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 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.

Note about Vertical Bars

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:

Your answer should be \(J=\rho^2\sin\phi\). Consequently, \(dV=\rho^2\sin\phi\,d\rho\,d\phi\,d\theta\), as promised!