Editing Tensor field (section)

== Definition ==

Let <math>M</math> be a [[manifold]], for instance the [[Euclidean space]] <math>\R^n</math>.

{{blockquote|'''Definition.'''  A '''tensor field''' of type <math>(p,q)</math> is a section
<math display="block">
  T\ \in\ \Gamma(M, V^{\otimes p}\otimes (V^*)^{\otimes q})
</math>
where <math>V=TM</math> to be the [[tangent bundle]] of <math>M</math> (whose sections are called vector fields or contra variant vector fields in Physics) and <math>V^* = T^*M</math> is its dual bundle, the cotangent space (whose sections are called 1 forms, or covariant vector fields in Physics), and <math>\otimes</math> is the [[Tensor product bundle|tensor product]] of vector bundles.}}  

Equivalently, a tensor field is a collection of elements <math>T_x\in V_x^{\otimes p}\otimes (V_x^*)^{\otimes q}</math> for every point <math>x\in M</math>, where <math>\otimes</math> now denotes the tensor product of vectors spaces, such that it constitutes a smooth map <math>T:M\rightarrow V^{\otimes p}\otimes (V^*)^{\otimes q}</math>. The elements <math>T_x</math> are called [[tensor]]s.

Locally in a coordinate neighbourhood <math>U</math> with coordinates <math>x^1, \ldots x^n</math> we have a local basis (Vielbein) of vector fields <math>\partial_1 = \frac{\partial}{\partial x^n} \ldots \partial_n = \frac{\partial}{\partial x_n}</math>, and a dual basis of 1 forms <math>dx^1, \ldots dx^n</math> so that 
<math>dx^i(\partial_j) = \partial_j x^i = \delta^i_j</math>. In the coordinate neighbourhood <math>U</math> we then have
<math display = "block">
        T_x = T^{i_1, \ldots i_p}_{j_1, \ldots, j_q}(x^1, \ldots, x^n) \partial_{i_1} \otimes \cdots \otimes \partial_{i_p}\otimes dx^{j_1} \otimes \cdots \otimes dx^{j_q}
</math>    
where here and below we use Einstein summation conventions. Note that if we choose different coordinate system <math>y^1 \ldots y^n</math> then <math>\frac\partial{\partial x^i} = \frac{\partial y^k}{\partial x^i}\frac\partial{\partial y^k}</math> and <math>dx^j = \frac{\partial x^j}{\partial y^\ell}dy^\ell</math> where the coordinates <math>(x^1, \ldots, x^n)</math> can be expressed in the coordinates <math>(y^1,\ldots y^n</math> and vice versa, so that
 
<math display = "block">
\begin{align}
        T_x &= T^{i_1, \ldots i_p}_{j_1, \ldots, j_q}(x^1, \ldots, x^n) \frac{\partial}{\partial x^{i_1}}\otimes \cdots \otimes \frac{\partial}{\partial x^{i_p}}\otimes dx^{j_1} \otimes \cdots  \otimes dx^{j_q} \\
            &= T^{i_1, \ldots i_p}_{j_1, \ldots, j_q}(x^1, \ldots, x^n) \frac{\partial y^{k_1}}{\partial x^{i_1}}\cdots\frac{\partial y^{k_p}}{\partial x^{i_p}}\frac{\partial x^{j_1}}{\partial y^{\ell_1}}\cdots\frac{\partial x^{j_q}}{\partial y^{\ell_q}}
\frac{\partial}{\partial y^{k_1}}\otimes \cdots \otimes \frac{\partial}{\partial y^{k_p}}\otimes dy^{\ell_1} \otimes \cdots \otimes dy^{\ell_q}\\
       &=T^{k_1, \ldots, k_p}_{\ell_1,\cdots \ell_q}(y^1, \ldots y^n) \frac{\partial}{\partial y^{k_1}}\otimes \cdots \otimes\frac{\partial}{\partial y^{k_p}}\otimes dy^{\ell_1} \otimes \cdots \otimes dy^{\ell_q}\\ 
\end{align} 
</math>
i.e. 
<math display = "block">
T^{k_1, \ldots, k_p}_{\ell_1,\cdots \ell_q}(y^1, \ldots y^n) = T^{i_1, \ldots i_p}_{j_1, \ldots, j_q}(x^1, \ldots, x^n) \frac{\partial y^{k_1}}{\partial x^{i_1}}\cdots\frac{\partial y^{k_p}}{\partial x^{i_p}}\frac{\partial x^{j_1}}{\partial y^{\ell_1}}\cdots\frac{\partial x^{j_q}}{\partial y^{\ell_q}}
</math>
The system of indexed functions <math>T^{i_1, \ldots i_p}_{j_1, \ldots, j_q}(x^1, \ldots, x^n)</math> (one system for each choice of coordinate system) connected by transformations as above are the tensors in the definitions below.  

'''Remark'''
One can, more generally, take 
<math>V</math> to be any [[vector bundle]] on <math>M</math>, and <math>V^*</math> its [[dual bundle]]. In that case can be a more general topological space. These sections are called tensors of <math>V</math> or tensors for short if no confusion is possible .