Some operation about vector and tensor

1 derivative of a vector

\[ Y = (y_1(x_1,x_2,x_3) ,y_2(x_1,x_2,x_3),y_3(x_1,x_2,x_3)) \]
\[ dy_1 = dy_1/dx_1 \cdot dx_1+ \ldots = \sum_i^3 \frac{dy_1}{dx_i} \cdot dx_i\]
\[ dy_i =  \sum_i^3 \frac{dy_1}{dx_i} dx_i \]

\begin{align}
dy_i = \begin{pmatrix}
dy_i/dx_1 &dy_i/dx_2 &dy_i/dx_3
\end{pmatrix} \,
\begin{pmatrix}
dx_1 \\
dx_2 \\
dx_3 \\
\end{pmatrix}
\end{align}

\begin{align}
\begin{pmatrix}
dy_1 \\
dy_2 \\
dy_3 \\
\end{pmatrix} =
\begin{pmatrix}
dy_1/dx_1 &dy_1/dx_2 &dy_1/dx_3 \\
dy_2/dx_1 &dy_2/dx_2 &dy_2/dx_3 \\
dy_3/dx_1 &dy_3/dx_2 &dy_3/dx_3 \\
\end{pmatrix} \,
\begin{pmatrix}
dx_1 \\
dx_2 \\
dx_3 \\
\end{pmatrix}
\end{align}

\begin{align}
\frac{d\vec{y}}{d\vec{x}} =
\begin{bmatrix}
dy_1/dx_1 &dy_1/dx_2 &dy_1/dx_3 \\
dy_2/dx_1 &dy_2/dx_2 &dy_2/dx_3 \\
dy_3/dx_1 &dy_3/dx_2 &dy_3/dx_3 \\
\end{bmatrix}
\end{align}

2 double dot product of two tensor
\[\vec{T}:\vec{U} = \sum_i\sum_j T_{ij}U_{ji}\]

3 the dot product of a tensor with a vector
\[ \vec{T}\cdot \vec{v} = \sum_i \vec{\delta}_i  (\sum_j T_{ij}v_j)  \]

4 the dot product of a vector with a tensor
\[\vec{v}\cdot{T} = \sum_i \vec{\delta}_i  (\sum_j T_{ji}) v_j \]

5. unit tensor 

\[ \nabla \cdot \vec{I} = 0 \]

\[ \vec{v} \cdot \vec{I} = \vec{v} \]

6. vector dot gradient of tensor

\[ \vec{A} \cdot \nabla \cdot \vec{T} = \nabla \cdot( \vec{A} cdot \vec{T}) - \vec{T}:\nabla \vec{A} \]

 

REFERENCE:
http://cs231n.stanford.edu/vecDerivs.pdf
http://www.polymerprocessing.com/notes/root92a.pdf
difference between matrix,bmatrix,pmatrix,vmatix,Vmatrix
https://math-linux.com/latex-26/faq/latex-faq/article/how-to-write-matrices-in-latex-matrix-pmatrix-bmatrix-vmatrix-vmatrix

Resource
The Poor Man’s Introduction to Tensors