The Gateaux Derivative of Map over Division Ring
Abstract
I consider differential of mapping f of continuous division ring as linear mapping the most close to mapping f. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux derivative of higher order and Taylor series. The Taylor series allow solving of simple differential equations. As an example of solution of differential equation I considered a model of exponent. I considered application of obtained theorems to complex field and quaternion algebra. In contrast to complex field in quaternion algebra congugation is linear function of original number a=a+iai+jaj+kak . In quaternion algebra this difference leads to the absence of analogue of the Cauchy Riemann equations that are well known in the theory of complex function.