Diagrams in ancient Egyptian geometry: Survey and assessment

This article surveys and catalogs the geometric diagrams that survive from ancient Egypt. These diagrams are often overspecified and some contain inaccuracies in their construction. The diagrams accompany algorithmic texts and support the mathematical programme of their authors. The study concludes with a brief comparison with the diagram traditions of ancient Babylon, early India, and Greece.     

Differentiability of implicit functions: Beyond the implicit function theorem

The implicit function theorem (IFT) can be used to deduce the differentiability of an implicit mapping S:u?y$S:u?y$ given by the equation e(y,u)=0$e(y,u)=0$. However, the IFT is not applicable when different norms are necessary for the differentiation of e w.r.t. y   and the invertibility of the partial derivative _ey(y,u)$e_y(y,u)$. We prove theorems ensuring the (twice) differentiability of the mapping S which can be applied in this case. We highlight the application of our results to quasilinear partial differential equations whose principal part depends nonlinearly on the gradient of the state ∇y.