14.3: Mathematical Identities
- Page ID
- 25049
\begin{aligned} \overline{\mathrm{A}} &=\hat{x} \mathrm{A}_{\mathrm{x}}+\hat{y} \mathrm{A}_{\mathrm{y}}+\hat{z} \mathrm{A}_{\mathrm{z}} \\[4pt] \overline{\mathrm{A}} \bullet \overline{\mathrm{B}} &=\mathrm{A}_{\mathrm{x}} \mathrm{B}_{\mathrm{x}}+\mathrm{A}_{\mathrm{y}} \mathrm{B}_{\mathrm{y}}+\mathrm{A}_{\mathrm{z}} \mathrm{B}_{\mathrm{z}}=\hat{\mathrm{a}} \times \hat{\mathrm{b}}|\overline{\mathrm{A}}||\overline{\mathrm{B}}| \cos \theta \\[4pt] \overline{\mathrm{A}} \times \overline{\mathrm{B}} &=\operatorname{det}\left|\begin{array}{ccc} \hat{x} & \hat{y} & \hat{z} \\[4pt] \mathrm{A}_{\mathrm{x}} & \mathrm{A}_{\mathrm{y}} & \mathrm{A}_{\mathrm{z}} \\[4pt] \mathrm{B}_{\mathrm{x}} & \mathrm{B}_{\mathrm{y}} & \mathrm{B}_{\mathrm{z}} \end{array}\right| \\[4pt] &=\hat{x}\left(\mathrm{A}_{\mathrm{y}} \mathrm{B}_{\mathrm{z}}-\mathrm{A}_{\mathrm{z}} \mathrm{B}_{\mathrm{y}}\right)+\hat{y}\left(\mathrm{A}_{\mathrm{z}} \mathrm{B}_{\mathrm{x}}-\mathrm{A}_{\mathrm{x}} \mathrm{B}_{\mathrm{z}}\right)+\hat{2}\left(\mathrm{A}_{\mathrm{x}} \mathrm{B}_{\mathrm{y}}-\mathrm{A}_{\mathrm{y}} \mathrm{B}_{\mathrm{x}}\right) \\[4pt] &=\hat{\mathrm{a}} \times \hat{\mathrm{b}}|\overline{\mathrm{A}}||\mathrm{\overline B}| \sin \theta \\[4pt] \overline{\mathrm{A}} \bullet(\overline{\mathrm{B}} \times \overline{\mathrm{C}}) &=\overline{\mathrm{B}} \bullet(\overline{\mathrm{C}} \times \overline{\mathrm{A}})=\overline{\mathrm{C}} \bullet(\overline{\mathrm{A}} \times \overline{\mathrm{B}}) \\[4pt] \overline{\mathrm{A}} \times(\overline{\mathrm{B}} \times \overline{\mathrm{C}}) &=(\overline{\mathrm{A}} \bullet \overline{\mathrm{C}}) \overline{\mathrm{B}}-(\overline{\mathrm{A}} \bullet \overline{\mathrm{B}}) \overline{\mathrm{C}} \\[4pt] (\overline{\mathrm{A}} \times \overline{\mathrm{B}}) \bullet(\overline{\mathrm{C}} \times \overline{\mathrm{D}}) &=(\overline{\mathrm{A}} \bullet \overline{\mathrm{C}})(\overline{\mathrm{B}} \bullet \overline{\mathrm{D}})-(\overline{\mathrm{A}} \bullet \overline{\mathrm{D}})(\overline{\mathrm{B}} \bullet \overline{\mathrm{C}}) \\[4pt] \nabla \times \nabla \Psi &=0 \\[4pt] \nabla \bullet(\nabla \times \overline{\mathrm{A}}) &=0 \\[4pt] \nabla \times(\nabla \times \overline{\mathrm{A}}) &=\nabla(\nabla \bullet \overline{\mathrm{A}})-\nabla^{2} \overline{\mathrm{A}} \\[4pt] -\overline{\mathrm{A}} \times(\nabla \times \overline{\mathrm{A}}) &=(\overline{\mathrm{A}} \bullet \nabla) \overline{\mathrm{A}}-\frac{1}{2} \nabla(\overline{\mathrm{A}} \bullet \overline{\mathrm{A}}) \\[4pt] \nabla(\Psi \Phi) &=\Psi \nabla \Phi+\Phi \nabla \Psi \\[4pt] \nabla \bullet(\Psi \overline{\mathrm{A}}) &=\overline{\mathrm{A}} \bullet \nabla \Psi+\Psi \nabla \bullet \overline{\mathrm{A}} \\[4pt] \nabla \times(\Psi \overline{\mathrm{A}}) &=\nabla \Psi \times \overline{\mathrm{A}}+\Psi \nabla \times \overline{\mathrm{A}} \\[4pt] \nabla^{2} \Psi &=\nabla \bullet \nabla \Psi \\[4pt] \nabla(\overline{\mathrm{A}} \bullet \overline{\mathrm{B}}) &=(\overline{\mathrm{A}} \bullet \nabla) \overline{\mathrm{B}}+(\overline{\mathrm{B}} \bullet \nabla) \overline{\mathrm{A}}+\overline{\mathrm{A}} \times(\nabla \times \overline{\mathrm{B}})+\overline{\mathrm{B}} \times(\nabla \times \overline{\mathrm{A}}) \\[4pt] \nabla \bullet(\overline{\mathrm{A}} \times \overline{\mathrm{B}}) &=\overline{\mathrm{B}} \bullet(\nabla \times \overline{\mathrm{A}})-\overline{\mathrm{A}} \bullet(\nabla \times \overline{\mathrm{B}}) \\[4pt] \nabla \times(\overline{\mathrm{A}} \times \overline{\mathrm{B}}) &=\overline{\mathrm{A}}(\nabla \bullet \overline{\mathrm{B}})-\overline{\mathrm{B}}(\nabla \bullet \overline{\mathrm{A}})+(\overline{\mathrm{B}} \bullet \nabla) \overline{\mathrm{A}}-(\overline{\mathrm{A}} \bullet \nabla) \overline{\mathrm{B}} \end{aligned}
Cartesian Coordinates (x,y,z):
\[\begin{aligned}
\nabla \Psi &=\hat{x} \frac{\partial \Psi}{\partial \mathrm{x}}+\hat{y} \frac{\partial \Psi}{\partial \mathrm{y}}+\hat{z} \frac{\partial \Psi}{\partial \mathrm{z}} \\[4pt]
\nabla \bullet \overline{\mathrm{A}} &=\frac{\partial \mathrm{A}_{\mathrm{x}}}{\partial \mathrm{x}}+\frac{\partial \mathrm{A}_{\mathrm{y}}}{\partial \mathrm{y}}+\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{z}} \\[4pt]
\nabla \times \overline{\mathrm{A}} &=\hat{x}\left(\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{y}}-\frac{\partial \mathrm{A}_{\mathrm{y}}}{\partial \mathrm{z}}\right)+\hat{y}\left(\frac{\partial \mathrm{A}_{\mathrm{x}}}{\partial \mathrm{z}}-\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{x}}\right)+\hat{z}\left(\frac{\partial \mathrm{A}_{\mathrm{y}}}{\partial \mathrm{x}}-\frac{\partial \mathrm{A}_{\mathrm{x}}}{\partial \mathrm{y}}\right) \\[4pt]
\nabla^{2} \Psi &=\frac{\partial^{2} \Psi}{\partial \mathrm{x}^{2}}+\frac{\partial^{2} \Psi}{\partial \mathrm{y}^{2}}+\frac{\partial^{2} \Psi}{\partial \mathrm{z}^{2}}
\end{aligned}\]
Cylindrical coordinates (r,φ,z):
\[\begin{aligned}
\nabla \Psi &=\hat{\rho} \frac{\partial \Psi}{\partial \mathrm{r}}+\hat{\phi} \frac{1}{\mathrm{r}} \frac{\partial \Psi}{\partial \mathrm{y}}+\hat{z} \frac{\partial \Psi}{\partial \mathrm{z}} \\[4pt]
\nabla \bullet \overline{\mathrm{A}} &=\frac{1}{\mathrm{r}} \frac{\partial\left(\mathrm{r} \mathrm{A}_{\mathrm{r}}\right)}{\partial \mathrm{r}}+\frac{1}{\mathrm{r}} \frac{\partial \mathrm{A}_{\phi}}{\partial \phi}+\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{z}} \\[4pt]
\nabla \times \overline{\mathrm{A}}=& \hat{r}\left(\frac{1}{\mathrm{r}} \frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \phi}-\frac{\partial \mathrm{A}_{\phi}}{\partial \mathrm{z}}\right)+\hat{\phi}\left(\frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \mathrm{z}}-\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{r}}\right)+\hat{z} \frac{1}{\mathrm{r}}\left(\frac{\partial\left(\mathrm{r} \mathrm{A}_{\phi}\right)}{\partial \mathrm{r}}-\frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \phi}\right)=\frac{1}{\mathrm{r}} \operatorname{det}\left|\begin{array}{ccc}
