In this work we show how to obtain a closed form expression of any isotropic tensor function F ( A ) $F\left (\boldsymbol{A}\right )$ and their associated derivatives with A $\boldsymbol{A}$ a second order tensor in a finite dimensional space. Our approach is based on a recent work of the author (SIAM Rev. 62(1):264–280, 2020 ) extending the Omega operator calculus, originally devised by MacMahon to describe partitions of natural numbers, to the realm of matrix analysis, namely, the Omega Matrix Calculus (OMC). The OMC is conceptually simple and useful in practice. Indeed, we show that the Cayley-Hamilton theorem and an improvement for low-rank second order tensors due to Segercrantz (Am. Math. Mon. 99(1):42–44, 1992 ), the representation of isotropic tensor functions and their first derivative of Itskov (Proc. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 459(2034):1449–1457, 2003 ), and Theorem 1.1 of Norris (Q. Appl. Math. 66(4):725–741, 2008 ) are all special cases of a general Omega expression introduced in this work.

中文翻译：

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通过 Omega 矩阵微积分的各向同性张量函数及其导数的方法

在这项工作中，我们展示了如何获得任何各向同性张量函数 F ( A ) $F\left (\boldsymbol{A}\right )$ 及其与 A $\boldsymbol{A}$ 相关的导数的闭式表达式有限维空间中的阶张量。我们的方法基于作者最近的工作 (SIAM Rev. 62(1):264–280, 2020 ) 将最初由 MacMahon 设计的 Omega 算子演算扩展到矩阵分析领域，即欧米茄矩阵演算（OMC）。OMC 在概念上很简单并且在实践中很有用。事实上，我们证明了 Cayley-Hamilton 定理和由于 Segercrantz (Am. Math. Mon. 99(1):42-44, 1992 )、各向同性张量函数的表示及其对低阶二阶张量的改进伊茨科夫 (Proc. R. Soc. Lond. Ser. A, Math. Phys. 英。科学。459(2034):1449–1457, 2003 ) 和 Theorem 1.1 of Norris (Q. Appl. Math. 66(4):725–741, 2008 ) 都是这项工作中引入的一般 Omega 表达式的特例。