If the equations $x\=x\left(u\,v\,w\right)\,y\=y\left(u\,v\,w\right)\,z\=z\left(u\,v\,w\right)$ define an appropriate change of coordinates whereby a region $R$ is mapped to a region $R\prime$, then the following equality of two iterated triple-integrals is valid,

$\int \int {\int }_{R}f\left(x\,y\,z\right) \mathrm{dv}$ = $\int \int {\int }_{R\prime }f\left(x\left(u\,v\,w\right)\,y\left(u\,v\,z\right)\,z\left(u\,v\,w\right)\right) \left|\frac{\partial \left(x\,y\,z\right)}{\partial \left(u\,v\,w\right)}\right| \mathrm{dv}\prime$ 

where $\mathrm{dv}$ is one of six possible orderings of the differentials $\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}$, and $\mathrm{dv}\prime$ is one of six possible orderings of the differentials $\mathrm{du}\,\mathrm{dv}\,\mathrm{dw}$. The integral on the right also contains the absolute value of the Jacobian

$\frac{\partial \left(x\,y\,z\right)}{\partial \left(u\,v\,w\right)}\=\left|\begin{array}{ccc}{\partial }_u\left[\begin{array}{c}x\\ y\\ z\end{array}\right]& {\partial }_v\left[\begin{array}{c}x\\ y\\ z\end{array}\right]& {\partial }_w\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\end{array}\right|$

Each column in the Jacobian matrix is differentiated with respect to a single variable, whereas the rows are gradients of the functions $x\left(u\,v\,w\right)\,y\left(u\,v\,w\right)\,z\left(u\,v\,w\right)$.

In addition, the Jacobian $\frac{\partial \left(x\,y\,z\right)}{\partial \left(u\,v\,w\right)}$ is the reciprocal of the Jacobian $\frac{\partial \left(u\,v\,w\right)}{\partial \left(x\,y\,z\right)}$, the Jacobian of the inverse transformation, because the corresponding Jacobian matrices are multiplicative inverses of each other.

(Versions of these three examples can be found as exercises in texts such as Calculus 6e (Early Transcendentals, Matrix Version) by Edwards & Penney, Prentice Hall, 2002; and Schaum's Outlines - Advanced Calculus by Spiegel, McGraw-Hill, 1996.)