Higher-Order Partial Differentiation

Noboru Endou; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2012)

  • Volume: 20, Issue: 2, page 113-124
  • ISSN: 1426-2630

Abstract

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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).

How to cite

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Noboru Endou, Hiroyuki Okazaki, and Yasunari Shidama. "Higher-Order Partial Differentiation." Formalized Mathematics 20.2 (2012): 113-124. <http://eudml.org/doc/267676>.

@article{NoboruEndou2012,
abstract = {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).},
author = {Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {partial differentiation in normed linear spaces},
language = {eng},
number = {2},
pages = {113-124},
title = {Higher-Order Partial Differentiation},
url = {http://eudml.org/doc/267676},
volume = {20},
year = {2012},
}

TY - JOUR
AU - Noboru Endou
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Higher-Order Partial Differentiation
JO - Formalized Mathematics
PY - 2012
VL - 20
IS - 2
SP - 113
EP - 124
AB - 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).
LA - eng
KW - partial differentiation in normed linear spaces
UR - http://eudml.org/doc/267676
ER -

References

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  1. [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990. 
  2. [2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990. 
  3. [3] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990. 
  4. [4] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. FormalizedMathematics, 1(3):529-536, 1990. 
  5. [5] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990. 
  6. [6] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990. 
  7. [7] Czesław Bylinski. The sum and product of finite sequences of real numbers. FormalizedMathematics, 1(4):661-668, 1990. 
  8. [8] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991. 
  9. [9] Noboru Endou and Yasunari Shidama. Completeness of the real Euclidean space. FormalizedMathematics, 13(4):577-580, 2005. 
  10. [10] Noboru Endou, Yasunari Shidama, and Keiichi Miyajima. Partial differentiation on normed linear spaces Rn. Formalized Mathematics, 15(2):65-72, 2007, doi:10.2478/v10037-007-0008-5.[Crossref] 
  11. [11] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990. 
  12. [12] Takao Inou´e, Noboru Endou, and Yasunari Shidama. Differentiation of vector-valued functions on n-dimensional real normed linear spaces. Formalized Mathematics, 18(4):207-212, 2010, doi: 10.2478/v10037-010-0025-7.[Crossref] 
  13. [13] Takao Inou´e, Adam Naumowicz, Noboru Endou, and Yasunari Shidama. Partial differentiation of vector-valued functions on n-dimensional real normed linear spaces. FormalizedMathematics, 19(1):1-9, 2011, doi: 10.2478/v10037-011-0001-x.[Crossref] Zbl1276.26033
  14. [14] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990. 
  15. [15] Keiichi Miyajima and Yasunari Shidama. Riemann integral of functions from R into Rn. Formalized Mathematics, 17(2):179-185, 2009, doi: 10.2478/v10037-009-0021-y.[Crossref] 
  16. [16] Keiko Narita, Artur Kornilowicz, and Yasunari Shidama. More on the continuity of real functions. Formalized Mathematics, 19(4):233-239, 2011, doi: 10.2478/v10037-011-0032-3.[Crossref] Zbl1276.26006
  17. [17] Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear spaces. Formalized Mathematics, 12(3):269-275, 2004. 
  18. [18] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990. 
  19. [19] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992. 
  20. [20] Jan Popiołek. Real normed space. Formalized Mathematics, 2(1):111-115, 1991. 
  21. [21] Yasunari Shidama. Banach space of bounded linear operators. Formalized Mathematics, 12(1):39-48, 2004. 
  22. [22] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990. 
  23. [23] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990. 

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