21. Multiple Integrals in Curvilinear Coordinates

d. Integrating in 2D Curvilinear Coordinates

3. The Jacobian

On the previous page we found that a double integral may be computed in a curvilinear coordinate system, \((u,v)\), as: \[ \iint_R f\,dA=\iint_R f(u,v)\,J\,du\,dv \] where the differential of area is \[ dA=J\,du\,dv \] and where \(J\) is the Jacobian factor: \[ J=|\vec{e}_u\times\vec{e}_v| \] However, there is a problem with this formula for the Jacobian, namely the cross product is only defined in \(3\) dimensions, but \(\vec{e}_u\) and \(\vec{e}_v\) are \(2\) dimensional vectors. To fix this, we simply let \(\vec{e}_u\) and \(\vec{e}_v\) have a third component of \(0\). Then: \[ \vec{e}_u\times\vec{e}_v =\begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat k \\[4pt] \dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} & 0 \\[8pt] \dfrac{\partial x}{\partial v} & \dfrac{\partial y}{\partial v} & 0 \end{vmatrix} =\hat{\imath}(0)-\hat{\jmath}(0)+\hat{k} \left(\dfrac{\partial x}{\partial u}\dfrac{\partial y}{\partial v} -\,\dfrac{\partial y}{\partial u}\dfrac{\partial x}{\partial v}\right) \] and: \[ J=|\vec{e}_u\times\vec{e}_v| =\sqrt{0^2+0^2 +\left(\dfrac{\partial x}{\partial u}\dfrac{\partial y}{\partial v} -\,\dfrac{\partial y}{\partial u}\dfrac{\partial x}{\partial v}\right)^2} =\left|\dfrac{\partial x}{\partial u}\dfrac{\partial y}{\partial v} -\,\dfrac{\partial y}{\partial u}\dfrac{\partial x}{\partial v}\right| \] Notice that this expression for the Jacobian factor is the absolute value of a determinant called the Jacobian determinant which is usually denoted by \[ \dfrac{\partial(x,y)}{\partial(u,v)}= \begin{vmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} \\[6pt] \dfrac{\partial x}{\partial v} & \dfrac{\partial y}{\partial v} \end{vmatrix} =\det \begin{pmatrix} \dfrac{\partial x}{\partial u} & \dfrac{\partial y}{\partial u} \\[6pt] \dfrac{\partial x}{\partial v} & \dfrac{\partial y}{\partial v} \end{pmatrix} \qquad \text{(*)} \]

Notice that the rows of the Jacobian determinant are the tangent vectors \(\vec{e}_u\) and \(\vec{e}_v\). \[ \dfrac{\partial(x,y)}{\partial(u,v)}= \begin{vmatrix} \leftarrow & \vec e_u & \rightarrow \\ \leftarrow & \vec e_v & \rightarrow \end{vmatrix} \]

Recall that the area 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: \[ J=\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right| \] and appears in the differential of area: \[ dA=J\,du\,dv \]

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 area element is \(dA=J\,du\,dv\).
The Jacobian determinant is useful in surface integtrals, which we study in the next chapter.
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 Polar Coordinates

Recompute the Jacobian factor for polar coordinates (again).

Recall that in polar coordinates, \(\vec{R}(r,\theta)=\left\langle r\cos\theta,r\sin\theta\right\rangle\). So the tangent vectors are: \[ \vec{e}_r=\left\langle \cos\theta,\sin\theta\right\rangle \qquad \text{and} \qquad \vec{e}_\theta=\left\langle -r\sin\theta,r\cos\theta\right\rangle \] So \[\begin{aligned} J&=\left|\dfrac{\partial(x,y)}{\partial(r,\theta)}\right| =\left| \begin{vmatrix} \dfrac{\partial x}{\partial r} & \dfrac{\partial y}{\partial r} \\[6pt] \dfrac{\partial x}{\partial \theta} & \dfrac{\partial y}{\partial \theta} \end{vmatrix} \right| =\left| \begin{vmatrix} \leftarrow & \vec e_r & \rightarrow \\ \leftarrow & \vec e_\theta & \rightarrow \end{vmatrix} \right| \\ &=\left| \begin{vmatrix} \cos\theta & \sin\theta \\ -r\sin\theta & r\cos\theta \end{vmatrix} \right| =|r\cos^2\theta-(-r\sin^2\theta)| =r \end{aligned}\] Using the general formulas for the Jacobian, we have just derived the Jacobian for polar coordinates, \(J=r\). Consequently, \(dA=r\,dr\,d\theta\).

Notice that this requires us to take \(r\) positive when we compute integrals.

On the next two pages we look at elliptic, and “ diamond shaped” coordinate systems and use them to compute integrals.