已知空间点（世界坐标系下）p=[XYZ]T\bf{p}=\begin{bmatrix}\it{X} & \it{Y} & \it{Z}\end{bmatrix}^\mathrm{T}p=[XYZ]T，记其重投影位置为x′=[u′v′]T\bf{x'}=\begin{bmatrix}\it{u'}&v'\end{bmatrix}^\mathrm{T}x′=[u′v′]T以及其对应实际像素坐标点x=[uv]T\bf{x}=\begin{bmatrix}\it{u}&v\end{bmatrix}^\mathrm{T}x=[uv]T，相机焦距为fxf_xfx，fyf_yfy，平移量为cxc_xcx，cyc_ycy，径向畸变系数k1k_1k1，k2k_2k2，旋转矩阵RRR对应的旋转向量w=[w0w1w2]T\bf{w}=\begin{bmatrix}\it{w_0}&w_1&w_2\end{bmatrix}^\mathrm{T}w=[w0w1w2]T，平移向量t=[t0t1t2]T\bf{t}=\begin{bmatrix}\it{t_0}&t_1&t_2\end{bmatrix}^\mathrm{T}t=[t0t1t2]T，则重投影误差e=[u−u′v−v′]T\bf{e}=\begin{bmatrix}\it{u-u'}&\it{v-v'}\end{bmatrix}^\mathrm{T}e=[u−u′v−v′]T关于空间点位置，相机内参，相机外参的雅可比矩阵推导如下：
首先，写出重投影误差e\bf{e}e的具体形式。记p\bf{p}p点在相机坐标系下的坐标为[x′y′z′]T\begin{bmatrix}x'&y'&z'\end{bmatrix}^\mathrm{T}[x′y′z′]T，有：
[x′y′z′]=R[XYZ]+[t0t1t2]
\begin{bmatrix}x'\\y'\\z'\end{bmatrix}=R\begin{bmatrix}X\\Y\\Z\end{bmatrix}+\begin{bmatrix}t_0\\t_1\\t_2\end{bmatrix}
⎣⎡x′y′z′⎦⎤=R⎣⎡XYZ⎦⎤+⎣⎡t0t1t2⎦⎤
记归一化平面坐标[xy]T\begin{bmatrix}x&y\end{bmatrix}^\mathrm{T}[xy]T，有：
[xy]=[x′z′y′z′]
\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}\displaystyle\frac{x'}{z'}\\\displaystyle\frac{y'}{z'}\end{bmatrix}
[xy]=⎣⎢⎡z′x′z′y′⎦⎥⎤
记畸变后对应坐标为[xdistortedydistorted]T\begin{bmatrix}x_{distorted}&y_{distorted}\end{bmatrix}^\mathrm{T}[xdistortedydistorted]T，r2=x2+y2r^2=x^2+y^2r2=x2+y2，有：
[xdistortedydistorted]=[x(1+k1r2+k2r4)y(1+k1r2+k2r4)]
\begin{bmatrix}x_{distorted}\\y_{distorted}\end{bmatrix}=\begin{bmatrix}x(1+k_1r^2+k_2r^4)\\y(1+k_1r^2+k_2r^4)\end{bmatrix}
[xdistortedydistorted]=[x(1+k1r2+k2r4)y(1+k1r2+k2r4)]
那么其重投影坐标x′\bf{x'}x′为：
[u′v′]=[fx∗xdistorted+cxfy∗ydistorted+cy]
\begin{bmatrix}u'\\v'\end{bmatrix}=\begin{bmatrix}f_x*x_{distorted}+c_x\\f_y*y_{distorted}+c_y\end{bmatrix}
[u′v′]=[fx∗xdistorted+cxfy∗ydistorted+cy]
故重投影误差e\bf{e}e为：
e=[e0e1]=[u−u′v−v′]=[u−fx∗x∗(1+k1r2+k2r4)−cxv−fy∗y∗(1+k1r2+k2r4)−cy]
\bf{e}=\begin{bmatrix}e_0\\e_1\end{bmatrix}=\begin{bmatrix}u-u'\\v-v'\end{bmatrix}=\begin{bmatrix}u-f_x*x*(1+k_1r^2+k_2r^4)-c_x\\v-f_y*y*(1+k_1r^2+k_2r^4)-c_y\end{bmatrix}
e=[e0e1]=[u−u′v−v′]=[u−fx∗x∗(1+k1r2+k2r4)−cxv−fy∗y∗(1+k1r2+k2r4)−cy]
故雅克比矩阵可写作（RRR用w\bf{w}w表示）：
∂e∂[fx,fy,k1,k2,w0,w1,w2,t0,t1,t2,X,Y,Z]=[∂e0∂fx,∂e0∂fy,∂e0∂k1,∂e0∂k2,∂e0∂w0,∂e0∂w1,∂e0∂w2,∂e0∂t0,∂e0∂t1,∂e0∂t2,∂e0∂X,∂e0∂Y,∂e0∂Z∂e1∂fx,∂e1∂fy,∂e1∂k1,∂e1∂k2,∂e1∂w0,∂e1∂w1,∂e1∂w2,∂e1∂t0,∂e1∂t1,∂e1∂t2,∂e1∂X,∂e1∂Y,∂e1∂Z]
\displaystyle\frac{\partial\bf{e}}{\partial\begin{bmatrix}f_x,f_y,k_1,k_2,w_0,w_1,w_2,t_0,t_1,t_2,X,Y,Z\end{bmatrix}}=\\\begin{bmatrix}\displaystyle\frac{\partial e_0}{\partial f_x},\displaystyle\frac{\partial e_0}{\partial f_y},\displaystyle\frac{\partial e_0}{\partial k_1},\displaystyle\frac{\partial e_0}{\partial k_2},\displaystyle\frac{\partial e_0}{\partial w_0},\displaystyle\frac{\partial e_0}{\partial w_1},\displaystyle\frac{\partial e_0}{\partial w_2},\displaystyle\frac{\partial e_0}{\partial t_0},\displaystyle\frac{\partial e_0}{\partial t_1},\displaystyle\frac{\partial e_0}{\partial t_2},\displaystyle\frac{\partial e_0}{\partial X},\displaystyle\frac{\partial e_0}{\partial Y},\displaystyle\frac{\partial e_0}{\partial Z}\\\displaystyle\frac{\partial e_1}{\partial f_x},\displaystyle\frac{\partial e_1}{\partial f_y},\displaystyle\frac{\partial e_1}{\partial k_1},\displaystyle\frac{\partial e_1}{\partial k_2},\displaystyle\frac{\partial e_1}{\partial w_0},\displaystyle\frac{\partial e_1}{\partial w_1},\displaystyle\frac{\partial e_1}{\partial w_2},\displaystyle\frac{\partial e_1}{\partial t_0},\displaystyle\frac{\partial e_1}{\partial t_1},\displaystyle\frac{\partial e_1}{\partial t_2},\displaystyle\frac{\partial e_1}{\partial X},\displaystyle\frac{\partial e_1}{\partial Y},\displaystyle\frac{\partial e_1}{\partial Z}\end{bmatrix}
