16.15.4 Tensor Volume Form Determinant Relation

The determinant of a tensor's volume form reveals how it scales volumes under linear transformations in multilinear algebra.

Tensor Volume Form Determinant Relation is the transformation law describing precisely how a volume form on a vector space responds when its arguments are passed through a linear map, stating that the pulled-back or pushed-forward volume equals the original volume scaled by the determinant of that map. It is the identity that makes the determinant the universal conversion factor between volumes measured before and after any linear transformation is applied.

Statement of the Relation

Pullback of a Volume Form

Given a linear map T from an n-dimensional vector space V to itself, and a volume form ω on V, the pullback of ω through T, evaluated on vectors v₁, ..., vₙ, is defined by evaluating ω on the images of those vectors:

(T^{*} ω) (v_{1}, \dots, v_{n}) = ω (T v_{1}, \dots, T v_{n})

The Determinant Scaling Identity

The core of the relation is that this pullback is always a scalar multiple of ω itself, and that scalar is exactly the determinant of T:

T^{*} ω = det (T) \cdot ω

This holds for every volume form ω on V, since all volume forms on the same space are scalar multiples of one another and the determinant scaling factor is independent of which particular volume form is chosen.

Derivation of the Relation

Reduction to the Top Exterior Power

The relation follows directly from the action of T on the top exterior power Λⁿ(V). Since this space is one-dimensional, the induced map Λⁿ(T) must act by multiplication by some scalar, and this scalar is by definition det(T):

Λ^{n} (T) (v_{1} \land \dots \land v_{n}) = T v_{1} \land \dots \land T v_{n} = det (T) \cdot (v_{1} \land \dots \land v_{n})

Since ω is a functional on Λⁿ(V), applying ω to both sides of this equation directly produces the volume form determinant relation.

Consistency With Coordinate Computation

In coordinates, if v₁, ..., vₙ form a matrix A and T is represented by matrix M, then Tv₁, ..., Tvₙ form the matrix MA, and:

det (M A) = det (M) \cdot det (A)

which is exactly the volume form determinant relation expressed using the multiplicativity of determinants, confirming the abstract argument through direct matrix computation.

Consequences of the Relation

Volume Scaling Factor

The relation gives det(T) its most direct geometric meaning: it is the factor by which T scales the signed volume of any region in V. A region of unsigned volume V₀ maps under T to a region of unsigned volume |det(T)| · V₀, with the sign of det(T) indicating whether orientation is preserved or reversed.

Invertibility Criterion

If det(T) is zero, the relation shows that T*ω is the zero form, meaning T collapses every nonzero volume to zero. This happens precisely when T is not invertible, since a noninvertible linear map sends some nonzero vector to zero and correspondingly collapses at least one dimension of the parallelepiped, forcing the volume to vanish.

Composition Behavior

For two linear maps S and T, applying the relation to their composition gives (ST)ω = det(ST) · ω, while composing the individual pullbacks gives T(Sω) = T(det(S) · ω) = det(S) · det(T) · ω, and consistency between these two expressions reproduces the multiplicative property det(ST) = det(S) · det(T) directly from the volume form perspective.

Significance of the Relation

The volume form determinant relation is the precise algebraic statement of what it means for the determinant to measure volume scaling. It reduces to the multiplicative property of determinants in coordinates, it identifies noninvertibility with total volume collapse, and it grounds the change-of-variables formula in calculus, where the Jacobian determinant plays exactly the role of this scaling factor for locally linear approximations of nonlinear coordinate changes.