Rate of deformation and spin tensors

I am studying a set of notes by Ellen Kuhl of Stanford university on continuum mechanics, where I encountered the rate of deformation and spin tensors, as discussed in this set of notes. This set of notes are used in a 2008 (graduate) course on continuum mechanics at Stanford. I am not sure if this set of notes have been turned into a book or not. I have a question regarding an equation in this set of note linked above:

I do not understand eq. (2.7.7), which looks like a simple chain rule. However, the LHS is the rate of change of the spatial deformation vector $t$ correponding to a fixed material vector $T$. Here $t$ is represented by one of the eigenvectors of the rate of deformation tensor $hat{n}_alpha$ at the current moment. But (2.7.7) seems to imply that this eigenvector changes with the spatial deformation vector $t$, so that one can take the derivative with respect to time of the representation $lambda_alpha hat{n}_alpha$ where $lambda_alpha=|t|$. However, I don’t see why $t$ should change with $hat{n}_alpha$. At the next moment, I think $t$ should have a new representation rather than assuming the same direction as $hat{n}_alpha$.

Later on, the notes claim that the spin tensor rotates the eigenvectors of the rate of deformation tensor. This seems to imply there should be some differential equation satisfied by the two tensors in general. I don’t understand: the two tensors should be able to assume arbitrary relationship in general, shouldn’t they?