博客主要介绍了矢量代换公式，包括矢量的加减、点乘、叉乘等运算。还阐述了散度、旋度相关公式，以及高斯定理、斯托克斯定理。同时给出了梯度、散度、旋度在不同坐标系下的表达式，最后介绍了格林第一和第二恒等式。
矢量代换公式
A⃗=A(e⃗xcosα+e⃗ycosβ+e⃗zcosγ)\vec{A}=A(\vec{e}_xcos\alpha+\vec{e}_ycos\beta+\vec{e}_zcos\gamma)A=A(excosα+eycosβ+ezcosγ)
A⃗±B⃗=e⃗x(Ax±Bx)+e⃗y(Ay±By)+e⃗z(Az±Bz)\vec{A}\pm\vec{B}=\vec{e}_x(A_x\pm B_x)+\vec{e}_y(A_y\pm B_y)+\vec{e}_z(A_z\pm B_z)A±B=ex(Ax±Bx)+ey(Ay±By)+ez(Az±Bz)
A⃗⋅B⃗=标量=ABcosθ=AxBx+AyBy+AzBz\vec{A}\cdot\vec{B}=标量=ABcos\theta=A_x B_x+A_y B_y+A_z B_zA⋅B=标量=ABcosθ=AxBx+AyBy+AzBz
A⃗×B⃗=矢量=e⃗nABsinθ\vec{A}\times \vec{B}=矢量=\vec{e}_nABsin\thetaA×B=矢量=enABsinθ
A⃗×B⃗=−B⃗×A⃗\vec{A}\times \vec{B}=-\vec{B}\times \vec{A}A×B=−B×A
A⃗×B⃗=e⃗x(AyBz−AzBy)+e⃗y(AzBx−AxBz)+e⃗z(AxBy−AyBx)\vec{A}\times \vec{B}=\vec{e}_x(A_y B_z - A_z B_y)+\vec{e}_y(A_z B_x - A_x B_z)+\vec{e}_z(A_x B_y - A_y B_x)A×B=ex(AyBz−AzBy)+ey(AzBx−AxBz)+ez(AxBy−AyBx)
A⃗×B⃗=∣e⃗xe⃗ye⃗zAxAyAzBxByBz∣\vec{A}\times \vec{B}=\left| \begin{matrix} {\vec{e}_x} & {\vec{e}_y} & {\vec{e}_z}\\ {A_x} & {A_y} & {A_z}\\ {B_x} & {B_y} & {B_z} \end{matrix} \right|A×B=∣∣∣∣∣∣exAxBxeyAyByezAzBz∣∣∣∣∣∣
A⃗⋅(A⃗×B⃗)=0\vec{A}\cdot(\vec{A} \times \vec{B})=0A⋅(A×B)=0
(A⃗+B⃗)⋅C⃗=A⃗⋅C⃗+B⃗⋅C⃗(\vec{A}+\vec{B})\cdot\vec{C}=\vec{A}\cdot\vec{C}+\vec{B}\cdot\vec{C}(A+B)⋅C=A⋅C+B⋅C
(A⃗+B⃗)×C⃗=A⃗×C⃗+B⃗×C⃗(\vec{A}+\vec{B})\times\vec{C}=\vec{A}\times\vec{C}+\vec{B}\times\vec{C}(A+B)×C=A×C+B×C
A⃗⋅(B⃗×C⃗)=(A⃗×B⃗)⋅C⃗=(A⃗×C⃗)⋅B⃗\vec{A}\cdot(\vec{B}\times \vec{C})=(\vec{A} \times \vec{B})\cdot\vec{C}=(\vec{A} \times \vec{C})\cdot\vec{B}A⋅(B×C)=(A×B)⋅C=(A×C)⋅B
A⃗×(B⃗×C⃗)=B⃗(A⃗⋅C⃗)−C⃗(A⃗⋅B⃗)\vec{A}\times(\vec{B} \times \vec{C})=\vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})A×(B×C)=B(A⋅C)−C(A⋅B)
散度相关公式
1.∇⋅C⃗=0(C⃗为常矢量)\nabla\cdot \vec{C}=0(\vec{C}为常矢量)∇⋅C=0(C为常矢量)
2.∇⋅(C⃗f)=C⃗⋅∇f\nabla\cdot (\vec{C}f)=\vec{C}\cdot \nabla f∇⋅(Cf)=C⋅∇f
3.∇⋅(kF⃗)=k∇⋅F⃗(k为常量)\nabla\cdot (k\vec{F})=k\nabla\cdot\vec{F}(k为常量)∇⋅(kF)=k∇⋅F(k为常量)
4.∇⋅(fF⃗)=f∇⋅F⃗+F⃗⋅∇f\nabla\cdot (f\vec{F})=f\nabla\cdot \vec{F}+\vec{F}\cdot\nabla f∇⋅(fF)=f∇⋅F+F⋅∇f
5.∇⋅(F⃗+G⃗)=∇⋅F⃗+∇⋅G⃗\nabla\cdot (\vec{F}+\vec{G})=\nabla\cdot \vec{F}+\nabla\cdot \vec{G}∇⋅(F+G)=∇⋅F+∇⋅G
