where volume (for very small regions) is given by [[Lebesgue measure]] in <math>m</math>-dimensional [[tangent space]]. By making the regions infinitessimally small, the factor relating the two densities is the ratio of volumes, which we term the '''differential volume ratio'''.
To obtain concrete formulas for volume on the <math>m</math>-dimensional manifold, we contruct <math>U</math> by mapping an <math>m</math>-dimensional rectangle in (local) coordimnatecoordinate space to the manifold via a smooth embedding function: <math>\R^m\to\R^n</math>. At very small scale, the embedding function becomes essentially linear so that <math>U</math> is a [[Parallelepiped#Parallelotope|parallelotope]] (multidimensional generalization of a parallelogram). Similarly, the flow transform, <math>f</math> becomes linear, so that the image, <math>V=f(U)</math> is also a parallelotope. In <math>\R^m</math>, we can represent an <math>m</math>-dimensional parallelotope with an <math>m\text{-by-}m</math> matrix whose colum-vectors are a set of edges (meeting at a common vertex) that span the paralellotope. The [[Determinant#Volume_and_Jacobian_determinant|volume is given by the absolute value of the determinant]] of this matrix. If more generally (as is the case here), an <math>m</math>-dimensional paralellotope is embedded in <math>\R^n</math>, it can be represented with a (tall) <math>n\text{-by-}m</math> matrix, say <math>\mathbf V</math>. Denoting the parallelotope as <math>/\mathbf V\!/</math>, its volume is then given by the square root of the [[Gram determinant]]:
