Résumé
Mathematical models are sometime given as functions of independent input
variables and equations or inequations connecting the input variables. A
probabilistic characterization of such models results in treating them as
functions with non-independent variables. Using the distribution function or
copula of such variables that comply with such equations or inequations, we
derive two types of partial derivatives of functions with non-independent
variables (i.e., actual and dependent derivatives) and argue in favor of the
latter. The dependent partial derivatives of functions with non-independent
variables rely on the dependent Jacobian matrix of dependent variables, which
is also used to define a tensor metric. The differential geometric framework
allows for deriving the gradient, Hessian and Taylor-type expansion of
functions with non-independent variables.