dFij

dFij is a notation used in differential calculus to denote the differential of the ij-th component of a matrix-valued function F. It is not a single universally standardized symbol, but a convenient shorthand that appears in contexts where F takes values in a matrix space and its individual components are studied separately.

Definition and usage. If F maps a domain X into matrices of size n by m, with entries Fij, then the differential dF is the matrix of differential 1-forms with entries dFij. In local coordinates xk on X, each component has dFij = sum_k (∂Fij/∂xk) dxk, and the full differential is written as the matrix dF = [dFij]. This treats F as a field whose components vary infinitesimally, component by component.

Example. Let F(x, y) = [[f11(x, y), f12(x, y)], [f21(x, y), f22(x, y)]]. Then dF = [[df11, df12], [df21, df22]], where df11 = ∂f11/∂x dx + ∂f11/∂y dy, and similarly for the other components. The expression dFij thus captures the infinitesimal change of the ij-th entry of F.

Relation to other concepts. dFij is related to the broader idea of matrix-valued differential forms and to the exterior derivative applied componentwise. It should not be confused with the deformation gradient F used in continuum mechanics, which is a separate object describing local deformations, nor with the exterior derivative of scalar functions.

See also. differential, differential form, matrix-valued function, deformation gradient, tensor calculus.