OK. A problem is simple and probably very stupid... In many (most of) maths books things like Euclidean metric, norm, scalar product etc. are defined in Cartesian coordinate system (that is one which is orthonormal). But what would happen if one were to define all these quantities in non-Cartesian system? For example in many cases (books) the dot product is defined as sum of coordinates for two vectors. Such expression is definitely not true for non-orthogonal systems. Can anyone comment on such issues, perhaps some can propose some detailed reading?