A compact tensor algebra expression of the law of propagation of uncertainty

Compact expressions for the GUM's 'law of propagation of uncertainty' and its extensions, including the possible effects of corrections for non-linearity and including the effects of correlation in the variables, are obtained in terms of simple manipulations on three basic tensors: the gradient, the Hessian and the third order third derivative tensor of the measurement function. These formulae are an alternative to those involving mixed matrix and differential algebra. They provide a systematic and reproducible way to arrange the terms provided by the derivatives of various orders of a measurement function. The proposed formulae, which appear mathematically complicated, are surprisingly easy to implement in spreadsheet programs and interpreted matrix oriented programming languages. This is illustrated by means of an example of the propagation of uncertainty for the ratio of two normal and correlated variables.