Introduction to Mathematical Physics/Dual of a vectorial space

__toc__ Dual of a vectorial space Definition 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. chaptens Tensors 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: or 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 eqcov we have the following relation between components of in and of in : eqcontra This comes from the identification of and 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. center | frame |Covariant and contravariant components of a vector .} figcovcontra 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 . secformultens 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: secappendgreeneq Green's theorem 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: