电磁场与电磁波第一章公式总结
矢量代換公式
A?=A(e?xcosα+e?ycosβ+e?zcosγ)\vec{A}=A(\vec{e}_xcos\alpha+\vec{e}_ycos\beta+\vec{e}_zcos\gamma)A=A(ex?cosα+ey?cosβ+ez?cosγ)
A?±B?=e?x(Ax±Bx)+e?y(Ay±By)+e?z(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?)
A??B?=標(biāo)量=ABcosθ=AxBx+AyBy+AzBz\vec{A}\cdot\vec{B}=標(biāo)量=ABcos\theta=A_x B_x+A_y B_y+A_z B_zA?B=標(biāo)量=ABcosθ=Ax?Bx?+Ay?By?+Az?Bz?
A?×B?=矢量=e?nABsinθ\vec{A}\times \vec{B}=矢量=\vec{e}_nABsin\thetaA×B=矢量=en?ABsinθ
A?×B?=?B?×A?\vec{A}\times \vec{B}=-\vec{B}\times \vec{A}A×B=?B×A
A?×B?=e?x(AyBz?AzBy)+e?y(AzBx?AxBz)+e?z(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?(Ay?Bz??Az?By?)+ey?(Az?Bx??Ax?Bz?)+ez?(Ax?By??Ay?Bx?)
A?×B?=∣e?xe?ye?zAxAyAzBxByBz∣\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=∣∣∣∣∣∣?ex?Ax?Bx??ey?Ay?By??ez?Az?Bz??∣∣∣∣∣∣?
A??(A?×B?)=0\vec{A}\cdot(\vec{A} \times \vec{B})=0A?(A×B)=0
(A?+B?)?C?=A??C?+B??C?(\vec{A}+\vec{B})\cdot\vec{C}=\vec{A}\cdot\vec{C}+\vec{B}\cdot\vec{C}(A+B)?C=A?C+B?C
(A?+B?)×C?=A?×C?+B?×C?(\vec{A}+\vec{B})\times\vec{C}=\vec{A}\times\vec{C}+\vec{B}\times\vec{C}(A+B)×C=A×C+B×C
A??(B?×C?)=(A?×B?)?C?=(A?×C?)?B?\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
A?×(B?×C?)=B?(A??C?)?C?(A??B?)\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)
散度相關(guān)公式
1.??C?=0(C?為常矢量)\nabla\cdot \vec{C}=0(\vec{C}為常矢量)??C=0(C為常矢量)
2.??(C?f)=C???f\nabla\cdot (\vec{C}f)=\vec{C}\cdot \nabla f??(Cf)=C??f
3.??(kF?)=k??F?(k為常量)\nabla\cdot (k\vec{F})=k\nabla\cdot\vec{F}(k為常量)??(kF)=k??F(k為常量)
4.??(fF?)=f??F?+F???f\nabla\cdot (f\vec{F})=f\nabla\cdot \vec{F}+\vec{F}\cdot\nabla f??(fF)=f??F+F??f
5.??(F?+G?)=??F?+??G?\nabla\cdot (\vec{F}+\vec{G})=\nabla\cdot \vec{F}+\nabla\cdot \vec{G}??(F+G)=??F+??G
旋度相關(guān)公式
1.?C?=0(C?為常矢量)\nabla \vec{C}=0(\vec{C}為常矢量)?C=0(C為常矢量)
2.?×(C?f)=?f×C?\nabla\times (\vec{C}f)=\nabla f\times\vec{C}?×(Cf)=?f×C
3.?×(fF?)=f?×C?+?f×C?\nabla\times (f\vec{F})=f\nabla\times\vec{C}+\nabla f\times\vec{C}?×(fF)=f?×C+?f×C
4.?×(F?+G?)=?×F?+?×G?\nabla\times (\vec{F}+\vec{G})=\nabla\times \vec{F}+\nabla\times \vec{G}?×(F+G)=?×F+?×G
5.??(F?×G?)=G???×F?+F???×G?\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.??(?×F?)=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??F?dV=∮sF??dS?\int_V \nabla\cdot \vec{F}dV=\oint_s \vec{F}\cdot d\vec{S}∫V???FdV=∮s?F?dS
(矢量場(chǎng)在空間任意閉合曲面的通量等于該閉合曲面所包含體積中矢量場(chǎng)的散度的體積分)
斯托克斯定理
∮CF??dl?=∫S?×F??dS?\oint_C \vec{F}\cdot d\vec{l}=\int_S \nabla\times\vec{F}\cdot d\vec{S}∮C?F?dl=∫S??×F?dS
(矢量場(chǎng)沿任意閉合曲線的環(huán)流等于矢量場(chǎng)的旋度在該閉合曲線所圍的曲面的通量)
梯度Δu\Delta uΔu
直角面坐標(biāo)系：?u\nabla u?u=e?x?u?x+e?y?u?y+e?z?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?
圓柱面坐標(biāo)系：?u\nabla u?u=e?ρ?u?ρ+e??1ρ?u??+e?z?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?+e??ρ1????u?+ez??z?u?
球面坐標(biāo)系：?u\nabla u?u=e?r?u?r+e?θ1r?u?θ+e??1rsinθ?u??\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?+eθ?r1??θ?u?+e??rsinθ1????u?
散度??F?\nabla\cdot \vec{F}??F
直角坐標(biāo)系 ：??F?\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??
柱面坐標(biāo)系：??F\nabla\cdot F??F=1ρ??ρ(ρFρ)+1ρ?F???+?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????F???+?z?Fz??
球面坐標(biāo)系：??F\nabla\cdot F??F=1r2??r(r2Fr)+1rsinθ??θ(sinθFθ)+1rsinθ?F???\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????F???
旋度?×F?\nabla\times \vec{F}?×F
直角坐標(biāo)系：?×F?=e?x(?Fz?y??Fy?z)+e?y(?Fx?z??Fz?x)+e?z(?Fy?x??Fx?y)=∣e?xe?ye?z??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??Fx??ey??y??Fy??ez??z??Fz??∣∣∣∣∣∣?
圓柱坐標(biāo)系：?×F?=1ρ∣e?ρρe??e?z??ρ?????zFρρF?Fz∣\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??Fρ??ρe??????ρF???ez??z??Fz??∣∣∣∣∣∣?
球坐標(biāo)系：?×F?=1r2sinθ∣e?rre?θrsinθe????r??θ???FrrFθrsinθF?∣\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??Fr??reθ??θ??rFθ??rsinθe??????rsinθF???∣∣∣∣∣∣?
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=∮S?φ?n?ψ?dS
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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