====================================================================================分割线
假设：
X = (x1, x2, x3)
Y=2*X = (2*x1, 2*x2, 2*x3) = (y1, y2, y3)
实际上有：
y1=f(x1,x2,x3)=2*x1
y2=f(x1,x2,x3)=2*x2
y3=f(x1,x2,x3)=2*x3
---------------------------------------------------------------------------------------------------------------------------------------------------分割线
J = ( ? y 1 ? x 1 ? y 1 ? x 2 ? y 1 ? x 3 ? y 2 ? x 1 ? y 2 ? x 2 ? y 2 ? x 3 ? y 3 ? x 1 ? y 3 ? x 2 ? y 3 ? x 3 ) \begin{equation*} J= \begin{pmatrix} \dfrac{\partial y_1}{\partial x_1}&\dfrac{\partial y_1}{\partial x_2}&\dfrac{\partial y_1}{\partial x_3} \\[2.5ex] \dfrac{\partial y_2}{\partial x_1}&\dfrac{\partial y_2}{\partial x_2}&\dfrac{\partial y_2}{\partial x_3} \\[2.5ex] \dfrac{\partial y_3}{\partial x_1}&\dfrac{\partial y_3}{\partial x_2}&\dfrac{\partial y_3}{\partial x_3} \\[1.5ex] \end{pmatrix} \end{equation*} J= ??x1??y1???x1??y2???x1??y3????x2??y1???x2??y2???x2??y3????x3??y1???x3??y2???x3??y3??? ??
d Y d x 1 = ? y 1 ? x 1 + ? y 1 ? x 1 + ? y 1 ? x 1 d Y d x 2 = ? y 1 ? x 2 + ? y 2 ? x 2 + ? y 3 ? x 2 d Y d x 3 = ? y 1 ? x 3 + ? y 2 ? x 3 + ? y 3 ? x 3 \dfrac{dY}{dx_1}= \dfrac{\partial y_1}{\partial x_1}+\dfrac{\partial y_1}{\partial x_1}+\dfrac{\partial y_1}{\partial x_1}\\[2.5ex] \dfrac{dY}{dx_2}= \dfrac{\partial y_1}{\partial x_2}+\dfrac{\partial y_2}{\partial x_2}+\dfrac{\partial y_3}{\partial x_2}\\[2.5ex] \dfrac{dY}{dx_3}= \dfrac{\partial y_1}{\partial x_3}+\dfrac{\partial y_2}{\partial x_3}+\dfrac{\partial y_3}{\partial x_3}\\[2.5ex] dx1?dY?=?x1??y1??+?x1??y1??+?x1??y1??dx2?dY?=?x2??y1??+?x2??y2??+?x2??y3??dx3?dY?=?x3??y1??+?x3??y2??+?x3??y3??
d Y d X = ( d Y d x 1 d Y d x 2 d Y d x 3 ) \dfrac{dY}{dX}= \begin{pmatrix} \dfrac{dY}{dx_1}&\dfrac{dY}{dx_2}&\dfrac{dY}{dx_3} \\ \end{pmatrix} dXdY?=(dx1?dY??dx2?dY??dx3?dY??)
若是v=（m，n，q），则偏导数计算过程中，偏导数前应该乘上分量对应投影值
比如，若v=(1,2,3),则在表示在偏导计算过程中，对应分量x1，x2，x3应该乘上对应的投影值
以??
d
Y
d
x
1
=
1
?
?
y
1
?
x
1
+
2
?
?
y
1
?
x
1
+
3
?
?
y
1
?
x
1
?为例
以 \ \ \dfrac{dY}{dx_1}= 1*\dfrac{\partial y_1}{\partial x_1}+2*\dfrac{\partial y_1}{\partial x_1}+3*\dfrac{\partial y_1}{\partial x_1} \ 为例
以??dx1?dY?=1??x1??y1??+2??x1??y1??+3??x1??y1???为例
更广泛的：
d
Y
j
d
x
i
=
m
?
?
y
j
?
x
i
+
n
?
?
y
j
?
x
i
+
q
?
?
y
j
?
x
i
\dfrac{dY_j}{dx_i}= m*\dfrac{\partial y_j}{\partial x_i}+n*\dfrac{\partial y_j}{\partial x_i}+q*\dfrac{\partial y_j}{\partial x_i}
dxi?dYj??=m??xi??yj??+n??xi??yj??+q??xi??yj??
所以当v=(1,1,1)时，有无投影没区别
===================================
如果哪里有错误，请在评论区指出，虚心听取