[Forget the high number] Jacobi Matrix and Jacobi determinant

Updated: 5 June 2016

"Vector value function" \ (Y=\textbf{f} (X): \omega\subset\mathbb{r}^n\rightarrow\mathbb{r}^m\)

Can be thought of as M-component functions

\ (y_1=f_1 (x_1,x_2,\cdots,x_n) \)

\ (y_2=f_2 (x_1,x_2,\cdots,x_n) \)

......

\ (y_m=f_m (x_1,x_2,\cdots,x_n) \)

"Jacobi Matrix"

\ (J\textbf{f} (X) =\begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \dfrac{\partial f_1}{\partial x_2} & \ Cdots &\dfrac{\partial f_1}{\partial x_n} \ \dfrac{\partial f_2}{\partial x_1} & \dfrac{\partial f_2}{\partial x _2} &\cdots& \dfrac{\partial f_2}{\partial x_n} \ \vdots& & & \vdots \ \dfrac{\partial f_m}{\partial  X_1} &\dfrac{\partial f_m}{\partial x_2} & \cdots &\dfrac{\partial f_m}{\partial x_n} \end{bmatrix} \) \ (J\textbf{f} (X) =\dfrac{\partial (f_1,f_2,\cdots,f_m)}{\partial (x_1,x_2,\cdots,x_n)}\)

The index of the independent variable in each row is incremented, and the subscript of the component function in each column is incremented. is a matrix of m rows n columns \ (M\times n\).

Each row in the Jacobi matrix is considered an n-dimensional vector, which is the gradient function of the line component function.

The Jacobi matrix is the derivative of the vector-valued function \ (f\) at the \ (x\) point: \ (\delta y=\textbf{f} (X_0+\delta x)-\textbf{f} (X_0) =a\delta X+\textbf{o} (\delta x), \ Qquad a=j\textbf{f} (x_0) \)

"Jacobi determinant"

If the vector value function \ (m=n\), then the determinant of its Jacobi matrix is the Jacobi determinant, recorded as \ (\dfrac{d (f_1,f_2,\cdots,f_m)}{d (x_1,x_2,\cdots,x_n)}\)

If the Jacobi determinant of continuous-function f at P-point is not 0, then it has an inverse function near that point.

[Forget the high number] Jacobi Matrix and Jacobi determinant