ABSTRACT
Contravariant base vectors may be defined by, and tensors can be resolved along either the covariant or contravariant base vectors using the formalism. The calculus of multivariable tensor fields whose spatial points are labeled with curvilinear coordinates is a generalization of that developed for Cartesian coordinates. The covariant and contravariant base vectors and various components of the metric and permutation tensors are covariantly constant. If indexed quantities are related by the appropriate transformation laws for a change of curvilinear coordinates, they can be regarded as the components of a extensor. Conversely, some indexed quantities such as the partial derivatives and the Christoffel symbols do not obey the correct transformation laws. Many of the mathematical formulas that we have developed for tensor fields of three variables in arbitrary curvilinear coordinates may be specialized to surface tensor fields of two variables simply by replacing the Latin indices by Greek indices.
