where A 0 = A ( 0 ) . It can shown then that the matrix S = A − − 1 L P transforms the vector A 0 − − 1 x ( 0 ) linearly to the same vector at any later time A − − 1 x , i.e., A − − 1 x = S A 0 − − 1 x ( 0 ) . From the definition of A , we can realize the vector A − − 1 x represents a unit normal on the surface of the ellipsoid (true only at the boundary) since a fluid element on the surface moves with the surface. Therefore, we see that S transforms one unit vector on the boundary to another unit vector on the boundary, in other words, it is orthogonal, i.e., S S T = S T S = I . In a similar manner as before, we can define another anti-symmetric matrix as