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 \overline 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.
Вход в систему
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