∂[fx,fy,k1,k2,w0,w1,w2,t0,t1,t2,X,Y,Z]∂e=⎣⎢⎢⎡∂fx∂e0,∂fy∂e0,∂k1∂e0,∂k2∂e0,∂w0∂e0,∂w1∂e0,∂w2∂e0,∂t0∂e0,∂t1∂e0,∂t2∂e0,∂X∂e0,∂Y∂e0,∂Z∂e0∂fx∂e1,∂fy∂e1,∂k1∂e1,∂k2∂e1,∂w0∂e1,∂w1∂e1,∂w2∂e1,∂t0∂e1,∂t1∂e1,∂t2∂e1,∂X∂e1,∂Y∂e1,∂Z∂e1⎦⎥⎥⎤
下面求每项偏导数的具体形式。
1.关于相机内参的偏导数
∂e0∂fx=−x(1+k1r2+k2r4) \displaystyle\frac{\partial e_0}{\partial f_x}=-x(1+k_1r^2+k_2r^4) ∂fx∂e0=−x(1+k1r2+k2r4)
∂e1∂fx=0 \displaystyle\frac{\partial e_1}{\partial f_x}=0 ∂fx∂e1=0
∂e0∂fy=0 \displaystyle\frac{\partial e_0}{\partial f_y}=0 ∂fy∂e0=0
∂e1∂fy=−y(1+k1r2+k2r4) \displaystyle\frac{\partial e_1}{\partial f_y}=-y(1+k_1r^2+k_2r^4) ∂fy∂e1=−y(1+k1r2+k2r4)
∂e0∂k1=−fxxr2 \displaystyle\frac{\partial e_0}{\partial k_1}=-f_xxr^2 ∂k1∂e0=−fxxr2
∂e1∂k1=−fyyr2 \displaystyle\frac{\partial e_1}{\partial k_1}=-f_yyr^2 ∂k1∂e1=−fyyr2
∂e0∂k2=−fxxr4 \displaystyle\frac{\partial e_0}{\partial k_2}=-f_xxr^4 ∂k2∂e0=−fxxr4
∂e1∂k2=−fyyr4 \displaystyle\frac{\partial e_1}{\partial k_2}=-f_yyr^4 ∂k2∂e1=−fyyr4
2.关于旋转的偏导数
考虑畸变时，e\bf{e}e关于旋转的偏导数形式较为复杂，故在推导之前，预先求出部分中间变量（后面的结果均会用到这几个量）：
∂x∂x′=1z′,∂x∂y′=0,∂x∂z′=−x′z′2∂y∂x′=0,∂y∂y′=1z′,∂y∂z′=−y′z′2∂e0∂x=−fx(1+k1r2+k2r4)−fxx(2k1x+4k2xr2)∂e0∂y=−fxx(2k1y+4k2yr2)∂e1∂x=−fyy(2k1x+4k2xr2)∂e1∂y=−fy(1+k1r2+k2r4)−fyy(2k1y+4k2yr2)
\displaystyle\frac{\partial x}{\partial x'}=\frac{1}{z'},\displaystyle\frac{\partial x}{\partial y'}=0,\displaystyle\frac{\partial x}{\partial z'}=-\frac{x'}{z'^2}\\
\displaystyle\frac{\partial y}{\partial x'}=0,\displaystyle\frac{\partial y}{\partial y'}=\frac{1}{z'},\displaystyle\frac{\partial y}{\partial z'}=-\frac{y'}{z'^2}\\
\displaystyle\frac{\partial e_0}{\partial x}=-f_x(1+k_1r^2+k_2r^4)-f_xx(2k_1x+4k_2xr^2)\\
\displaystyle\frac{\partial e_0}{\partial y}=-f_xx(2k_1y+4k_2yr^2)\\
\displaystyle\frac{\partial e_1}{\partial x}=-f_yy(2k_1x+4k_2xr^2)\\
\displaystyle\frac{\partial e_1}{\partial y}=-f_y(1+k_1r^2+k_2r^4)-f_yy(2k_1y+4k_2yr^2)
∂x′∂x=z′1,∂y′∂x=0,∂z′∂x=−z′2x′∂x′∂y=0,∂y′∂y=z′1,∂z′∂y=−z′2y′∂x∂e0=−fx(1+k1r2+k2r4)−fxx(2k1x+4k2xr2)∂y∂e0=−fxx(2k1y+4k2yr2)∂x∂e1=−fyy(2k1x+4k2xr2)∂y∂e1=−fy(1+k1r2+k2r4)−fyy(2k1y+4k2yr2)
旋转向量w=θn{\bf{w}}=\theta nw=θn,其中θ\thetaθ代表旋转角度，nnn代表旋转轴，由罗德里格斯公式有：
R=cosθI+(1−cosθ)nnT+sinθn∧
R=\cos\theta {\boldsymbol{I}}+(1-\cos\theta)nn^\mathrm{T}+\sin\theta n^\wedge
R=cosθI+(1−cosθ)nnT+sinθn∧
当旋转一个很小的角度时，θ→0\theta\to0θ→0，有：
ΔR=I+θn∧=I+w∧=[100010001]+[0−w2w1w20−w0−w1w00]
\Delta R={\boldsymbol{I}}+\theta n^\wedge={\boldsymbol{I}}+{\bf{w}}^\wedge=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}+\begin{bmatrix}0&-w_2&w_1\\w_2&0&-w_0\\-w_1&w_0&0\end{bmatrix}
ΔR=I+θn∧=I+w∧=⎣⎡100010001⎦⎤+⎣⎡0w2−w1−w20w0w1−w00⎦⎤
记世界坐标系下空间点p\bf{p}p在相机坐标系下为p′\bf{p'}p′，则p′=Rp+t{\bf{p'}}=R{\bf{p}}+tp′=Rp+t，则基于左扰动模型有：
∂p′∂w=limw→0ΔR∗Rp+t−Rp−tw=limw→0(I+w∧)Rp−Rpw=limw→0w∧Rpw=limw→0−(Rp)∧ww=−(Rp)∧
\begin{aligned}\displaystyle\frac{\partial\bf{p'}}{\partial \bf{w}}&=\lim\limits_{\bf{w}\to0}\displaystyle\frac{{\Delta R*R{\bf{p}}+t}-R{\bf{p}}-t}{\bf{w}}
