Exact expansions of arbitrary tensor functions \(\mathbf F(\mathbf A)\) and their derivatives. (Q1429537)

Let \(\mathbf A\) be a second order tensor in an \(n\)-dimensional Euclidean space, and \(\mathbf F(\mathbf A)\) a tensor function defined by the power series. A method is presented for determining the exact expansions of \(\mathbf F(\mathbf A)\). It is shown that \(\mathbf F(\mathbf A)\) can be obtained by differentiating a scalar function of the eigenvalues of \(\mathbf A\). In two- and three-dimensional cases, closed-form, singularity-free expressions of arbitrary tensor functions and their first derivatives are deduced.