial z}{\partial u}\frac{\partial y}{\partial v} \\ \frac{\partial z}{\partial u}\frac{\partial x}{\partial v}-\frac{\partial x}{\partial u}\frac{\partial z}{\partial v} \\ \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}-\frac{\partial y}{\partial u}\frac{\partial x}{\partial v} \\ \end{pmatrix} \ \cdot\ \begin{pmatrix} \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial w} \\ \end{pmatrix} \end{vmatrix}\Delta u\Delta v\Delta w \\ & = \Bigg| \frac{\partial x}{\partial w}\frac{\partial y}{\partial u}\frac{\partial z}{\partial v}-\frac{\partial x}{\partial w}\frac{\partial y}{\partial v}\frac{\partial z}{\partial u}+ \frac{\partial x}{\partial v}\frac{\partial y}{\partial w}\frac{\partial z}{\partial u}-\frac{\partial x}{\partial u}\frac{\partial y}{\partial w}\frac{\partial z}{\partial v}+ \frac{\partial x}{\partial u}\frac{\partial y}{\partial v}\frac{\partial z}{\partial w}-\frac{\partial x}{\partial v}\frac{\partial y}{\partial u}\frac{\partial z}{\partial w} \Bigg| \Delta u\Delta v\Delta w & \quad\text{dot product}\\ & = \Bigg| \frac{\partial x}{\partial u}\bigg( \frac{\partial y}{\partial v}\frac{\partial z}{\partial w}-\frac{\partial y}{\partial w}\frac{\partial z}{\partial v} \bigg) + \frac{\partial x}{\partial v}\bigg( \frac{\partial y}{\partial w}\frac{\partial z}{\partial u}-\frac{\partial y}{\partial u}\frac{\partial z}{\partial w} \bigg) + \frac{\partial x}{\partial w}\bigg( \frac{\partial y}{\partial u}\frac{\partial z}{\partial v}-\frac{\partial y}{\partial v}\frac{\partial z}{\partial u} \bigg) \Bigg| \Delta u\Delta v\Delta w & \quad\text{rearrangement}\\ & = \begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \\ \end{vmatrix} \end{vmatrix} \Delta u\Delta v\Delta w & \quad\text{cross product}\\ \end{align}}

Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\Delta V_1}{\Delta V_2}=\frac{\begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \\ \end{vmatrix} \end{vmatrix} \Delta u\Delta v\Delta w}{\Delta u\Delta v\Delta w}= \begin{vmatrix} \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \\ \end{vmatrix} \end{vmatrix}} .

The Jacobian of the transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=g(u,v)} and $y=h(u,v)$ is:

${\frac {\partial (x,y,z)}{\partial (u,v,w)}}={\begin{vmatrix}{\frac {\partial x}{\partial u}}&{\frac {\partial x}{\partial v}}&{\frac {\partial x}{\partial w}}\\{\frac {\partial y}{\partial u}}&{\frac {\partial y}{\partial v}}&{\frac {\partial y}{\partial w}}\\{\frac {\partial z}{\partial u}}&{\frac {\partial z}{\partial v}}&{\frac {\partial z}{\partial w}}\\\end{vmatrix}}$

The absolute value is added to prevent a negative volume.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \iiint\limits_Rf(x,y,z)dV & \approx \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x_i,y_j,z_k)\Delta V \\ & \approx \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x(u_i,v_j,w_k),y(u_i,v_j,w_k),z(u_i,v_j,w_k))\Delta V_2 \\ \text{Since } & \Delta V_2 \approx \Bigg| \frac{\partial (x,y,z)}{\partial (u,v,w)} \Bigg|\Delta u\Delta v\Delta w \\ \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x(u_i,v_j,w_k),y(u_i,v_j,w_k),z(u_i,v_j,w_k))\Delta V_2 & \approx \sum_{i=1}^m \sum_{j=1}^n \sum_{k=1}^p f(x(u_i,v_j,w_k),y(u_i,v_j,w_k),z(u_i,v_j,w_k))\ \Bigg| \frac{\partial (x,y,z)}{\partial (u,v,w)} \Bigg|\Delta u\Delta v\Delta w \\ & \approx\iiint\limits_S f(x(u,v,w),y(u,v,w),z(u,v,w))\ \Bigg| \frac{\partial (x,y,z)}{\partial (u,v,w)} \Bigg|\ du\ dv\ dw \\ \end{align}}

Change of variables in a triple integral Suppose the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C^1} transformation whose Jacobian is nonzero and that maps a region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle uvw} -space onto a region Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle xyz} -space.

Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint\limits_Rf(x,y,z)dV=\iiint\limits_Sf(x(u,v,w),y(u,v,w),z(u,v,w))\begin{vmatrix} \frac{\partial(x,y,z)}{\partial(u,v,w)} \\ \end{vmatrix}\ du\ dv\ dw}

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.

Examples

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (r,\theta)} into a Cartesian one, the transformation is as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=r\cos\theta\quad y=r\sin\theta}

Now, assume there is a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x,y)} and an integral:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint\limits_R f(x,y)\ dA}

We want to change the variables in the function from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r,\theta} . According to what we have learned so far in this chapter, the integral after the change should look like this:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint\limits_S f(r\cos\theta,r\sin\theta)\ \bigg| \frac{\partial(x,y)}{\partial(r,\theta)} \bigg|\ dr\ d\theta}

Then we evaluate the Jacobian.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial(x,y)}{\partial(r,\theta)} & = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \\ \end{vmatrix} \\ \text{Since}\quad & \frac{\partial x}{\partial r}=\cos\theta\quad \frac{\partial x}{\partial \theta}=-r\sin\theta \\ & \frac{\partial y}{\partial r}=\sin\theta\quad \frac{\partial y}{\partial \theta}=r\cos\theta \\ \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \\ \end{vmatrix} & = r\cos^2\theta-(-r\sin^2\theta) \\ & = r(\cos^2\theta+\sin^2\theta) \\ & = r \\ \end{align}}

So, we yield:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint\limits_R f(x,y)\ dA=\iint\limits_S f(r\cos\theta,r\sin\theta)\ r\ dr\ d\theta}

which is the integral for polar coordinates.

In the case for spherical coordinates, recall that the relations between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y,z} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho,\phi,\theta} are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} x=\rho\sin\phi\cos\theta \\ y=\rho\sin\phi\sin\theta \\ z=\rho\cos\phi \\ \end{cases}}

Thus the integral Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint\limits_R f(x,y,z)\ dV} can be changed into

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint\limits_S f(\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi)\ \bigg| \frac{\partial (x,y,z)}{\partial (\rho,\phi,\theta)}\bigg|\ d\rho\ d\phi\ d\theta}

Then we evaluate the Jacobian.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)} & = \begin{vmatrix} \frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \phi} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \phi} & \frac{\partial y}{\partial \theta} \\ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \phi} & \frac{\partial z}{\partial \theta} \\ \end{vmatrix} \\ \text{Since}\quad & \frac{\partial x}{\partial \rho}=\sin\phi\cos\theta\quad \frac{\partial x}{\partial \phi}=\rho\cos\phi\cos\theta\quad \frac{\partial x}{\partial \theta}=-\rho\sin\phi\sin\theta \\ & \frac{\partial y}{\partial \rho}=\sin\phi\sin\theta\quad \frac{\partial y}{\partial \phi}=\rho\cos\phi\sin\theta\quad \frac{\partial y}{\partial \theta}=\rho\sin\phi\cos\theta \\ & \frac{\partial z}{\partial \rho}=\cos\phi\quad \frac{\partial z}{\partial \phi}=-\rho\sin\phi\quad \frac{\partial z}{\partial \rho}=0 \\ \begin{vmatrix} \frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \phi} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \phi} & \frac{\partial y}{\partial \theta} \\ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \phi} & \frac{\partial z}{\partial \theta} \\ \end{vmatrix} & = -\rho^2\sin\phi \\ \end{align}}

As a result,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigg| \frac{\partial(x,y,z)}{\partial (\rho,\phi,\theta)} \bigg|=\rho^2\sin\phi}

We can yield:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint\limits_R f(x,y,z)\ dV=\iiint\limits_S f(\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi)\ \rho^2\sin\phi\ d\rho\ d\phi\ d\theta}

which is the integral for spherical coordinates.

Resources

• Change of Variables and the Jacobian, Wikibooks: Calculus
• Change of Variables in Multiple Integrals, Mathematics LibreTexts
• Change of Variables in Multiple Integrals, OpenStax
• Change of Variables, Paul's Online Notes