  This paper is an edited and shortened version of Chapter 6 from the thesis of
the author. First the one dimensional orthogonal derivative will be extended to
the two-dimensional case. In the two-dimensional case we have to define the
region of integration. In this paper we treat the integration over the square
region and over the triangle region where in the last case we use biorthogonal
polynomials expressed in terms of Appell functions. Next the two-dimensional
orthogonal derivative will be extended to the two-dimensional fractional
orthogonal derivative. The results are highly dependent on the $F_1$,\ $F_2$
and $F_3$ Appell functions.
