Derivatives of curvilinear coordinate unit vectors: A concise matrix-vector approach

The time evolution of a curvilinear coordinate system can be identified with an angular velocity matrix or vector, which contains the complete differentiation rule for all unit coordinate vectors of the curvilinear system, providing a concise method for their computations. When generalized the technique provides a powerful tool for differentiation, with respect to any parameter, of not only unit coordinate vectors but any vector that remains fixed in a general orthogonal curvilinear coordinate system. Several curvilinear systems, as well as the normal and tangential coordinate system are examined.