\\&=\lim\limits_{\bf{w}\to0}\displaystyle\frac{{(\boldsymbol{I}+{\bf{w}}^\wedge)R{\bf{p}}}-R{\bf{p}}}{\bf{w}}
\\&=\lim\limits_{\bf{w}\to0}\displaystyle\frac{{\bf{w}}^\wedge R{\bf{p}}}{\bf{w}}
=\lim\limits_{\bf{w}\to0}\displaystyle\frac{-(R{\bf{p}})^\wedge\bf{w}}{\bf{w}}=-(R\bf{p})^\wedge
\end{aligned}
∂w∂p′=w→0limwΔR∗Rp+t−Rp−t=w→0limw(I+w∧)Rp−Rp=w→0limww∧Rp=w→0limw−(Rp)∧w=−(Rp)∧
p′=[x′y′z′]T\bf{p'}=\begin{bmatrix}x'&y'&z'\end{bmatrix}^\mathrm{T}p′=[x′y′z′]T，w=[w0w1w2]T\bf{w}=\begin{bmatrix}\it{w_0}&w_1&w_2\end{bmatrix}^\mathrm{T}w=[w0w1w2]T，记RRR的行向量为r0,r1,r2\boldsymbol{r_0},\boldsymbol{r_1},\boldsymbol{r_2}r0,r1,r2则上式写作：
∂[x′y′z′]T∂[w0w1w2]T=[∂x′∂w0∂x′∂w1∂x′∂w2∂y′∂w0∂y′∂w1∂y′∂w2∂z′∂w0∂z′∂w1∂z′∂w2]=[0r2p−r1p−r2p0r0pr1p−r0p0]
\displaystyle\frac{\partial \begin{bmatrix}x'&y'&z'\end{bmatrix}^\mathrm{T}}{\partial \begin{bmatrix}\it{w_0}&w_1&w_2\end{bmatrix}^\mathrm{T}}
=\begin{bmatrix}\displaystyle\frac{\partial x'}{\partial w_0}&\displaystyle\frac{\partial x'}{\partial w_1}&\displaystyle\frac{\partial x'}{\partial w_2}\\\displaystyle\frac{\partial y'}{\partial w_0}&\displaystyle\frac{\partial y'}{\partial w_1}&\displaystyle\frac{\partial y'}{\partial w_2}\\\displaystyle\frac{\partial z'}{\partial w_0}&\displaystyle\frac{\partial z'}{\partial w_1}&\displaystyle\frac{\partial z'}{\partial w_2}\end{bmatrix}
=\begin{bmatrix}0&\boldsymbol{r_2}{\bf{p}}&-\boldsymbol{r_1}{\bf{p}}\\-\boldsymbol{r_2}{\bf{p}}&0&\boldsymbol{r_0}{\bf{p}}\\\boldsymbol{r_1}{\bf{p}}&-\boldsymbol{r_0}{\bf{p}}&0\end{bmatrix}
∂[w0w1w2]T∂[x′y′z′]T=⎣⎢⎢⎢⎢⎢⎡∂w0∂x′∂w0∂y′∂w0∂z′∂w1∂x′∂w1∂y′∂w1∂z′∂w2∂x′∂w2∂y′∂w2∂z′⎦⎥⎥⎥⎥⎥⎤=⎣⎡0−r2pr1pr2p0−r0p−r1pr0p0⎦⎤
归一化平面坐标[xy]T\begin{bmatrix}x&y\end{bmatrix}^\mathrm{T}[xy]T是关于p′\bf{p'}p′的函数，由链式求导有：
∂e0∂w0=∂e0∂x(∂x∂x′∂x′∂w0+∂x∂y′∂y′∂w0+∂x∂z′∂z′∂w0)+∂e0∂y(∂y∂x′∂x′∂w0+∂y∂y′∂y′∂w0+∂y∂z′∂z′∂w0)=∂e0∂x∂x∂z′∂z′∂w0+∂e0∂y(∂y∂y′∂y′∂w0+∂y∂z′∂z′∂w0)
\begin{aligned}
\displaystyle\frac{\partial e_0}{\partial w_0}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_0}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial w_0}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_0})+
\displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial w_0}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_0}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_0})
\\&=\displaystyle\frac{\partial e_0}{\partial x}\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_0}+
\displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_0}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_0})
\end{aligned}
∂w0∂e0=∂x∂e0(∂x′∂x∂w0∂x′+∂y′∂x∂w0∂y′+∂z′∂x∂w0∂z′)+∂y∂e0(∂x′∂y∂w0∂x′+∂y′∂y∂w0∂y′+∂z′∂y∂w0∂z′)=∂x∂e0∂z′∂x∂w0∂z′+∂y∂e0(∂y′∂y∂w0∂y′+∂z′∂y∂w0∂z′)
∂e0∂w1=∂e0∂x(∂x∂x′∂x′∂w1+∂x∂y′∂y′∂w1+∂x∂z′∂z′∂w1)+∂e0∂y(∂y∂x′∂x′∂w1+∂y∂y′∂y′∂w1+∂y∂z′∂z′∂w1)=∂e0∂x(∂x∂x′∂x′∂w1+∂x∂z′∂z′∂w1)+∂e0∂y∂y∂z′∂z′∂w1 \begin{aligned} \displaystyle\frac{\partial e_0}{\partial w_1}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_1}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial w_1}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_1})+ \displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial w_1}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_1}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_1}) \\&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_1}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_1})+ \displaystyle\frac{\partial e_0}{\partial y}\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_1} \end{aligned} ∂w1∂e0=∂x∂e0(∂x′∂x∂w1∂x′+∂y′∂x∂w1∂y′+∂z′∂x∂w1∂z′)+∂y∂e0(∂x′∂y∂w1∂x′+∂y′∂y∂w1∂y′+∂z′∂y∂w1∂z′)=∂x∂e0(∂x′∂x∂w1∂x′+∂z′∂x∂w1∂z′)+∂y∂e0∂z′∂y∂w1∂z′
