detF

detF denotes the determinant of a square matrix F. As a scalar function of the entries of F, it equals the signed volume scaling factor of the linear transformation represented by F: applying F to a shape changes its volume by a factor detF, and a negative detF indicates a reversal of orientation. In many contexts, F is the deformation gradient, the linear map that relates differential elements in a reference configuration to those in the current configuration.

If χ maps reference coordinates X to current coordinates x = χ(X), then F = ∂χ/∂X and detF is the Jacobian determinant J. The local volume change is dV = detF dV0, where dV and dV0 are differential volumes in current and reference configurations, respectively. The quantity detF is therefore central in problems of continuum mechanics and finite deformation.

Key properties include the multiplicativity det(AB) = detA detB for square matrices A and B, and det(F^{-1}) = 1/detF when F is invertible. If F is singular, detF = 0. The sign of detF indicates orientation preservation (positive) or reversal (negative). For a diagonal F = diag(a, b, c), detF = abc, illustrating how detF aggregates scaling along principal directions.

Applications include incompressibility constraints, where detF = 1, and the change of variables formula in integration: ∫_Ω f(x) dx = ∫_{Ω0} f(χ(X)) detF dX. In continuum mechanics, detF is often denoted J and appears in constitutive models and energy formulations, linking deformation to volume change and material response.