]> $$\overrightarrow{A}$$ $$\overrightarrow{B}$$ $$\overrightarrow{A}·\overrightarrow{B}=\left|A\right|\left|B\right|\cos \theta =A_x{B}_x+A_y{B}_y+A_z{B}_z$$ $$\overrightarrow{A}\times \overrightarrow{B}=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ A_x& A_y& A_z\\ B_x& B_y& B_z\end{array}\right|$$ $$=\overrightarrow{i}\left(A_y{B}_z-A_z{B}_y\right)+\overrightarrow{j}\left(A_z{B}_x-A_x{B}_z\right)+\overrightarrow{k}\left(A_x{B}_y-A_y{B}_x\right)$$ $$\overrightarrow{A}\times \overrightarrow{B}=-\overrightarrow{B}\times \overrightarrow{A}$$ $$\left(\overrightarrow{A}\times \overrightarrow{B}\right)·\overrightarrow{C}=\overrightarrow{A}·\left(\overrightarrow{B}\times \overrightarrow{C}\right)=\left(\overrightarrow{C}\times \overrightarrow{A}\right)·\overrightarrow{B}$$ $$\overrightarrow{A}\times \left(\overrightarrow{B}\times \overrightarrow{C}\right)=\overrightarrow{B}\left(\overrightarrow{A}·\overrightarrow{C}\right)-\overrightarrow{C}\left(\overrightarrow{A}·\overrightarrow{B}\right)$$ $$$$ $$\iint f\left(x,y\right)dA=\underset{{\theta }_1}{\overset{{\theta }_2}{\int }}\underset{r_1}{\overset{r_2}{\int }}f\left(r\cos \theta ,r\sin \theta \right)r·drd\theta$$ $$\iiint f\left(x,y,z\right)dV=\underset{{\theta }_1}{\overset{{\theta }_2}{\int }}\underset{{\rho }_1}{\overset{{\rho }_2}{\int }}\underset{z_1}{\overset{z_2}{\int }}f\left(\rho \cos \theta ,\rho \sin \theta ,z\right)\rho ·dzd\rho d\theta$$ $$\iiint f\left(x,y,z\right)dV=\underset{{\theta }_1}{\overset{{\theta }_2}{\int }}\underset{{\phi }_1}{\overset{{\phi }_2}{\int }}\underset{R_1}{\overset{R_2}{\int }}f\left(R\cos \phi \sin \theta ,R\sin \phi \sin \theta ,R\cos \theta \right)R^2\sin \theta ·dRd\phi d\theta$$ $$$$ $$dr=rd\theta$$ $$\stackrel{‸}{r}=\frac{\overrightarrow{r}}{\left|r\right|}$$ $$$$ $$\mathrm{grad}f=\nabla f=\overrightarrow{i}\frac{\partial f}{\partial x}+\overrightarrow{j}\frac{\partial f}{\partial y}+\overrightarrow{k}\frac{\partial f}{\partial z}$$ $$=\stackrel{‸}{\rho }\frac{\partial f}{d\rho }+\stackrel{‸}{\phi }\frac{1}{\rho }\frac{\partial f}{\partial \phi }+\stackrel{\stackrel{‸}{‸}}{z}\frac{\partial f}{dz}$$ $$=\stackrel{‸}{R}\frac{\partial f}{\partial r}+\stackrel{‸}{\theta }\frac{1}{R}\frac{\partial f}{\partial \theta }+\stackrel{‸}{\phi }\frac{1}{R\sin \theta }\frac{\partial f}{d\phi }$$ $$$$ $$\mathrm{div}\overrightarrow{E}=\nabla ·\overrightarrow{E}=\frac{\partial E_x}{\partial x}+\frac{\partial E_y}{\partial y}+\frac{\partial E_z}{\partial z}$$ $$=\frac{1}{\rho }\frac{\partial \left(\rho E_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial E_{\phi }}{\partial \phi }+\frac{\partial E_z}{\partial z}$$ $$=\frac{1}{R^2}\frac{\partial \left(R^2{E}_R\right)}{\partial R}+\frac{1}{R\sin \theta }\frac{\partial E_{\theta }\sin \theta }{\partial \theta }+\frac{1}{R\sin \theta }\frac{\partial A_{\phi }}{\partial \phi }$$ $$$$ $$\mathrm{curl}\overrightarrow{E}=\mathrm{rot}\overrightarrow{E}=\nabla \times \overrightarrow{E}=\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ E_x& E_y& E_z\end{array}\right|$$ $$=\overrightarrow{i}\left(\frac{\partial E_z}{\partial y}-\frac{\partial E_y}{\partial z}\right)+\overrightarrow{j}\left(\frac{\partial E_x}{\partial z}-\frac{\partial E_z}{\partial x}\right)+\overrightarrow{k}\left(\frac{\partial E_y}{\partial x}-\frac{\partial E_x}{\partial y}\right)$$ $$=\stackrel{‸}{\rho }\left(\frac{1}{\rho }\frac{\partial E_z}{\partial \phi }-\frac{\partial E_{\phi }}{\partial z}\right)+\stackrel{‸}{\phi }\left(\frac{\partial E_{\rho }}{\partial z}-\frac{\partial E_z}{\partial \rho }\right)+\stackrel{‸}{z}\frac{1}{\rho }\left(\frac{\partial \left(\rho E_{\phi }\right)}{\partial \rho }-\frac{\partial E_{\rho }}{\partial \phi }\right)$$ $$=\stackrel{‸}{R}\frac{1}{R\sin \theta }\left(\frac{\partial \left(E_{\phi }\sin \theta \right)}{\partial \theta }-\frac{\partial E_{\theta }}{\partial \phi }\right)+\stackrel{‸}{\theta }\frac{1}{R}\left(\frac{1}{\sin \theta }\frac{\partial E_R}{\partial \phi }-\frac{\partial \left(R{E}_{\phi }\right)}{\partial R}\right)$$ $$$$ $$\text{Laplacian}f={\nabla }^{2}f=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}$$ $$=\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(\rho \frac{\partial f}{\partial \rho }\right)+\frac{1}{{\rho }^2}\frac{{\partial }^{2}f}{\partial {\phi }^2}+\frac{{\partial }^{2}f}{\partial z^2}$$ $$=\frac{1}{R^2}\frac{\partial }{\partial R}\left(R^2\frac{\partial f}{dR}\right)+\frac{1}{R^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\frac{1}{R^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\phi }^2}$$ $$$$ $${\nabla }^2\overrightarrow{A}=\stackrel{‸}{x}·{\nabla }^2{A}_x+\stackrel{‸}{y}·{\nabla }^2{A}_y+\stackrel{‸}{z}·{\nabla }^2{A}_z$$ $$=\stackrel{‸}{\rho }·{\nabla }^2{A}_{\rho }+\stackrel{‸}{\phi }·{\nabla }^2{A}_{\phi }+\stackrel{‸}{z}{\nabla }^2{A}_z$$ $$$$ $$\mathrm{div}\mathrm{curl}\overrightarrow{F}=0$$ $$\mathrm{curl}\mathrm{grad}f=0$$ $$\mathrm{curl}\mathrm{curl}\overrightarrow{F}=\mathrm{grad}\mathrm{div}\overrightarrow{F}-{\nabla }^2\overrightarrow{F}$$ $$$$ $$\nabla \left(au+bv\right)=a\nabla u+b\nabla v$$ $$\nabla \left(uv\right)=u\nabla v+v\nabla u$$ $$\nabla \left(\frac{u}{v}\right)=\frac{\left(v\nabla u-u\nabla v\right)}{v^2}$$ $$\nabla u^n=n{u}^{n-1}\nabla u$$ $$$$ $$\nabla ·\left(f\overrightarrow{A}\right)=\left(\nabla f·\overrightarrow{A}\right)+f\left(\nabla ·\overrightarrow{A}\right)$$ $$\nabla \times \left(f\overrightarrow{A}\right)=\left(\nabla f\times \overrightarrow{A}\right)+f\left(\nabla \times \overrightarrow{A}\right)$$ $$\mathrm{div}\left(\overrightarrow{A}\times \overrightarrow{B}\right)=\overrightarrow{B}\mathrm{rot}\overrightarrow{A}-\overrightarrow{A}\mathrm{rot}\overrightarrow{B}$$ $$\mathrm{rot}\left(\overrightarrow{A}\times \overrightarrow{B}\right)=\overrightarrow{A}\left(\nabla ·\overrightarrow{B}\right)-\overrightarrow{B}\left(\nabla ·\overrightarrow{A}\right)+\overrightarrow{B}\left(\nabla ·\overrightarrow{A}\right)-\left(\overrightarrow{A}\nabla \right)\overrightarrow{B}$$ $$$$ $$\overrightarrow{R}=\stackrel{‸}{x}x+\stackrel{‸}{y}y+\stackrel{‸}{z}z$$ $$\mathrm{div}\overrightarrow{R}=3$$ $$\mathrm{rot}\overrightarrow{R}=\overrightarrow{0}$$ $$\mathrm{rot}\left(\phi \left(R\right)\overrightarrow{R}\right)=\overrightarrow{0}$$ $$\nabla ·\left(\frac{\overrightarrow{R}}{R^3}\right)=0$$ $$\nabla \times \left(-\frac{\overrightarrow{R}}{R^3}\right)=\overrightarrow{0}$$ $$\nabla \left(\frac{1}{R}\right)=-\frac{\overrightarrow{R}}{R^3}$$ $$\nabla \left(\overrightarrow{R}·[\overrightarrow{a}\times \overrightarrow{b}]\right)=[\overrightarrow{a}\times \overrightarrow{b}]$$ $$$$ $$\overrightarrow{F}={\overrightarrow{F}}_0{e}^{j\left(\overrightarrow{k}·\overrightarrow{r}\right)}$$ $$\nabla ·\overrightarrow{F}=j\overrightarrow{k}·\overrightarrow{F}$$ $$\nabla \times \overrightarrow{F}=j\overrightarrow{K}\times \overrightarrow{F}$$ $${\nabla }^2\overrightarrow{F}=-k^2\overrightarrow{F}$$ $$v=v_0{e}^{j\left(\overrightarrow{k}·\overrightarrow{r}\right)}$$ $$\nabla v=j\overrightarrow{k}v$$