In Section 1.4b we considered the n-tuple of partial derivatives of a single function ∂F/∂xj and we noticed that this n-tuple does not transform in the same way as the n-tuple of components of a vector. These components ∂F/∂xj transform as a new type of “vector.” In this chapter we shall talk of the general notion of “tensor” that will include both notions of vector and a whole class of objects characterized by a transformation law generalizing 1.6. We shall, however, strive to define these objects and operations on them “intrinsically,” that is, in a basis-free fashion. We shall also be very careful in our use of sub- and superscripts when we express components in terms of bases; the notation is designed to help us recognize intrinsic quantities when they are presented in component form and to help prevent us from making blatant errors.

Covectors and Riemannian Metrics

Linear Functionals and the Dual Space

Let E be a real vector space. Although for some purposes E may be infinite-dimensional, we are mainly concerned with the finite-dimensional case. Although ℝn as the space of real n-tuples (x1, …, xn), comes equipped with a distinguished basis (1, 0, 0, …, 0)T, …, the general n-dimensional vector space E has no basis prescribed.