Stretching and Spin Tensors Relations

Can someone help me to understand this please:

The stretching tensor (D) is the symmetric part of the velocity gradient (L), and if I derive D from L with using the polar decomposition of the deformation gradient (F=RU) I get the relation:

D=0.5 R (Udot Uinv + Uinv Udot) R' ----(Eq.1) where Udot is the time derivative of U, Uinv is the inverse of U, and the prime indicates the tranpose of the matrix.

I derive D also from the Lagrangian Green strain tensor E=0.5(F^TF-I)=0.5(U²-I), and i get the relation:

D=R Udot Uinv R'-----(Eq.2)

I am not sure if Eq.1 and Eq.2 are always equivalent, if yes is that mean the spin tensor (skew-symmetric part of L) is always equal to W= Rdot R' (because in some book, I found W=Rdot R' + 0.5 R (Udot Uinv - Uinv Udot) R^')

Thanks alot