Second-Order Partial Differentiation of Real Binary Functions

For simplicity, we adopt the following convention: x, x0, y, y0, r are real numbers, z, z0 are elements of R2, f , f1, f2 are partial functions from R2 to R, R is a rest, and L is a linear function. Let us note that every rest is total. Let f be a partial function from R2 to R and let z be an element of R2. The functor pdiff1(f, z) yielding a function from R2 into R is defined as follows: (Def. 1) For every z such that z ∈ R2 holds (pdiff1(f, z))(z) = partdiff1(f, z). The functor pdiff2(f, z) yields a function from R2 into R and is defined as follows: