There is a spatial scalar field depending on the three Cartesian coordinates, for example, the Young modulus, E (x) = E (x1, x2, x3), and a reference point. There are two reference frames, local and lobal, and the global-to-local rotation matrix A. Both frames have their origins at the reference point. In the global frame, at the reference point, the gradient "nabla E" is given (a vector of length 3) and the Hessian "nabla nabla E" (a symmetric matrix of size 3x3). The rotation matrix is spatially varying A = A (x). The gradient and the Hessian of the rotation matrix are known; these are 3d and 4d arrays, 3x3x3 and 3x3x3x3. I need to compute the gradient and Hessian of the scalar field E in the local frame. Should rotation matrix A be independent of x, this is a trivial problem:

nabla E (local) = nabla E (global) . A

nabla nabla E (local) = A . nabla nabla E (global) . AT

where nabla means gradient and nabla nabla means Hessian.

However, the rotation matrix is not constant, it is spatially dependent, and additional terms may be needed on the right-hand sides of both equations, involving the spatial derivatives of the rotation matrix. Colleagues, did anybody face this issue?