Introduction to Mathematical Physics/Dual of a vectorial space

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

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 $$E$$ be a finite dimension vectorial space. Let $$e_i$$ be a basis of $$E$$. A vector $$X$$ of $$E$$ can be referenced by its components $$x^i$$ is the basis $$e_i$$:

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 $$E$$ corresponds a space $$E^*$$ called the dual of $$E$$. A element of $$E^*$$ is a linear form on $$E$$: it is a linear mapping $$p$$ that maps any vector $$Y$$ of $$E$$ to a real. $$p$$ is defined by a set of number $$x_i$$ because the most general form of a linear form on $$E$$ is:

A basis $$e^i$$ of $$E^*$$ can be defined by the following linear form

where $$\delta_i^j$$ is one if $$i=j$$ and zero if not. Thus to each vector $$X$$ of $$E$$ of components $$x^i$$ can be associated a dual vector in $$E^*$$ of components $$x_i$$:

The quantity

is an invariant. It is independent on the basis chosen. On another hand, the expression of the components of vector $$X$$ depend on the basis chosen. If $$\omega_k^i$$ defines a transformation that maps basis $$e_i$$ to another basis $$e'_i$$

we have the following relation between components $$x_i$$ of $$X$$ in $$e_i$$ and $$x'_i$$ of $$X$$ in $$e'_i$$:

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. $$x_i$$ are such variables. Contravariant variables that are transformed like the components of a vector on this basis. Using a physicist vocabulary $$x_i$$ is called a covariant vector and $$x^i$$ a contravariant vector. Let $$x_i$$ and $$y_j$$ two vectors of two vectorial spaces $$E_1$$ and $$E_2$$. The tensorial product space $$E_1\otimes E_2$$ is the vectorial space such that there exist a unique isomorphism between the space of the bilinear forms of $$E_1\times E_2$$ and the linear forms of $$E_1\otimes E_2$$. A bilinear form of $$E_1\times E_2$$ is:

It can be considered as a linear form of $$E_1\otimes E_2$$ using application $$\otimes$$ from $$E_1\times E_2$$ to $$E_1\otimes E_2$$ that is linear and distributive with respect to $$+$$. If $$e_i$$ is a basis of $$E_1$$ and $$f_j$$ a basis of $$E_2$$, then

$$e_i\otimes e_j$$ is a basis of $$E_1\otimes E_2$$. Thus tensor $$x_iy_j=T_{ij}$$ is an element of $$E_1\otimes E_2$$. A second order covariant tensor is thus an element of $$E^*\otimes E^*$$. In a change of basis, its components $$a{ij}$$ 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 $$a$$ of $$E^*\otimes E^*\otimes E$$ and noted $$a_{ij}^k$$.

A second order tensor is called symmetric if $$a_{ij}=a_{ji}$$. It is called antisymmetric is $$a_{ij}=-a_{ji}$$.

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

where $$det(\omega)$$ is the determinant of transformation $$\omega$$.

Let us introduce two particular tensors. {{IMP/eq|$$e_{ijk}=\left\{ \begin{array}{ll} 1& \mbox{if permutation } ijk \mbox{ of } 1,2,3 \mbox{ is even}\\ -1& \mbox{if permutation } ijk \mbox{ of } 1,2,3 \mbox{ is  odd}\\ 0 & \mbox{if } ijk \mbox{ is not a permutation of } 1,2,3 \end{array}\right.$$}}
 * The Kronecker symbol $$\delta_{ij}$$ is defined by:  {{IMP/eq|$$  \delta_{ij}=\left\{\begin{array}{ll}   1& \mbox{ if  }i=j\\   0& \mbox{ if  }i\neq j  \end{array}\right.  $$}}  It is the only second order tensor invariant in $$R^3$$ by rotations.
 * The signature of permutations tensor $$e_{ijk}$$ is defined by:

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

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


 * Scalar product $$a.b$$ is the contraction of vectors $$a$$ and $$b$$ :
 * vectorial product of two vectors $$a$$ and $$b$$ is:

From those definitions, following formulas can be showed: $$\begin{matrix} a.(b \wedge c)&=&a_i(b\wedge c)_i\\ &=&a_i\epsilon_{ijk}b_jc_k\\ &=&\epsilon_{ijk}a_ib_jc_k\\ &=&\left|\begin{array}{ccc}a_1&a_2&a_3\\ b_1&b_2&b_3\\ c_1&c_2&c_3\\ \end{array} \right| \end{matrix}$$ Here is useful formula:

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

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