y=f(x1,⋯ ,xi,⋯ ,xn)y = f(x_1,\cdots,x_i,\cdots,x_n) y=f(x1,⋯,xi,⋯,xn)
X=[x1,⋯ ,xi,⋯ ,xn]X = [x_1,\cdots,x_i,\cdots,x_n] X=[x1,⋯,xi,⋯,xn]
∂y∂X=[∂f∂x1,⋯ ,∂f∂xi,⋯ ,∂f∂xn]\frac {\partial y}{\partial X} = [\frac {\partial f}{\partial x_1},\cdots,\frac {\partial f}{\partial x_i},\cdots,\frac {\partial f}{\partial x_n}] ∂X∂y=[∂x1∂f,⋯,∂xi∂f,⋯,∂xn∂f]
Y=[f1(x1,⋯ ,xi,⋯ ,xn),⋯ ,fi(x1,⋯ ,xi,⋯ ,xn),⋯ ,fm(x1,⋯ ,xi,⋯ ,xn)]Y = [f_1(x_1,\cdots,x_i,\cdots,x_n),\cdots, f_i(x_1,\cdots,x_i,\cdots,x_n),\cdots, f_m(x_1,\cdots,x_i,\cdots,x_n)] Y=[f1(x1,⋯,xi,⋯,xn),⋯,fi(x1,⋯,xi,⋯,xn),⋯,fm(x1,⋯,xi,⋯,xn)]
∂Y∂X=[∂f1∂x1∂f2∂x1⋯∂fm∂x1∂f1∂x2∂f2∂x2⋯∂fm∂x1⋮⋮⋱⋮∂f1∂xn⋯⋯∂fm∂xn]\frac {\partial Y}{\partial X} = \begin{bmatrix} \frac {\partial f_1}{\partial x_1} & \frac {\partial f_2}{\partial x_1} & \cdots & \frac {\partial f_m}{\partial x_1}\cr \frac {\partial f_1}{\partial x_2} & \frac {\partial f_2}{\partial x_2} & \cdots & \frac {\partial f_m}{\partial x_1}\cr \vdots & \vdots & \ddots & \vdots\cr \frac {\partial f_1}{\partial x_n} & \cdots & \cdots & \frac {\partial f_m}{\partial x_n}\cr \end{bmatrix} ∂X∂Y=∂x1∂f1∂x2∂f1⋮∂xn∂f1∂x1∂f2∂x2∂f2⋮⋯⋯⋯⋱⋯∂x1∂fm∂x1∂fm⋮∂xn∂fm
这是一个n行m列的矩阵,有时也会写成m行n列,都是一样的,区别在于加不加转置
