ABSTRACT
This chapter begins by an investigation of the symmetric derivatives and follow where they lead. The first question that would obviously arise in a study of the symmetric derivatives is to ask how far properties of ordinary derivatives extend to symmetric derivatives. The symmetric derivative has been defined as a kind of odd derivative. In any discussion of generalized derivatives, of which the symmetric derivative may be considered one of the simplest, it is natural to ask to what extent the usual rules of derivation apply. One of the most interesting and immediate applications of Riemann’s theory of trigonometric series is to the problem of uniqueness. The first and second symmetric derivatives can also be viewed as the initial steps in a hierarchy of symmetric derivatives.
