Introduction to Mathematical Physics/Dual of a vectorial space

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Dual of a vectorial space[edit]

Definition[edit]

Definition:

Let E be a vectorial space on a commutative field K. The vectorial space {\mathcal L}(E,K)of the linear forms on E is called the dual of E and is noted E^*.

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


chaptens

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 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:


X=x^ie_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:


x^ie_i=\sum_i x^ie_i

or


a_{ijk}b_{ikm}=\sum_i\sum_k a_{ijk}b_{ikm}

To the vectorial space E correspond 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:


p(Y)=x_iy^i

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


e^je_i=\delta_i^j

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:


X=x_ie^i

The quantity


|X|=x_ix^i

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

eqcov

e'_i=\omega^k_ie_k

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

eqcontra

x^i=\omega^i_kx'^k

This comes from the identification of


X=x'^ie'_i=x'^i\omega^k_ie_k

and


X=x^ie_i

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.

Covariant and contravariant components of a vector X.}
figcovcontra

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:


b(x,y)=b^{ij}x_ix_j

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

x\otimes y=x_iy_j e_i\otimes e_j.

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:


a'_{ij}=\omega^k_i\omega^l_j a_{kl}

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:


a'_{ij}=det(\omega)\omega^k_i\omega^l_j a_{kl}

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

secformultens

Let us introduce two particular tensors.

  • The Kronecker symbol \delta_{ij} is defined by:

  \delta_{ij}=\left\{\begin{array}{ll}   1& \mbox{ si  }i=j\\   0& \mbox{ si  }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:

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.


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

  e_{ijk}e_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}

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


  • Scalar product a.b is the contraction of vectors a and b :

  a.b=a_ib_i

  • vectorial product of two vectors a and b is:

  (a\wedge b)_i=e_{ijk}a_jb_k

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:

a\wedge (b\wedge c)=d(a.c)-c(a.b)


secappendgreeneq

Green's theorem[edit]

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

Theorem:

Let \omega be a bounded domain of R^p with a regular boundary. Let \vec n be the unitary vector normal to hypersurface \partial \omega (oriented towards the exterior of \omega). Let t_{ij\dots q} be a tensor, continuously derivable in \omega, then:\index{Green's theorem}


\int_{\omega}t_{ij\dots q,r}dv=\int_{\partial\omega}t_{ij\dots q}n_r ds

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


\int_{\omega}\mbox{ grad }\phi dv=\int_{\partial\omega}\phi \vec n\vec{ds}


\int_{\omega}\mbox{ rot }\vec{u}dv=\int_{\partial\omega}(\vec{n}\wedge\vec{u})\vec{ds}


\int_{\omega}f_{,i}gdv+\int_{\omega}fg_{,i}dv=\int_{\partial\omega}
fg\vec{\nu}_i\vec{ds}