Exercise 1786Calculus › Multivariable calculus › Multiple integralsL1Added 3 Nov 2026

3D change of variables

#change of variables#3D#Jacobian

Statement

Compute the volume of the parallelepiped spanned by $\vec u=(1,0,0)$ , $\vec v=(1,1,0)$ , $\vec w=(1,1,1)$ .

Hint

$V=|\det(\vec u,\vec v,\vec w)|$ .

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Determinant.
$\det\begin{pmatrix}1&1&1\\0&1&1\\0&0&1\end{pmatrix}=1.$

Volume. $V=1$ .

Conclusion. Volume $=1$ .