旋度相关公式
1.∇C⃗=0(C⃗为常矢量)\nabla \vec{C}=0(\vec{C}为常矢量)∇C=0(C为常矢量)
2.∇×(C⃗f)=∇f×C⃗\nabla\times (\vec{C}f)=\nabla f\times\vec{C}∇×(Cf)=∇f×C
3.∇×(fF⃗)=f∇×C⃗+∇f×C⃗\nabla\times (f\vec{F})=f\nabla\times\vec{C}+\nabla f\times\vec{C}∇×(fF)=f∇×C+∇f×C
4.∇×(F⃗+G⃗)=∇×F⃗+∇×G⃗\nabla\times (\vec{F}+\vec{G})=\nabla\times \vec{F}+\nabla\times \vec{G}∇×(F+G)=∇×F+∇×G
5.∇⋅(F⃗×G⃗)=G⃗⋅∇×F⃗+F⃗⋅∇×G⃗\nabla\cdot (\vec{F}\times\vec{G})=\vec{G}\cdot\nabla\times \vec{F}+\vec{F}\cdot\nabla\times \vec{G}∇⋅(F×G)=G⋅∇×F+F⋅∇×G
6.∇⋅(∇×F⃗)=0;\nabla\cdot(\nabla\times\vec{F})=0;∇⋅(∇×F)=0;
7.∇×(∇u)=0;\nabla\times(\nabla u)=0;∇×(∇u)=0;
高斯定理
∫V∇⋅F⃗dV=∮sF⃗⋅dS⃗\int_V \nabla\cdot \vec{F}dV=\oint_s \vec{F}\cdot d\vec{S}∫V∇⋅FdV=∮sF⋅dS
(矢量场在空间任意闭合曲面的通量等于该闭合曲面所包含体积中矢量场的散度的体积分)
斯托克斯定理
∮CF⃗⋅dl⃗=∫S∇×F⃗⋅dS⃗\oint_C \vec{F}\cdot d\vec{l}=\int_S \nabla\times\vec{F}\cdot d\vec{S}∮CF⋅dl=∫S∇×F⋅dS
(矢量场沿任意闭合曲线的环流等于矢量场的旋度在该闭合曲线所围的曲面的通量)
梯度Δu\Delta uΔu
直角面坐标系：∇u\nabla u∇u=e⃗x∂u∂x+e⃗y∂u∂y+e⃗z∂u∂z\vec{e}_x\frac{\partial u}{\partial x}+\vec{e}_y\frac{\partial u}{\partial y}+\vec{e}_z\frac{\partial u}{\partial z}ex∂x∂u+ey∂y∂u+ez∂z∂u
圆柱面坐标系：∇u\nabla u∇u=e⃗ρ∂u∂ρ+e⃗ϕ1ρ∂u∂ϕ+e⃗z∂u∂z\vec{e}_{\rho}\frac{\partial u}{\partial \rho}+\vec{e}_{\phi}\frac{1}{\rho}\frac{\partial u}{\partial \phi}+\vec{e}_z\frac{\partial u}{\partial z}eρ∂ρ∂u+eϕρ1∂ϕ∂u+ez∂z∂u
球面坐标系：∇u\nabla u∇u=e⃗r∂u∂r+e⃗θ1r∂u∂θ+e⃗ϕ1rsinθ∂u∂ϕ\vec{e}_r\frac{\partial u}{\partial r}+\vec{e}_{\theta}\frac{1}{r}\frac{\partial u}{\partial \theta}+\vec{e}_{\phi}\frac{1}{rsin\theta}\frac{\partial u}{\partial {\phi}}er∂r∂u+eθr1∂θ∂u+eϕrsinθ1∂ϕ∂u
散度∇⋅F⃗\nabla\cdot \vec{F}∇⋅F
直角坐标系 ：∇⋅F⃗\nabla\cdot \vec{F}∇⋅F=∂Fx∂x+∂Fy∂y+∂Fz∂z\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}∂x∂Fx+∂y∂Fy+∂z∂Fz
柱面坐标系：∇⋅F\nabla\cdot F∇⋅F=1ρ∂∂ρ(ρFρ)+1ρ∂Fϕ∂ϕ+∂Fz∂z\frac{1}{\rho}\frac{\partial }{\partial \rho}(\rho F_\rho)+\frac{1}{\rho}\frac{\partial F_\phi}{\partial \phi}+\frac{\partial F_z}{\partial z}ρ1∂ρ∂(ρFρ)+ρ1∂ϕ∂Fϕ+∂z∂Fz
