Tensors

Let \(\overrightarrow{\rm u}\) defined in base (x,y,z) and \(\overrightarrow{\rm u’} \) defined in base (x’,y’,z’)
\( \overrightarrow{\rm u’} = P^{-1}. \overrightarrow{\rm u}\) where P is the transition matrix from the old to the new base.

Tensor definition

Geometric entity that has \(3^n\) components in a 3-dimensional coordinate system, where n is the rank (or order) of the tensor.
Rank 0 tensor: \(3^0=1\), example: temperature
Rank 1 tensor: \(3^1=3\), example: speed
Rank 2 tensor: \(3^2=9\), example: spatial derivatives of velocity
Thus the vectors are tensors of rank (or order) 1.

Tensor product

The tensor product of 2 tensors of rank n and m gives 1 tensor of rank n+m
Example: tensor of rank 1 \(a_i\) and \(b_j\), the tensor \(c_{ij}\) of rank 2 defined as:
$$c_{ij} = a_i . b_j \text{, in matrix: } [c]=\left[ \begin{array}{cc} a_1.b_1 & a_1.b_2 & a_1.b_3 \\ a_2.b_1 & a_2.b_2 & a_2.b_3 \\ a_3.b_1 & a_3.b_2 & a_3.b_3 \end{array} \right]$$ $$\overrightarrow{\rm a} \otimes \overrightarrow{\rm b} = [c] = \overline{\overline{\rm c}} $$

Properties

Distributivity of addition:
$$[U \otimes(V_1+V_2 )](x,y)= [U \otimes V_1 + U \otimes V_2 ](x,y)$$ Distributivity of multiplication:
$$\lambda.[U \otimes V](x,y)= [(\lambda.U) \otimes V](x,y)= [U \otimes (\lambda.V)](x,y) $$ Non-commutativity:
The tensors are not commutative.

Usual definitions

Identity tensor (or metric):
$$\overline{\overline{\rm I}} = \delta_{ij} (\overline{\rm e}_i \otimes \overline{\rm e}_j )\rightarrow \text{ Tensor whose components are equal to 1 if i=j , and 0 otherwise; therefore on the diagonal.}$$
Inverse tensor: of \(\overline{\rm T}\) denoted \(\overline{\rm e}^{-1}\), such as:
$$T_{ik}.T_{kj}^{-1}= \delta_{ij}$$ Unit tensor: \(\delta_{ij}\) (or Kronecker symbol):
$$\delta_{ij} = 1 \text{ for i=j ;} [\delta]=I=\left[ \begin{array}{cc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]$$ Trace:
$$tr[c]=c_{ij}.\delta_{ij}=c_{ii}$$ Transpose:
– Vector Product:
$$(\overrightarrow{\rm a} \otimes \overrightarrow{\rm b})^T = \overrightarrow{\rm b} \otimes \overrightarrow{\rm a}$$ – Tensor:
$$\overline{\overline{\rm T}}^T=T_{ji} (\overline{\rm e}_i \otimes \overline{\rm e}_j)$$