= Solution
Let the new orthonormal <basis> satisfy $e'_i=R_{ij}e_j$. Then an $n$th-rank tensor transforms by
$$
T'_{i_1\ldots i_n}=R_{i_1j_1}\cdots R_{i_nj_n}T_{j_1\ldots j_n}.
$$
Since $\partial/\partial x'_i=R_{ij}\partial/\partial x_j$, the Hessian of a <scalar> obeys this law with two factors of $R$.
An isotropic tensor is unchanged under every rotation. In three dimensions the most general isotropic tensors of ranks zero through three are respectively
$$
\boxed{a,\qquad0,\qquad b\delta_{ij},\qquad c\epsilon_{ijk}.}
$$
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