we obtain, by a double application of the theorem of multiplication of determinants,
We therefore get the invariant,
Formation of Tensors by Differentiation. Although the algebraic operations of tensor formation have proved to be as simple as in the special case of invariance with respect to linear orthogonal transformations, nevertheless in the general case, the invariant differential operations are, unfortunately, considerably more complicated. The reason for this is as follows. If
is a contra-variant vector, the coefficients of its transformation,
, are independent of position only if the transformation is a linear one. For then the vector components,