球面坐标系：∇⋅F\nabla\cdot F∇⋅F=1r2∂∂r(r2Fr)+1rsinθ∂∂θ(sinθFθ)+1rsinθ∂Fϕ∂ϕ\frac{1}{r^2}\frac{\partial }{\partial r}(r^2 F_r)+\frac{1}{rsin\theta}\frac{\partial }{\partial \theta}(sin\theta F_\theta)+\frac{1}{rsin\theta}\frac{\partial F_\phi}{\partial \phi}r21∂r∂(r2Fr)+rsinθ1∂θ∂(sinθFθ)+rsinθ1∂ϕ∂Fϕ
旋度∇×F⃗\nabla\times \vec{F}∇×F
直角坐标系：∇×F⃗=e⃗x(∂Fz∂y−∂Fy∂z)+e⃗y(∂Fx∂z−∂Fz∂x)+e⃗z(∂Fy∂x−∂Fx∂y)=∣e⃗xe⃗ye⃗z∂∂x∂∂y∂∂zFxFyFz∣\nabla\times \vec{F}=\vec{e}_x(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z})+\vec{e}_y(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x})+\vec{e}_z(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y})=\left| \begin{matrix} {\vec{e}_x} & {\vec{e}_y} & {\vec{e}_z}\\ {\frac{\partial}{\partial x}} & {\frac{\partial}{\partial y}} & {\frac{\partial}{\partial z}}\\ {F_x} & {F_y} & {F_z} \end{matrix} \right|∇×F=ex(∂y∂Fz−∂z∂Fy)+ey(∂z∂Fx−∂x∂Fz)+ez(∂x∂Fy−∂y∂Fx)=∣∣∣∣∣∣ex∂x∂Fxey∂y∂Fyez∂z∂Fz∣∣∣∣∣∣
圆柱坐标系：∇×F⃗=1ρ∣e⃗ρρe⃗ϕe⃗z∂∂ρ∂∂ϕ∂∂zFρρFϕFz∣\nabla\times \vec{F}=\frac{1}{\rho}\left| \begin{matrix} {\vec{e}_{\rho}} & {\rho\vec{e}_{\phi}} & {\vec{e}_z}\\ {\frac{\partial}{\partial \rho}} & {\frac{\partial}{\partial \phi}} & {\frac{\partial}{\partial z}}\\ {F_\rho} & {\rho F_\phi} & {F_z} \end{matrix} \right|∇×F=ρ1∣∣∣∣∣∣eρ∂ρ∂Fρρeϕ∂ϕ∂ρFϕez∂z∂Fz∣∣∣∣∣∣
球坐标系：∇×F⃗=1r2sinθ∣e⃗rre⃗θrsinθe⃗ϕ∂∂r∂∂θ∂∂ϕFrrFθrsinθFϕ∣\nabla\times \vec{F}=\frac{1}{r^2sin\theta}\left| \begin{matrix} {\vec{e}_{r}} & {r\vec{e}_{\theta}} & {rsin\theta \vec{e}_\phi}\\ {\frac{\partial}{\partial r}} & {\frac{\partial}{\partial \theta}} & {\frac{\partial}{\partial \phi}}\\ {F_r} & {r F_\theta} & {rsin\theta F_\phi} \end{matrix} \right|∇×F=r2sinθ1∣∣∣∣∣∣er∂r∂Frreθ∂θ∂rFθrsinθeϕ∂ϕ∂rsinθFϕ∣∣∣∣∣∣
格林第一恒等式
∫V(∇φ⋅∇ψ+φ∇2ψ)dV=∮Sφ∂ψ∂ndS\int_V(\nabla\varphi\cdot\nabla\psi+\varphi\nabla^2\psi)dV=\oint_S\varphi\frac{\partial\psi}{\partial n}dS∫V(∇φ⋅∇ψ+φ∇2ψ)dV=∮Sφ∂n∂ψdS
格林第二恒等式
∫V(φ∇2ψ−ψ∇2φ)dV=∮S(φ∂ψ∂n−ψ∂φ∂n)dS\int_V(\varphi\nabla^2\psi-\psi\nabla^2\varphi)dV=\oint_S(\varphi\frac{\partial\psi}{\partial n}-\psi\frac{\partial\varphi}{\partial n})dS∫V(φ∇2ψ−ψ∇2φ)dV=∮S(φ∂n∂ψ−ψ∂n∂φ)dS
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