∂e0∂w2=∂e0∂x(∂x∂x′∂x′∂w2+∂x∂y′∂y′∂w2+∂x∂z′∂z′∂w2)+∂e0∂y(∂y∂x′∂x′∂w2+∂y∂y′∂y′∂w2+∂y∂z′∂z′∂w2)=∂e0∂x∂x∂x′∂x′∂w2+∂e0∂y∂y∂y′∂y′∂w2 \begin{aligned} \displaystyle\frac{\partial e_0}{\partial w_2}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_2}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial w_2}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_2})+ \displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial w_2}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_2}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_2}) \\&=\displaystyle\frac{\partial e_0}{\partial x}\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_2}+ \displaystyle\frac{\partial e_0}{\partial y}\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_2} \end{aligned} ∂w2∂e0=∂x∂e0(∂x′∂x∂w2∂x′+∂y′∂x∂w2∂y′+∂z′∂x∂w2∂z′)+∂y∂e0(∂x′∂y∂w2∂x′+∂y′∂y∂w2∂y′+∂z′∂y∂w2∂z′)=∂x∂e0∂x′∂x∂w2∂x′+∂y∂e0∂y′∂y∂w2∂y′
∂e1∂w0=∂e1∂x(∂x∂x′∂x′∂w0+∂x∂y′∂y′∂w0+∂x∂z′∂z′∂w0)+∂e1∂y(∂y∂x′∂x′∂w0+∂y∂y′∂y′∂w0+∂y∂z′∂z′∂w0)=∂e1∂x∂x∂z′∂z′∂w0+∂e1∂y(∂y∂y′∂y′∂w0+∂y∂z′∂z′∂w0) \begin{aligned} \displaystyle\frac{\partial e_1}{\partial w_0}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_0}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial w_0}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_0})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial w_0}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_0}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_0}) \\&=\displaystyle\frac{\partial e_1}{\partial x}\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_0}+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_0}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_0}) \end{aligned} ∂w0∂e1=∂x∂e1(∂x′∂x∂w0∂x′+∂y′∂x∂w0∂y′+∂z′∂x∂w0∂z′)+∂y∂e1(∂x′∂y∂w0∂x′+∂y′∂y∂w0∂y′+∂z′∂y∂w0∂z′)=∂x∂e1∂z′∂x∂w0∂z′+∂y∂e1(∂y′∂y∂w0∂y′+∂z′∂y∂w0∂z′)
∂e1∂w1=∂e1∂x(∂x∂x′∂x′∂w1+∂x∂y′∂y′∂w1+∂x∂z′∂z′∂w1)+∂e1∂y(∂y∂x′∂x′∂w1+∂y∂y′∂y′∂w1+∂y∂z′∂z′∂w1)=∂e1∂x(∂x∂x′∂x′∂w1+∂x∂z′∂z′∂w1)+∂e1∂y∂y∂z′∂z′∂w1 \begin{aligned} \displaystyle\frac{\partial e_1}{\partial w_1}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_1}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial w_1}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_1})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial w_1}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_1}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_1}) \\&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_1}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_1})+ \displaystyle\frac{\partial e_1}{\partial y}\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_1} \end{aligned} ∂w1∂e1=∂x∂e1(∂x′∂x∂w1∂x′+∂y′∂x∂w1∂y′+∂z′∂x∂w1∂z′)+∂y∂e1(∂x′∂y∂w1∂x′+∂y′∂y∂w1∂y′+∂z′∂y∂w1∂z′)=∂x∂e1(∂x′∂x∂w1∂x′+∂z′∂x∂w1∂z′)+∂y∂e1∂z′∂y∂w1∂z′
