calculation2The Mathematical Study of Mollusk Shells
Derivatives of rotation matrices
Suppose R = [aij] is a 3x3rotation matrix. This means that R preserves lengthsand angles, and preserves orientation. Since R timesthe first basis vector
1 a11R  = [a21] = first column of R, etc., 0 a31
the statement"R preserves lengths and angles" implies that the three columns ofR have unit length and are pairwise orthogonal. This is equivalent to the statement that R RT = I: thatR times the transpose of R (transpose means rows andcolumns are interchanged) is the identity matrix I.
Suppose now Ra is a smooth curve of rotationmatrices, with R0=I. Differentiating the equationRaRaT = I with respectto a givesR'aRaT + RaR'aT = 0. Setting a=0 then givesR'0 + R'0T = 0, i.e thematrix R'0 is equal to minus its transpose. So