Differentiability of Lipschitz mappings into metric spaces: area and co-area formulas

We give a self-contained exposition of metric differentiability for Lipschitz mappings from sets in Euclidean spaces into arbitrary metric spaces, together with the corresponding area and co-area formulas. The novelty is a new metric implicit function theorem for mappings into metric spaces, which is then used to give a direct and geometric proof of the co-area formula for Lipschitz mappings into metric spaces.