∂e1∂w2=∂e1∂x(∂x∂x′∂x′∂w2+∂x∂y′∂y′∂w2+∂x∂z′∂z′∂w2)+∂e1∂y(∂y∂x′∂x′∂w2+∂y∂y′∂y′∂w2+∂y∂z′∂z′∂w2)=∂e1∂x∂x∂x′∂x′∂w2+∂e1∂y∂y∂y′∂y′∂w2 \begin{aligned} \displaystyle\frac{\partial e_1}{\partial w_2}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_2}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial w_2}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial w_2})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial w_2}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_2}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial w_2}) \\&=\displaystyle\frac{\partial e_1}{\partial x}\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial w_2}+ \displaystyle\frac{\partial e_1}{\partial y}\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial w_2} \end{aligned} ∂w2∂e1=∂x∂e1(∂x′∂x∂w2∂x′+∂y′∂x∂w2∂y′+∂z′∂x∂w2∂z′)+∂y∂e1(∂x′∂y∂w2∂x′+∂y′∂y∂w2∂y′+∂z′∂y∂w2∂z′)=∂x∂e1∂x′∂x∂w2∂x′+∂y∂e1∂y′∂y∂w2∂y′
3.关于平移的偏导数
由p′=Rp+t{\bf{p'}}=R{\bf{p}}+tp′=Rp+t，即：
[x′y′z′]=[r0p+t0r1p+t1r2p+t2]
\begin{bmatrix} x'\\y'\\z'\end{bmatrix}=\begin{bmatrix}\boldsymbol{r_0}{\bf{p}}+t_0\\\boldsymbol{r_1}{\bf{p}}+t_1\\\boldsymbol{r_2}{\bf{p}}+t_2\end{bmatrix}
⎣⎡x′y′z′⎦⎤=⎣⎡r0p+t0r1p+t1r2p+t2⎦⎤
故有：
∂x′∂t0=1,∂y′∂t1=1,∂z′∂t2=1
\displaystyle\frac{\partial x'}{\partial t_0}=1,\displaystyle\frac{\partial y'}{\partial t_1}=1,\displaystyle\frac{\partial z'}{\partial t_2}=1
∂t0∂x′=1,∂t1∂y′=1,∂t2∂z′=1
同样地，由链式求导有：
∂e0∂t0=∂e0∂x(∂x∂x′∂x′∂t0+∂x∂y′∂y′∂t0+∂x∂z′∂z′∂t0)+∂e0∂y(∂y∂x′∂x′∂t0+∂y∂y′∂y′∂t0+∂y∂z′∂z′∂t0)=∂e0∂x∂x∂x′
\begin{aligned}
\displaystyle\frac{\partial e_0}{\partial t_0}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial t_0}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial t_0}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial t_0})+
\displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial t_0}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial t_0}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial t_0})
\\&=\displaystyle\frac{\partial e_0}{\partial x}\displaystyle\frac{\partial x}{\partial x'}
\end{aligned}
∂t0∂e0=∂x∂e0(∂x′∂x∂t0∂x′+∂y′∂x∂t0∂y′+∂z′∂x∂t0∂z′)+∂y∂e0(∂x′∂y∂t0∂x′+∂y′∂y∂t0∂y′+∂z′∂y∂t0∂z′)=∂x∂e0∂x′∂x
∂e0∂t1=∂e0∂x(∂x∂x′∂x′∂t1+∂x∂y′∂y′∂t1+∂x∂z′∂z′∂t1)+∂e0∂y(∂y∂x′∂x′∂t1+∂y∂y′∂y′∂t1+∂y∂z′∂z′∂t1)=∂e0∂y∂y∂y′ \begin{aligned} \displaystyle\frac{\partial e_0}{\partial t_1}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial t_1}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial t_1}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial t_1})+ \displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial t_1}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial t_1}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial t_1}) \\&=\displaystyle\frac{\partial e_0}{\partial y}\displaystyle\frac{\partial y}{\partial y'} \end{aligned} ∂t1∂e0=∂x∂e0(∂x′∂x∂t1∂x′+∂y′∂x∂t1∂y′+∂z′∂x∂t1∂z′)+∂y∂e0(∂x′∂y∂t1∂x′+∂y′∂y∂t1∂y′+∂z′∂y∂t1∂z′)=∂y∂e0∂y′∂y
∂e0∂t2=∂e0∂x(∂x∂x′∂x′∂t2+∂x∂y′∂y′∂t2+∂x∂z′∂z′∂t2)+∂e0∂y(∂y∂x′∂x′∂t2+∂y∂y′∂y′∂t2+∂y∂z′∂z′∂t2)=∂e0∂x∂x∂z′+∂e0∂y∂y∂z′ \begin{aligned} \displaystyle\frac{\partial e_0}{\partial t_2}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial t_2}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial t_2}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial t_2})+ \displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial t_2}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial t_2}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial t_2}) \\&=\displaystyle\frac{\partial e_0}{\partial x}\displaystyle\frac{\partial x}{\partial z'}+\displaystyle\frac{\partial e_0}{\partial y}\displaystyle\frac{\partial y}{\partial z'} \end{aligned} ∂t2∂e0=∂x∂e0(∂x′∂x∂t2∂x′+∂y′∂x∂t2∂y′+∂z′∂x∂t2∂z′)+∂y∂e0(∂x′∂y∂t2∂x′+∂y′∂y∂t2∂y′+∂z′∂y∂t2∂z′)=∂x∂e0∂z′∂x+∂y∂e0∂z′∂y
