Tensors
Indicial Notation
Note that we always adopt the Einstein summation convention unless otherwise specified.
Two types of indices : free index (live index) is that which only appears once in a term of the expression. The number of the free index indicates the tensor order. dummy index(summation index) is that which is repeated only twice in a term of the expression, and indicates summation.
Kronecker Delta
Definition :
\[\delta_{ij} = \begin{cases}
&1, \textit{iff} \quad i = j\\
& 0, \textit{iff} \quad i \ne j
\end{cases}\]
Matrix Form :
\[\delta_{ij} = \hat{\mathbf{e_i}} \cdot \hat{\mathbf{e_j}} = I\]
Substitution Operator :\(\delta_{ij}A_{ik} = A_{jk}\) \(\delta_{ij} \delta_{ji} = \delta_{ii} = \delta_{jj}\) \(\delta_{ji}a_{ji} = a_{ii}\)
Permutation Symbol
Definition :(also known as Levi-Civita symbol or alternating symbol )
\[\epsilon_{ijk} = \begin{cases}&+1, \textit{if} \quad (i,j,k) \in \{(1,2,3),(2,3,1),(3,1,2)\}\\
&-1,\textit{if}\quad (i,j,k) \in \{(1,3,2),(3,2,1),(2,1,3)\}\\
&0, \textit{for the remaining cases}
\end{cases}\]
Mnemonic device for permutation symbol:
Property :\(\epsilon_{ijk} = \dfrac12(i-j)(j-k)(k-i)\)
\[\begin{aligned}
\epsilon_{ijk} &= \epsilon_{lmn}\delta_{li}\delta_{mj}\delta_{nk}\\
& = \begin{vmatrix}\delta_{1i}&\delta_{1j}&\delta_{1k}\\\delta_{2i}&\delta_{2j}&\delta_{2k}\\\delta_{3i}&\delta_{3j}&\delta_{3k}\end{vmatrix}=\begin{vmatrix}\delta_{1i}&\delta_{2i}&\delta_{3i}\\\delta_{1j}&\delta_{2j}&\delta_{3j}\\\delta_{1k}&\delta_{2k}&\delta_{3k}\end{vmatrix}
\end{aligned}\]
\[\epsilon_{ijk}\epsilon_{pqr}=\begin{vmatrix}\delta_{ip}&\delta_{iq}&\delta_{ir}\\\delta_{jp}&\delta_{jq}&\delta_{jr}\\\delta_{kp}&\delta_{kq}&\delta_{kr}\end{vmatrix}\]
Vector Product :
- component:
\[(\vec{\mathbf{a}} \wedge \vec{\mathbf{b}})_{i} = \epsilon_{ijk}a_jb_k \]
\[\vec{\mathbf{a}}\wedge\vec{\mathbf{b}}=\epsilon_{ijk}{a}_{j}{b}_{k}\hat{\mathbf{e}}_{i}\\{a}_{j}\hat{\mathbf{e}}_{j}\wedge{b}_{k}\hat{\mathbf{e}}_{k}={a}_{j}{b}_{k}\epsilon_{ijk}\hat{\mathbf{e}}_{i}\\{a}_{j}{b}_{k}(\hat{\mathbf{e}}_{j}\wedge\hat{\mathbf{e}}_{k})={a}_{j}{b}_{k}\epsilon_{ijk}\hat{\mathbf{e}}_{i}={a}_{j}{b}_{k}\epsilon_{jki}\hat{\mathbf{e}}_{i}\]
\[(\hat{\mathbf{e}}_j \wedge \hat{\mathbf{e}}_k) = \epsilon_{ijk}\hat{\mathbf{e}}_i\\(\hat{\mathbf{e}}_j \wedge \hat{\mathbf{e}}_k)\cdot\hat{\mathbf{e}}_k = \epsilon_{ijk}\]
Algebraic Operations with Tensors
The tensor product , made up of two vectors \(\vec{\mathbf{v}}\) and \(\vec{\mathbf{u}}\) , becomes a dyad , which is a particular case of a second-order tensor. The dyad is represented by
\[\vec{\mathbf{u}}\vec{\mathbf{v}}\equiv\vec{\mathbf{u}}\otimes\vec{\mathbf{v}}=\mathbf{A}\]
Scalar Product (dot product)
second-order tensor and a vector:
two second-oder tensor:
Double Scalar Product
