Bing Xie, Xiquan Liang, Xiuzhuan Shen (2009)
Formalized Mathematics
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In this article we define second-order partial differentiation of real binary functions and discuss the relation of second-order partial derivatives and partial derivatives defined in [17].
Takao Inoué (2010)
Formalized Mathematics
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In this article, we shall extend the result of [17] to discuss second-order partial differentiation of real ternary functions (refer to [7] and [14] for partial differentiation).
Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama (2012)
Formalized Mathematics
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In this article, we shall extend the formalization of [10] to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to [10], [12] and [13] for partial differentiation).
Chanapat Pacharapokin, Hiroshi Yamazaki, Yasunari Shidama, Yatsuka Nakamura (2009)
Formalized Mathematics
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For a complex valued function defined on its domain in complex numbers the differentiability in a single point and on a subset of the domain is presented. The main elements of differential calculus are developed. The algebraic properties of differential complex functions are shown.
Takao Inoué, Bing Xie, Xiquan Liang (2010)
Formalized Mathematics
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In this article, we shall extend the result of [19] to discuss partial differentiation of real ternary functions (refer to [8] and [16] for partial differentiation).
Keiko Narita, Artur Kornilowicz, Yasunari Shidama (2011)
Formalized Mathematics
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In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.
Hiroyuki Okazaki, Noboru Endou, Keiko Narita, Yasunari Shidama (2011)
Formalized Mathematics
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In this article, we formalize the differentiability of functions from the set of real numbers into a normed vector space [14].
Hiroshi Yamazaki, Yasunari Shidama, Yatsuka Nakamura, Chanapat Pacharapokin (2009)
Formalized Mathematics
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In this article we prove Cauchy-Riemann differential equations of complex functions. These theorems give necessary and sufficient condition for differentiable function.
Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama (2013)
Formalized Mathematics
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In this article, we formalize continuous differentiability of realvalued functions on n-dimensional real normed linear spaces. Next, we give a definition of the Ck space according to [23].
Keiko Narita, Artur Korniłowicz, Yasunari Shidama (2012)
Formalized Mathematics
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In control engineering, differentiable partial functions from R into Rn play a very important role. In this article, we formalized basic properties of such functions.
Xiquan Liang, Piqing Zhao, Ou Bai (2010)
Formalized Mathematics
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In this article, we first extend several basic theorems of the operation of vector in 3-dimensional Euclidean spaces. Then three unit vectors: e1, e2, e3 and the definition of vector function in the same spaces are introduced. By dint of unit vector the main operation properties as well as the differentiation formulas of vector function are shown [12].