∂e1∂t0=∂e1∂x(∂x∂x′∂x′∂t0+∂x∂y′∂y′∂t0+∂x∂z′∂z′∂t0)+∂e1∂y(∂y∂x′∂x′∂t0+∂y∂y′∂y′∂t0+∂y∂z′∂z′∂t0)=∂e1∂x∂x∂x′ \begin{aligned} \displaystyle\frac{\partial e_1}{\partial t_0}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial t_0}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial t_0}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial t_0})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial t_0}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial t_0}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial t_0}) \\&=\displaystyle\frac{\partial e_1}{\partial x}\displaystyle\frac{\partial x}{\partial x'} \end{aligned} ∂t0∂e1=∂x∂e1(∂x′∂x∂t0∂x′+∂y′∂x∂t0∂y′+∂z′∂x∂t0∂z′)+∂y∂e1(∂x′∂y∂t0∂x′+∂y′∂y∂t0∂y′+∂z′∂y∂t0∂z′)=∂x∂e1∂x′∂x
∂e1∂t1=∂e1∂x(∂x∂x′∂x′∂t1+∂x∂y′∂y′∂t1+∂x∂z′∂z′∂t1)+∂e1∂y(∂y∂x′∂x′∂t1+∂y∂y′∂y′∂t1+∂y∂z′∂z′∂t1)=∂e0∂y∂y∂y′ \begin{aligned} \displaystyle\frac{\partial e_1}{\partial t_1}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial t_1}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial t_1}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial t_1})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial t_1}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial t_1}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial t_1}) \\&=\displaystyle\frac{\partial e_0}{\partial y}\displaystyle\frac{\partial y}{\partial y'} \end{aligned} ∂t1∂e1=∂x∂e1(∂x′∂x∂t1∂x′+∂y′∂x∂t1∂y′+∂z′∂x∂t1∂z′)+∂y∂e1(∂x′∂y∂t1∂x′+∂y′∂y∂t1∂y′+∂z′∂y∂t1∂z′)=∂y∂e0∂y′∂y
∂e1∂t2=∂e1∂x(∂x∂x′∂x′∂t2+∂x∂y′∂y′∂t2+∂x∂z′∂z′∂t2)+∂e1∂y(∂y∂x′∂x′∂t2+∂y∂y′∂y′∂t2+∂y∂z′∂z′∂t2)=∂e1∂x∂x∂z′+∂e1∂y∂y∂z′ \begin{aligned} \displaystyle\frac{\partial e_1}{\partial t_2}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial t_2}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial t_2}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial t_2})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial t_2}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial t_2}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial t_2}) \\&=\displaystyle\frac{\partial e_1}{\partial x}\displaystyle\frac{\partial x}{\partial z'}+\displaystyle\frac{\partial e_1}{\partial y}\displaystyle\frac{\partial y}{\partial z'} \end{aligned} ∂t2∂e1=∂x∂e1(∂x′∂x∂t2∂x′+∂y′∂x∂t2∂y′+∂z′∂x∂t2∂z′)+∂y∂e1(∂x′∂y∂t2∂x′+∂y′∂y∂t2∂y′+∂z′∂y∂t2∂z′)=∂x∂e1∂z′∂x+∂y∂e1∂z′∂y
4.关于空间点的偏导数
由p′=Rp+t{\bf{p'}}=R{\bf{p}}+tp′=Rp+t，即：
[x′y′z′]=[R00R01R02R10R11R12R20R21R22][XYZ]+[t0t1t2]
\begin{bmatrix}x'\\y'\\z'\end{bmatrix}=\begin{bmatrix}R_{00}&R_{01}&R_{02}\\R_{10}&R_{11}&R_{12}\\R_{20}&R_{21}&R_{22}\end{bmatrix}\begin{bmatrix}X\\Y\\Z\end{bmatrix}+\begin{bmatrix}t_0\\t_1\\t_2\end{bmatrix}
⎣⎡x′y′z′⎦⎤=⎣⎡R00R10R20R01R11R21R02R12R22⎦⎤⎣⎡XYZ⎦⎤+⎣⎡t0t1t2⎦⎤
故有：
∂x′∂X=R00,∂x′∂Y=R01,∂x′∂Z=R02∂y′∂X=R10,∂y′∂Y=R11,∂y′∂Z=R12∂z′∂X=R20,∂z′∂Y=R21,∂z′∂Z=R22
\displaystyle\frac{\partial x'}{\partial X}=R_{00},\displaystyle\frac{\partial x'}{\partial Y}=R_{01},\displaystyle\frac{\partial x'}{\partial Z}=R_{02}\\
