Càlcul Tensorial

It is not necessarily true that there are only two kinds of tensors.

If V is a real vector space, there is a dual vector space V∗, consisting of the linear forms defined on V with real values. Tensors of type (r,s) are vectors belonging to the tensor products of r copies of V and s copies of V∗. Whens=0, your tensor is

contravariant, and when r=0 it is covariant. If both r and s are not 0, then the tensor is of mixed type, so they are neither covariant nor contravariant.

Covariant and contravariant tensors are distinct tensors, they are not components of the same tensor. They can be thought like this tough, assuming that V

is naturally endowed with a (nondegenerate) inner product on V, which will determine a natural isomorphism between V and V∗. In this case the tensors of type (r,s) can be considered of the same type (r+s,0) or (0,r+s), because we can identify V∗ with V, in particular we can raise or lower the indices of the components of vectors and tensors. Note that while you can always define an inner or scalar product on V, it is not necessary that V has a natural one, and it may have more than one.

If V is a complex vector space, you can associate three different other spaces to it. If V is complex, its (complex) dual is made of complex linear forms acting on V and having complex values. But there is also a space ¯V of complex anti-linear forms on V, and another one ¯V∗=¯V∗ of complex anti-linear forms on V∗. So in the complex case you will have four types of vectors, and consequently the tensors can be of type (r,s,p,q), which belong to a complex tensor product of r copies of V, s copies of V∗, p copies of ¯V, and q copies of ¯V∗.

“Tensor algebra and tensor analysis were developed by Riemann, Christoffel, Ricci, Levi-Civita and others in the nineteenth century. The special theory of relativity, as propounded by Einstein in 1905, was elegantly expressed by Minkowski in terms of tensor fields in a flat space-time. In 1915, Einstein formulated the general theory of relativity, in which the space-time manifold is curved. The theory is aesthetically and intellectually satisfying. The general theory of relativity involves tensor analysis in a pseudo- Riemannian manifold from the outset. Later, it was realized that even the pre-relativistic particle mechanics and continuum mechanics can be elegantly formulated in terms of tensor analysis in the three-dimensional Euclidean space. In recent decades, relativistic quantum field theories, gauge field theories, and various unified field theories have all used tensor algebra analysis exhaustively.”

The mathematics of tensors is particularly useful for de- scribing properties of substances which vary in direction– although that’s only one example of their use.

“Gradient a 1-form? How so? Hasn’t one always known the gradient as a vector? Yes, indeed, but only because one was not familiar with the more appropriate 1-form concept.” ~Charles Misner, Kip Thorne, and John Archibald Wheeler, in MTW