On Schwarz differentiability, III

On Pseudo-differentiability

We present some new relations between the pseudo-derivatives and parabolic epiderivatives recently introduced by Rockafellar, and also its infinite dimensional counterparts. Significant extensions of the most important known results are proven, which further clarify the range of applicability of this new theory.

On Concepts of Directional Differentiability

Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of G~teaux, Fr6chet, and Hadamard are singled out from the general framework of cr-directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equ...

On the Differentiability of Isometries

Let 9TC be an re-dimensional differentiable manifold of class Ck, k = l, • • • , oo, w (as usual O" means analytic, and we use the conventions * ±m= and w + w=w for integers m). An automorphism of 9TC is a one-to-one map of 911 onto itself which is of class C* and is nonsingular. Now let 9TC be given a Riemannian structure, i.e. a positive definite covariant tensor field of rank two, and su...

Schwarz boundary problem on a triangle

In this paper, the Schwarz boundary value problem (BVP) for the inhomogeneous Cauchy-Riemann equation in a triangle is investigated explicitly. Firstly, by the technique of parquetingreflection and the Cauchy-Pompeiu representation formula a modified Cauchy-Schwarz representation formula is obtained. Then, the solution of the Schwarz BVP is explicitly solved. In particular, the boundary behavio...

On Asymptotic Differentiability of Averages