I have proper rotation function $R$ over $\mathbb{R}^3$ that yields a $3 \times 3$ tensor $R(x,y,z)$ for every point (x,y,z) in space. If I differentiate this tensor with respect to position (x,y,z), I get a 3x3x3 tensor $T$ at every point (x,y,z).
Now, I know $R$ defines a rotation 'field' in space, and I can extract a scalar field such as rotation angle, or a vector field such as axis of rotation from $R$. The question is, can I use $T$ to extract a scalar field that represents the curvature at each point? If so, how?