21. Multiple Integrals in Curvilinear Coordinates
e. Integrating in 3D Curvilinear Coordinates
3. The Jacobian
On the previous page we found that a triple integral may be computed in a curvilinear coordinate system, \((u,v,w)\), as: \[ \iiint_R f\,dV=\iiint_R f(u,v,w)\,J\,du\,dv\,dw \] where the the differential of volume is \[ dV=J\,du\,dv\,dw \] and where \(J\) is the Jacobian factor: \[ J=\vec{e}_u\times\vec{e}_v\cdot\vec{e}_w \] The triple product can be computed either as a cross product followed by a dot product or as the determinant. \[ \vec{e}_u\times\vec{e}_v\cdot\vec{e}_w= \begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial y}{\partial u}&\dfrac{\partial z}{\partial u} \\[6pt] \dfrac{\partial x}{\partial v}&\dfrac{\partial y}{\partial v}&\dfrac{\partial z}{\partial v} \\[6pt] \dfrac{\partial x}{\partial w}&\dfrac{\partial y}{\partial w}&\dfrac{\partial z}{\partial w} \end{vmatrix} =\dfrac{\partial(x,y,z)}{\partial(u,v,w)} \qquad \text{(*)} \] This determinant is called the Jacobian determinant and the above formula defines the notation \(\dfrac{\partial(x,y,z)}{\partial(u,v,w)}\). Thus the Jacobian factor is the absolute value of the Jacobian determinant: \[ J=\left\dfrac{\partial(x,y,z)}{\partial(u,v,w)}\right \]
Notice that the rows of the Jacobian determinant are the tangent vectors \( \vec{e}_u\), \(\vec{e}_v\) and \(\vec{e}_w\).
Recall that the volume is always positive. Likewise the Jacobian factor is always positive and requires an absolute value.
In summary:
The Jacobian factor is the absolute value of the Jacobian determinant: \[ J=\left\dfrac{\partial(x,y,z) }{\partial(u,v,w) }\right \] and appears in the differential of volume: \[ dV=J\,du\,dv\,dw \]
3 Meanings of Jacobian
Actually, there are three different quantities which are frequently called the Jacobian. They are:

The Jacobian Matrix:
\(
\begin{pmatrix}
\dfrac{\partial x}{\partial u}&\dfrac{\partial x}{\partial v}&\dfrac{\partial x}{\partial w} \\
\dfrac{\partial y}{\partial u}&\dfrac{\partial y}{\partial v}&\dfrac{\partial y}{\partial w} \\
\dfrac{\partial z}{\partial u}&\dfrac{\partial z}{\partial v}&\dfrac{\partial z}{\partial w}
\end{pmatrix}
\)
Note: This is actually the transpose of the matrix displayed in (*). This transpose has no affect on the determinant. 
The Jacobian Determinant:
\(
\dfrac{\partial(x,y,z) }{\partial(u,v,w) }=
\begin{vmatrix}
\dfrac{\partial x}{\partial u}& \dfrac{\partial y}{\partial u}& \dfrac{\partial z}{\partial u} \\
\dfrac{\partial x}{\partial v}& \dfrac{\partial y}{\partial v}& \dfrac{\partial z}{\partial v} \\
\dfrac{\partial x}{\partial w}& \dfrac{\partial y}{\partial w}& \dfrac{\partial z}{\partial w}
\end{vmatrix}
\)
Note: This defines the symbol \(\dfrac{\partial(x,y,z)}{\partial(u,v,w)}\) as the Jacobian determinant. 
The Jacobian Factor:
\(
J=\left\dfrac{\partial(x,y,z) }{\partial(u,v,w)}\right
=\left
\begin{vmatrix}
\dfrac{\partial x}{\partial u}& \dfrac{\partial y}{\partial u}& \dfrac{\partial z}{\partial u} \\
\dfrac{\partial x}{\partial v}& \dfrac{\partial y}{\partial v}& \dfrac{\partial z}{\partial v} \\
\dfrac{\partial x}{\partial w}& \dfrac{\partial y}{\partial w}& \dfrac{\partial z}{\partial w}
\end{vmatrix}
\right
\)
Note:The double bars around the matrix indicate that we first take a determinant and then an absolute value.
The Jacobian factor is what appears in an integral.
Thus the volume element is \(dV=J\,du\,dv\,dw\).
The Jacobian determinant is useful in differential
equations, which you will study in a future course.
The Jacobian matrix is used frequently in linear
algebra, but not in this course.
So if someone refers to the Jacobian, you need to ask which one. However, in
this book, if we just say Jacobian, we will mean the Jacobian factor.
As you may have noticed in the previous forumlas, multiple vertical bars can
be quite confusing. A pair of vertical bars can have one of three meanings:
1) Absolute value
2) Length
3) Determinant
The meaning of the vertical bars is determined by the quantity inside them.
1) Vertical bars around a scalar mean the
absolute value of that scalar.
2) Vertical bars around a vector mean the
length of that vector.
3) Vertical bars around a matrix mean the
determinant of that matrix.
Notice that in formula for the Jacobian factor there are two pairs of vertical bars. The inner pair encloses a matrix. So this is a determinant which produces a scalar. So the outer pair of vertical bars is an absolute value.
The Jacobian for Cylindrical Coordinates
Recompute the Jacobian factor for cylindrical coordinates.
Recall that in cylindrical coordinates, \(\vec{R}(r,\theta,z)=\left\langle r\cos\theta,r\sin\theta,z\right\rangle\). So \[ \vec{e}_r=\left\langle \cos\theta,\sin\theta,0\right\rangle \qquad \vec{e}_\theta=\left\langle r\sin\theta,r\cos\theta,0\right\rangle \qquad \text{and} \qquad \vec{e}_z=\left\langle 0,0,1\right\rangle \] So, \[\begin{aligned} J&=\left\dfrac{\partial(x,y,z) }{\partial(u,v,w) }\right =\left \begin{vmatrix} \dfrac{\partial x}{\partial u}&\dfrac{\partial y}{\partial u}&\dfrac{\partial z}{\partial u} \\[6pt] \dfrac{\partial x}{\partial v}&\dfrac{\partial y}{\partial v}&\dfrac{\partial z}{\partial v} \\[6pt] \dfrac{\partial x}{\partial w}&\dfrac{\partial y}{\partial w}&\dfrac{\partial z}{\partial w} \end{vmatrix} \right=\left \begin{vmatrix} \cos\theta & \sin\theta & 0 \\ r\sin\theta & r\cos\theta & 0 \\ 0 & 0 & 1 \end{vmatrix} \right\\ &=r\cos^2\theta(r\sin^2\theta) =r \end{aligned}\]
Using the general formulas for the Jacobian, we have just derived the Jacobian for cylindrical coordinates, \(J=r\). Consequently, \(dV=r\,dr\,d\theta\,dz\).
Notice that this requires us to take \(r\) positive when we compute integrals.
The Jacobian for Spherical Coordinates
We previously gave a formula for the Jacobian factor for spherical coordinates, but we never derived it. Now we do:
Compute the Jacobian factor for spherical coordinates.
\(J=\rho^2\sin\phi\)
This is an exercise you should definitely do yourself just to see all the miraculous cancellations.
Your answer should be \(J=\rho^2\sin\phi\). Consequently, \(dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\), as promised!
Heading
Placeholder text: Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum Lorem ipsum