\hat{r} / \partial \mathrm{r} & \partial / \partial \phi & \partial(\partial \mathrm{z} \mid \\[4pt]
\mathrm{A}_{\mathrm{r}} & \mathrm{r} \mathrm{A}_{\phi} & \mathrm{A}_{\mathrm{z}}
\end{array}\right| \\[4pt]
\nabla^{2} \Psi &=\frac{1}{\mathrm{r}} \frac{\partial}{\partial \mathrm{r}}\left(\mathrm{r} \frac{\partial \Psi}{\partial \mathrm{r}}\right)+\frac{1}{\mathrm{r}^{2}} \frac{\partial^{2} \Psi}{\partial \phi^{2}}+\frac{\partial^{2} \Psi}{\partial \mathrm{z}^{2}}
\end{aligned}\]
Spherical coordinates (r,θ,φ):
\[\begin{aligned} \nabla \Psi &=\hat{r} \frac{\partial \Psi}{\partial \mathrm{r}}+\hat{\theta} \frac{1}{\mathrm{r}} \frac{\partial \Psi}{\partial \theta}+\hat{\phi} \frac{1}{\mathrm{r} \sin \theta} \frac{\partial \Psi}{\partial \phi} \\[4pt] \nabla \bullet \overline{\mathrm{A}} &=\frac{1}{\mathrm{r}^{2}} \frac{\partial\left(\mathrm{r}^{2} \mathrm{A}_{\mathrm{r}}\right)}{\partial \mathrm{r}}+\frac{1}{\mathrm{r} \sin \theta} \frac{\partial\left(\sin \theta \mathrm{A}_{\theta}\right)}{\partial \theta}+\frac{1}{\mathrm{r} \sin \theta} \frac{\partial \mathrm{A}_{\phi}}{\partial \phi} \\[4pt] \nabla \times \overline{\mathrm{A}} &=\hat{r} \frac{1}{\mathrm{r} \sin \theta}\left(\frac{\partial\left(\mathrm{r} \sin \theta \mathrm{A}_{\phi}\right)}{\partial \theta}-\frac{\partial \mathrm{A}_{\theta}}{\partial \phi}\right)+\hat{\theta}\left(\frac{1}{\mathrm{r} \sin \theta} \frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \phi}-\frac{1}{\mathrm{r}} \frac{\partial\left(\mathrm{r} \mathrm{A}_{\phi}\right)}{\partial \mathrm{r}}\right)+\hat{\phi} \frac{1}{\mathrm{r}}\left(\frac{\partial\left(\mathrm{r} \mathrm{A}_{\theta}\right)}{\partial \mathrm{r}}-\frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \theta}\right) \\[4pt] &=\frac{1}{\mathrm{r}^{2} \sin \theta} \operatorname{det}\left|\begin{array}{ccc} \hat{r} & r \hat{\theta} & \mathrm{r} \sin \theta \hat{\phi} \\[4pt] \partial / \partial \mathrm{r} & \partial / \partial \theta & \partial / \partial \phi \\[4pt] \mathrm{A}_{\mathrm{r}} & \mathrm{r} \mathrm{A}_{\theta} & \mathrm{r} \sin \theta \mathrm{A}_{\phi} \end{array}\right| \\[4pt] \nabla^{2} \Psi &=\mathrm{\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \Psi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \Psi}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} \Psi}{\partial \phi^{2}}} \end{aligned}\]
Gauss’ Divergence Theorem:
\[\int_{V} \nabla \bullet \overline{G} d v=\oint_{A} \overline{G} \bullet \hat{n} d a \nonumber\]
Stokes’ Theorem:
\[ \mathrm{\int_{A}(\nabla \times \overline{G}) \bullet \hat{n} \ d a=\oint_{C} \overline{G} \bullet d} \overline{\ell} \nonumber\]
Fourier Transforms for pulse signals h(t):
\[\begin{aligned} \mathrm{\underline{H}(f)} &= \mathrm{\int_{-\infty}^{\infty} h(t) e^{-j 2 \pi t} d t} \\[4pt] \mathrm{h(t)} &= \mathrm{\int_{-\infty}^{\infty} \underline{H}(f) e^{+j 2 \pi f t} d f} \end{aligned}\]