Tensors on flat spaces
Tensors are generalizations of vectors. The traditional way of introducing tensors is through their transformation properties. This chapter shows that tensors are linear relations between scalars, vectors, and even other tensors. It focuses on a notational device—the Schouten index convention—that simplifies tensor transformation equations. Coordinates in different coordinate systems are distinguished by primes attached to indices. Coordinate differentials in one coordinate system thus determine the coordinate differentials in another coordinate system. An invariant is a quantity that does not change under coordinate transformations. The simplest type of invariant is a scalar, a number, such as the spacetime separation. Contravariant vectors share the attributes of the displacement vector and should simply be called vectors. Totally antisymmetric tensors and determinants play an important role in differential geometry.