en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
1 correction found
the n-volume of a parallelepiped is the determinant of its edge vectors.
Ordinary geometric volume is the absolute value of the determinant, not the determinant itself. The determinant can be negative because it encodes orientation as well as size.
Full reasoning
For a parallelepiped spanned by edge vectors, the geometric (n)-volume is (|\det A|), where (A) is the matrix whose columns (or rows) are those edge vectors. The determinant itself is a signed quantity: changing orientation flips its sign without changing the actual volume.
That distinction matters here because the surrounding discussion is about the magnitude of the Jacobian determinant in the change-of-variables formula. So saying that the volume "is the determinant" is mathematically inaccurate; it should be the absolute value of the determinant.
Authoritative sources say this explicitly. UC Davis notes state that the volume of a parallelepiped is the absolute value of the determinant of the corresponding matrix. MIT OCW likewise states that the absolute value of the determinant gives the volume/hypervolume, while the determinant itself is a signed version of hypervolume.
2 sources
- Volume of a a parallelepiped
It can be shown that the volume of the parallelepiped is the absolute value of the determinant of the following matrix.
- 18.013A Calculus with Applications, Fall 2001, Online Textbook
the absolute value of the determinant of that matrix will be the volume of the parallelepiped... the determinant can be considered a linear and signed version of hypervolume.