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here T : Sym —> Sym is a given isotropic tensor function; Sym denotes the set of symmetric second-order tensors, and I is the identity tensor. The algebraic theorem asserts that for each isotropic function T( •), there are coefficients y,, y2, and y3, which are scalar functions of the principal invariants of B, such that Eq. (1.1) is valid for each B in Sym. Moreover, it is well known that the coefficients yi are uniquely determined by Eq. (1.1) only if the eigenvalues of B are distinct. A natural question on the continuity and differentiability properties of y( arises: suppose T( •) is of class C" (n > 0); can we choose the coefficients yi so that they are of class Ck (0 < k < «)? In 1959, Serrin [4] gave a counterexample in which T( •) is differentiable but there is no way to choose the yt such as to remain continuous. He went on to prove in the same article that the yt can be chosen to be continuous if T( •) is of class C . Recently Man [5] improved Serrin's sufficient 3 2 condition of smoothness from C to C , but that is all that we currently know about the analytical aspects of the representation formula (l.l).1 Except for the analogue of (1.1) in two-dimensional space (cf. Man [5]) and a major memoir of Ball [6] on scalar-valued functions, as far as we are aware, nothing else has been published on the analytical aspects of other representation formulae for isotropic tensor functions. As an explanation for such a deplorable situation, there is the argument that the continuity and differentiability of representation formulae are ignored because they are irrelevant in many applications. We do not accept this argument. Even if we set

Differentiability of the scalar coefficients in two representation formulae for isotropic tensor functions in two dimensions