\displaystyle\frac{\partial y'}{\partial X}=R_{10},\displaystyle\frac{\partial y'}{\partial Y}=R_{11},\displaystyle\frac{\partial y'}{\partial Z}=R_{12}\\
\displaystyle\frac{\partial z'}{\partial X}=R_{20},\displaystyle\frac{\partial z'}{\partial Y}=R_{21},\displaystyle\frac{\partial z'}{\partial Z}=R_{22}
∂X∂x′=R00,∂Y∂x′=R01,∂Z∂x′=R02∂X∂y′=R10,∂Y∂y′=R11,∂Z∂y′=R12∂X∂z′=R20,∂Y∂z′=R21,∂Z∂z′=R22
同样地，由链式求导有：
∂e0∂X=∂e0∂x(∂x∂x′∂x′∂X+∂x∂y′∂y′∂X+∂x∂z′∂z′∂X)+∂e0∂y(∂y∂x′∂x′∂X+∂y∂y′∂y′∂X+∂y∂z′∂z′∂X)=∂e0∂x(∂x∂x′∂x′∂X+∂x∂z′∂z′∂X)+∂e0∂y(∂y∂y′∂y′∂X+∂y∂z′∂z′∂X)
\begin{aligned}
\displaystyle\frac{\partial e_0}{\partial X}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial X}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial X}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial X})+
\displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial X}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial X}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial X})
\\&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial X}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial X})+
\displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial X}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial X})
\end{aligned}
∂X∂e0=∂x∂e0(∂x′∂x∂X∂x′+∂y′∂x∂X∂y′+∂z′∂x∂X∂z′)+∂y∂e0(∂x′∂y∂X∂x′+∂y′∂y∂X∂y′+∂z′∂y∂X∂z′)=∂x∂e0(∂x′∂x∂X∂x′+∂z′∂x∂X∂z′)+∂y∂e0(∂y′∂y∂X∂y′+∂z′∂y∂X∂z′)
∂e0∂Y=∂e0∂x(∂x∂x′∂x′∂Y+∂x∂y′∂y′∂Y+∂x∂z′∂z′∂Y)+∂e0∂y(∂y∂x′∂x′∂Y+∂y∂y′∂y′∂Y+∂y∂z′∂z′∂Y)=∂e0∂x(∂x∂x′∂x′∂Y+∂x∂z′∂z′∂Y)+∂e0∂y(∂y∂y′∂y′∂Y+∂y∂z′∂z′∂Y) \begin{aligned} \displaystyle\frac{\partial e_0}{\partial Y}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial Y}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial Y}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial Y})+ \displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial Y}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial Y}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial Y}) \\&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial Y}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial Y})+ \displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial Y}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial Y}) \end{aligned} ∂Y∂e0=∂x∂e0(∂x′∂x∂Y∂x′+∂y′∂x∂Y∂y′+∂z′∂x∂Y∂z′)+∂y∂e0(∂x′∂y∂Y∂x′+∂y′∂y∂Y∂y′+∂z′∂y∂Y∂z′)=∂x∂e0(∂x′∂x∂Y∂x′+∂z′∂x∂Y∂z′)+∂y∂e0(∂y′∂y∂Y∂y′+∂z′∂y∂Y∂z′)
∂e0∂Z=∂e0∂x(∂x∂x′∂x′∂Z+∂x∂y′∂y′∂Z+∂x∂z′∂z′∂Z)+∂e0∂y(∂y∂x′∂x′∂Z+∂y∂y′∂y′∂Z+∂y∂z′∂z′∂Z)=∂e0∂x(∂x∂x′∂x′∂Z+∂x∂z′∂z′∂Z)+∂e0∂y(∂y∂y′∂y′∂Z+∂y∂z′∂z′∂Z) \begin{aligned} \displaystyle\frac{\partial e_0}{\partial Z}&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial Z}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial Z}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial Z})+ \displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial Z}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial Z}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial Z}) \\&=\displaystyle\frac{\partial e_0}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial Z}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial Z})+ \displaystyle\frac{\partial e_0}{\partial y}(\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial Z}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial Z}) \end{aligned} ∂Z∂e0=∂x∂e0(∂x′∂x∂Z∂x′+∂y′∂x∂Z∂y′+∂z′∂x∂Z∂z′)+∂y∂e0(∂x′∂y∂Z∂x′+∂y′∂y∂Z∂y′+∂z′∂y∂Z∂z′)=∂x∂e0(∂x′∂x∂Z∂x′+∂z′∂x∂Z∂z′)+∂y∂e0(∂y′∂y∂Z∂y′+∂z′∂y∂Z∂z′)
