除环上映射的Gateaux导数
The Gateaux Derivative of Map over Division Ring
摘要：
本文认为连续除环的映射$f$的微分是最接近于映射$f$的线性映射。与已知导数定义对应的不同表达式是补充。探讨了高阶的加托导数和泰勒级数。泰勒级数允许解简单的微分方程。作为求解微分方程的一个例子，我考虑了一个指数模型。我考虑了所得到的定理在复数域和四元数代数中的应用。与四元数代数中的复数域不同，共轭是以原数\bar a=a+iai+jaj+kak为线性函数的。在四元数代数中，这种差异导致了在复变函数理论中众所周知的柯西-黎曼方程的类似物的缺失。
英文摘要：
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 \bar 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.
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