A rotation tensor R has the property that RT=R−1, and rotates a vector v by right-multiplication: v′=Rv.
Given the basis system ei and a rotated basis system gi=Rei, where R=Rijei⊗ej, the coordinate transformation tensor from ei to gi is
Q=(gi⋅ej)gi⊗ej=RijTgi⊗ej=I Single contraction from the left of each base vector with the rotation/coordinate transformation tensor. In practice, left contraction with each free index.
Consider a rotation as a function of time, i.e. R(t). Then
∂t∂RTR=0[RT∂t∂R]T=∂t∂RTR+RT∂t∂R=−RT∂t∂R⇒Skew symmetric If we now consider rotation of vector v=R(t)⋅v0 with time, starting at v0, we have
v˙v˙=∂t∂Rv0=∂t∂RRTv,v0=R−1v=RTv=ωv The skew symmetric tensor ω is often called the spin tensor.