∂e1∂X=∂e1∂x(∂x∂x′∂x′∂X+∂x∂y′∂y′∂X+∂x∂z′∂z′∂X)+∂e1∂y(∂y∂x′∂x′∂X+∂y∂y′∂y′∂X+∂y∂z′∂z′∂X)=∂e1∂x(∂x∂x′∂x′∂X+∂x∂z′∂z′∂X)+∂e1∂y(∂y∂y′∂y′∂X+∂y∂z′∂z′∂X) \begin{aligned} \displaystyle\frac{\partial e_1}{\partial X}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial X}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial X}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial X})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial X}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial X}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial X}) \\&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial X}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial X})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial X}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial X}) \end{aligned} ∂X∂e1=∂x∂e1(∂x′∂x∂X∂x′+∂y′∂x∂X∂y′+∂z′∂x∂X∂z′)+∂y∂e1(∂x′∂y∂X∂x′+∂y′∂y∂X∂y′+∂z′∂y∂X∂z′)=∂x∂e1(∂x′∂x∂X∂x′+∂z′∂x∂X∂z′)+∂y∂e1(∂y′∂y∂X∂y′+∂z′∂y∂X∂z′)
∂e1∂Y=∂e1∂x(∂x∂x′∂x′∂Y+∂x∂y′∂y′∂Y+∂x∂z′∂z′∂Y)+∂e1∂y(∂y∂x′∂x′∂Y+∂y∂y′∂y′∂Y+∂y∂z′∂z′∂Y)=∂e1∂x(∂x∂x′∂x′∂Y+∂x∂z′∂z′∂Y)+∂e1∂y(∂y∂y′∂y′∂Y+∂y∂z′∂z′∂Y) \begin{aligned} \displaystyle\frac{\partial e_1}{\partial Y}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial Y}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial Y}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial Y})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial Y}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial Y}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial Y}) \\&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial Y}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial Y})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial Y}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial Y}) \end{aligned} ∂Y∂e1=∂x∂e1(∂x′∂x∂Y∂x′+∂y′∂x∂Y∂y′+∂z′∂x∂Y∂z′)+∂y∂e1(∂x′∂y∂Y∂x′+∂y′∂y∂Y∂y′+∂z′∂y∂Y∂z′)=∂x∂e1(∂x′∂x∂Y∂x′+∂z′∂x∂Y∂z′)+∂y∂e1(∂y′∂y∂Y∂y′+∂z′∂y∂Y∂z′)
∂e1∂Z=∂e1∂x(∂x∂x′∂x′∂Z+∂x∂y′∂y′∂Z+∂x∂z′∂z′∂Z)+∂e1∂y(∂y∂x′∂x′∂Z+∂y∂y′∂y′∂Z+∂y∂z′∂z′∂Z)=∂e1∂x(∂x∂x′∂x′∂Z+∂x∂z′∂z′∂Z)+∂e1∂y(∂y∂y′∂y′∂Z+∂y∂z′∂z′∂Z) \begin{aligned} \displaystyle\frac{\partial e_1}{\partial Z}&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial Z}+\displaystyle\frac{\partial x}{\partial y'}\frac{\partial y'}{\partial Z}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial Z})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial x'}\frac{\partial x'}{\partial Z}+\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial Z}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial Z}) \\&=\displaystyle\frac{\partial e_1}{\partial x}(\displaystyle\frac{\partial x}{\partial x'}\frac{\partial x'}{\partial Z}+\displaystyle\frac{\partial x}{\partial z'}\frac{\partial z'}{\partial Z})+ \displaystyle\frac{\partial e_1}{\partial y}(\displaystyle\frac{\partial y}{\partial y'}\frac{\partial y'}{\partial Z}+\displaystyle\frac{\partial y}{\partial z'}\frac{\partial z'}{\partial Z}) \end{aligned} ∂Z∂e1=∂x∂e1(∂x′∂x∂Z∂x′+∂y′∂x∂Z∂y′+∂z′∂x∂Z∂z′)+∂y∂e1(∂x′∂y∂Z∂x′+∂y′∂y∂Z∂y′+∂z′∂y∂Z∂z′)=∂x∂e1(∂x′∂x∂Z∂x′+∂z′∂x∂Z∂z′)+∂y∂e1(∂y′∂y∂Z∂y′+∂z′∂y∂Z∂z′)
至此，雅可比矩阵中各项已给出。
参考：https://zhuanlan.zhihu.com/p/44651602
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