Tensor calculus

In classical mechanics, and especially in Newtonian mechanics, physical effects result from forces acting on solid bodies. As a mathematical object, a force is a vector. There is an intrinsic, purely operative definition of vectors as elements of a vector space E over a body K (article Calcul matriciel in the present treatise). We shall see 1.1 that there is another definition of vectors, more satisfactory for the physicist, and indeed more fruitful of inspiration for the mathematician. Some fields of physics, in particular continuum mechanics (see [A 303] Deformation and stresses in a continuous medium and other articles in the Structural Calculus section of this treatise), favour other mathematical concepts: in particular, the concept of tension.

There are two equivalent definitions of tensors in finite dimension (in the rest of this article, we will restrict ourselves to tensor calculus on finite dimensional spaces):

• intrinsic tensor calculus, which is the introduction of formal multiplication on a vector space;

• the tensor calculus of physicists: a tensor is an array of numbers attached to a particular basis of the vector space E, and transforms according to a law given by a change of basis.

This article comprises four paragraphs:

• The first paragraph describes the concepts of covariance and contravariance for vectors and shapes;

• a second paragraph, inspired by the previous example, defines tensors and establishes their equivalence;

• a third paragraph deals specifically with the external product and the definition of determinants;

• paragraph 4