$$ \begin{aligned} A&\equiv \text{Scalar} \\ A_i&\equiv \text{Vector} \\ A_{i_1\cdots i_n}&\equiv \text{Tensor} \\ \end{aligned} $$
$$ \begin{aligned} \sum_iA_iB_i&=A_iB_i \\ {\color{529CCA}\textbf{Example:}}\quad C_i=\sum_j B_{ij}A_{j}&=B_{ij}A_j\quad\quad\qquad \end{aligned} $$
$$ \begin{aligned} A_iB_i&=\vec{A}^T\vec{B}=\vec{A}\cdot\vec{B}= \begin{bmatrix}A_1 & \cdots & A_N \end{bmatrix} \begin{bmatrix}B_1 \\ \vdots \\ B_N \end{bmatrix}=A_1B_1+\cdots+A_NB_N \\ A_iB_j&=\vec{A}\vec{B}^T= \begin{bmatrix} A_1 \\ \vdots \\ A_N \end{bmatrix} \begin{bmatrix} B_1 & \cdots & B_N \end{bmatrix} =\begin{bmatrix} A_1B_1&\cdots&A_1B_N \\ \vdots & \ddots & \vdots \\ A_NB_1 & \cdots & A_NB_N \end{bmatrix} \end{aligned} $$
<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/postal-horn_1f4ef.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/postal-horn_1f4ef.png" width="40px" /> Vectors: Something with magnitude and direction, its components change with the coordinate system
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<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/nesting-dolls_1fa86.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/nesting-dolls_1fa86.png" width="40px" /> Tensors: are mathematical objects that can be thought of as a multi-dimensional array of numbers. Example: this a representation of a second-order tensor.
$$ \begin{aligned} \sigma_{ij}&=\left[\vec{T}^{(e_1)},\vec{T}^{(e_2)},\vec{T}^{(e_3)}\right] \\ &=\begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \\ \end{bmatrix} \end{aligned} $$
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$$ \begin{aligned} \small\bullet \;\text{ Kronecker delta}\qquad\qquad\;\;\delta_{ij}&=\left\{\begin{matrix} 1 & i=j \\ 0 & i\ne j \end{matrix} \right. \;=\begin{bmatrix} 1 & 0 & \cdots &\cdots&0 \\ 0 &1 &0 &\cdots & 0 \\ \vdots& \vdots & \ddots & \vdots & \vdots \\ 0 & \cdots &\cdots&\cdots &1 \end{bmatrix} \\ \small{\bullet} \;\text{ Levi-Civita symbol}\qquad\quad \epsilon_{ijk}&=\frac{1}{2}(i-k)(j-k)(k-i)\\ \epsilon_{123}&=\epsilon_{312}=\epsilon_{231}=1\\ \epsilon_{321}&=\epsilon_{132}=\epsilon_{213}=-1 \\ \text{if any } i,j,k \text{ are equal }\epsilon_{ijk}&=0 \end{aligned} $$
<aside> <img src="https://em-content.zobj.net/source/microsoft-teams/337/zany-face_1f92a.png" alt="https://em-content.zobj.net/source/microsoft-teams/337/zany-face_1f92a.png" width="40px" /> Isotropic: Tensors that are the same in all frames, $\delta_{ij}$ and $\epsilon_{ijk}$ are isotropic
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$$ \begin{aligned} \vec{A}\cdot\vec{B}&=\delta_{ij}A_iB_i=A_iB_i \\ \vec{A}\times\vec{B}&=\vec{e}i\cdot\epsilon{ijk}A_jB_k \\ (\vec{A}\times\vec{B})i&=\epsilon{ijk}A_jB_k \end{aligned} $$
$$ \vec{e}_i\cdot\vec{e}j=\delta{ij}\qquad;\qquad\vec{e}_i\times\vec{e}j=\epsilon{ijk}\vec{e}_k $$
$$ A'k=\overbrace{L{ik}}^{\small{\vec{e}i\cdot\vec{e}k'}}\times A_i \\ {\color{red}|}\vec{A}'{\color{cyan}|}^2=\underbrace{L{ik}L{jk}}{\delta{ij}}A_iA_j $$
