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