Introduction to Mathematical Physics/Dual of a vectorial space

Dual of a vectorial space edit

Definition edit


Let   be a vectorial space on a commutative field  . The vectorial space  of the linear forms on   is called the dual of   and is noted  .

When   has a finite dimension, then   has also a finite dimension and its dimension is equal to the dimension of  . If   has an infinite dimension,   has also an infinite dimension but the two spaces are not isomorphic.


Tensors edit

In this appendix, we introduce the fundamental notion of tensor\index{tensor} in physics. More information can be found in ([#References|references]) for instance. Let   be a finite dimension vectorial space. Let   be a basis of  . A vector   of   can be referenced by its components   is the basis  :


In this chapter the repeated index convention (or {\bf Einstein summing convention}) will be used. It consists in considering that a product of two quantities with the same index correspond to a sum over this index. For instance:




To the vectorial space   corresponds a space   called the dual of  . A element of   is a linear form on  : it is a linear mapping   that maps any vector   of   to a real.   is defined by a set of number   because the most general form of a linear form on   is:


A basis   of   can be defined by the following linear form


where   is one if   and zero if not. Thus to each vector   of   of components   can be associated a dual vector in   of components  :


The quantity


is an invariant. It is independent on the basis chosen. On another hand, the expression of the components of vector   depend on the basis chosen. If   defines a transformation that maps basis   to another basis  



we have the following relation between components   of   in   and   of   in  :



This comes from the identification of




Equations eqcov and eqcontra define two types of variables: covariant variables that are transformed like the vector basis.   are such variables. Contravariant variables that are transformed like the components of a vector on this basis. Using a physicist vocabulary   is called a covariant vector and   a contravariant vector.

Covariant and contravariant components of a vector  .}

Let   and   two vectors of two vectorial spaces   and  . The tensorial product space   is the vectorial space such that there exist a unique isomorphism between the space of the bilinear forms of   and the linear forms of  . A bilinear form of   is:


It can be considered as a linear form of   using application   from   to   that is linear and distributive with respect to  . If   is a basis of   and   a basis of  , then


  is a basis of  . Thus tensor   is an element of  . A second order covariant tensor is thus an element of  . In a change of basis, its components   are transformed according the following relation:


Now we can define a tensor on any rank of any variance. For instance a tensor of third order two times covariant and one time contravariant is an element   of   and noted  .

A second order tensor is called symmetric if  . It is called antisymmetric is  .

Pseudo tensors are transformed slightly differently from ordinary tensors. For instance a second order covariant pseudo tensor is transformed according to:


where   is the determinant of transformation  .


Let us introduce two particular tensors.

  • The Kronecker symbol   is defined by:
    It is the only second order tensor invariant in   by rotations.
  • The signature of permutations tensor   is defined by:

It is the only pseudo tensor of rank 3 invariant by rotations in  . It verifies the equality:


Let us introduce two tensor operations: scalar product, vectorial product.

  • Scalar product   is the contraction of vectors   and   :
  • vectorial product of two vectors   and   is:

From those definitions, following formulas can be showed:


Here is useful formula:



Green's theorem edit

Green's theorem allows one to transform a volume calculation integral into a surface calculation integral.


Let   be a bounded domain of   with a regular boundary. Let   be the unitary vector normal to hypersurface   (oriented towards the exterior of  ). Let   be a tensor, continuously derivable in  , then:\index{Green's theorem}


Here are some important Green's formulas obtained by applying Green's theorem: