14.3: Mathematical Identities
- Page ID
- 25049
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) 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\(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\begin{aligned} \overrightarrow{\mathrm{A}} &=\hat{x} \mathrm{A}_{\mathrm{x}}+\hat{y} \mathrm{A}_{\mathrm{y}}+\hat{z} \mathrm{A}_{\mathrm{z}} \\[4pt] \overrightarrow{\mathrm{A}} \bullet \overrightarrow{\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}}|\overrightarrow{\mathrm{A}}||\overrightarrow{\mathrm{B}}| \cos \theta \\[4pt] \overrightarrow{\mathrm{A}} \times \overrightarrow{\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}}|\overrightarrow{\mathrm{A}}||\mathrm{\vec B}| \sin \theta \\[4pt] \overrightarrow{\mathrm{A}} \bullet(\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}}) &=\overrightarrow{\mathrm{B}} \bullet(\overrightarrow{\mathrm{C}} \times \overrightarrow{\mathrm{A}})=\overrightarrow{\mathrm{C}} \bullet(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}) \\[4pt] \overrightarrow{\mathrm{A}} \times(\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}}) &=(\overrightarrow{\mathrm{A}} \bullet \overrightarrow{\mathrm{C}}) \overrightarrow{\mathrm{B}}-(\overrightarrow{\mathrm{A}} \bullet \overrightarrow{\mathrm{B}}) \overrightarrow{\mathrm{C}} \\[4pt] (\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}) \bullet(\overrightarrow{\mathrm{C}} \times \overrightarrow{\mathrm{D}}) &=(\overrightarrow{\mathrm{A}} \bullet \overrightarrow{\mathrm{C}})(\overrightarrow{\mathrm{B}} \bullet \overrightarrow{\mathrm{D}})-(\overrightarrow{\mathrm{A}} \bullet \overrightarrow{\mathrm{D}})(\overrightarrow{\mathrm{B}} \bullet \overrightarrow{\mathrm{C}}) \\[4pt] \nabla \times \nabla \Psi &=0 \\[4pt] \nabla \bullet(\nabla \times \overrightarrow{\mathrm{A}}) &=0 \\[4pt] \nabla \times(\nabla \times \overrightarrow{\mathrm{A}}) &=\nabla(\nabla \bullet \overrightarrow{\mathrm{A}})-\nabla^{2} \overrightarrow{\mathrm{A}} \\[4pt] -\overrightarrow{\mathrm{A}} \times(\nabla \times \overrightarrow{\mathrm{A}}) &=(\overrightarrow{\mathrm{A}} \bullet \nabla) \overrightarrow{\mathrm{A}}-\frac{1}{2} \nabla(\overrightarrow{\mathrm{A}} \bullet \overrightarrow{\mathrm{A}}) \\[4pt] \nabla(\Psi \Phi) &=\Psi \nabla \Phi+\Phi \nabla \Psi \\[4pt] \nabla \bullet(\Psi \overrightarrow{\mathrm{A}}) &=\overrightarrow{\mathrm{A}} \bullet \nabla \Psi+\Psi \nabla \bullet \overrightarrow{\mathrm{A}} \\[4pt] \nabla \times(\Psi \overrightarrow{\mathrm{A}}) &=\nabla \Psi \times \overrightarrow{\mathrm{A}}+\Psi \nabla \times \overrightarrow{\mathrm{A}} \\[4pt] \nabla^{2} \Psi &=\nabla \bullet \nabla \Psi \\[4pt] \nabla(\overrightarrow{\mathrm{A}} \bullet \overrightarrow{\mathrm{B}}) &=(\overrightarrow{\mathrm{A}} \bullet \nabla) \overrightarrow{\mathrm{B}}+(\overrightarrow{\mathrm{B}} \bullet \nabla) \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{A}} \times(\nabla \times \overrightarrow{\mathrm{B}})+\overrightarrow{\mathrm{B}} \times(\nabla \times \overrightarrow{\mathrm{A}}) \\[4pt] \nabla \bullet(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}) &=\overrightarrow{\mathrm{B}} \bullet(\nabla \times \overrightarrow{\mathrm{A}})-\overrightarrow{\mathrm{A}} \bullet(\nabla \times \overrightarrow{\mathrm{B}}) \\[4pt] \nabla \times(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}) &=\overrightarrow{\mathrm{A}}(\nabla \bullet \overrightarrow{\mathrm{B}})-\overrightarrow{\mathrm{B}}(\nabla \bullet \overrightarrow{\mathrm{A}})+(\overrightarrow{\mathrm{B}} \bullet \nabla) \overrightarrow{\mathrm{A}}-(\overrightarrow{\mathrm{A}} \bullet \nabla) \overrightarrow{\mathrm{B}} \end{aligned}
Cartesian Coordinates (x,y,z):
\[\begin{aligned}
\nabla \Psi &=\hat{x} \frac{\partial \Psi}{\partial \mathrm{x}}+\hat{y} \frac{\partial \Psi}{\partial \mathrm{y}}+\hat{z} \frac{\partial \Psi}{\partial \mathrm{z}} \\[4pt]
\nabla \bullet \overrightarrow{\mathrm{A}} &=\frac{\partial \mathrm{A}_{\mathrm{x}}}{\partial \mathrm{x}}+\frac{\partial \mathrm{A}_{\mathrm{y}}}{\partial \mathrm{y}}+\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{z}} \\[4pt]
\nabla \times \overrightarrow{\mathrm{A}} &=\hat{x}\left(\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{y}}-\frac{\partial \mathrm{A}_{\mathrm{y}}}{\partial \mathrm{z}}\right)+\hat{y}\left(\frac{\partial \mathrm{A}_{\mathrm{x}}}{\partial \mathrm{z}}-\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{x}}\right)+\hat{z}\left(\frac{\partial \mathrm{A}_{\mathrm{y}}}{\partial \mathrm{x}}-\frac{\partial \mathrm{A}_{\mathrm{x}}}{\partial \mathrm{y}}\right) \\[4pt]
\nabla^{2} \Psi &=\frac{\partial^{2} \Psi}{\partial \mathrm{x}^{2}}+\frac{\partial^{2} \Psi}{\partial \mathrm{y}^{2}}+\frac{\partial^{2} \Psi}{\partial \mathrm{z}^{2}}
\end{aligned}\]
Cylindrical coordinates (r,φ,z):
\[\begin{aligned}
\nabla \Psi &=\hat{\rho} \frac{\partial \Psi}{\partial \mathrm{r}}+\hat{\phi} \frac{1}{\mathrm{r}} \frac{\partial \Psi}{\partial \mathrm{y}}+\hat{z} \frac{\partial \Psi}{\partial \mathrm{z}} \\[4pt]
\nabla \bullet \overrightarrow{\mathrm{A}} &=\frac{1}{\mathrm{r}} \frac{\partial\left(\mathrm{r} \mathrm{A}_{\mathrm{r}}\right)}{\partial \mathrm{r}}+\frac{1}{\mathrm{r}} \frac{\partial \mathrm{A}_{\phi}}{\partial \phi}+\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{z}} \\[4pt]
\nabla \times \overrightarrow{\mathrm{A}}=& \hat{r}\left(\frac{1}{\mathrm{r}} \frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \phi}-\frac{\partial \mathrm{A}_{\phi}}{\partial \mathrm{z}}\right)+\hat{\phi}\left(\frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \mathrm{z}}-\frac{\partial \mathrm{A}_{\mathrm{z}}}{\partial \mathrm{r}}\right)+\hat{z} \frac{1}{\mathrm{r}}\left(\frac{\partial\left(\mathrm{r} \mathrm{A}_{\phi}\right)}{\partial \mathrm{r}}-\frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \phi}\right)=\frac{1}{\mathrm{r}} \operatorname{det}\left|\begin{array}{ccc}
\hat{r} / \partial \mathrm{r} & \partial / \partial \phi & \partial(\partial \mathrm{z} \mid \\[4pt]
\mathrm{A}_{\mathrm{r}} & \mathrm{r} \mathrm{A}_{\phi} & \mathrm{A}_{\mathrm{z}}
\end{array}\right| \\[4pt]
\nabla^{2} \Psi &=\frac{1}{\mathrm{r}} \frac{\partial}{\partial \mathrm{r}}\left(\mathrm{r} \frac{\partial \Psi}{\partial \mathrm{r}}\right)+\frac{1}{\mathrm{r}^{2}} \frac{\partial^{2} \Psi}{\partial \phi^{2}}+\frac{\partial^{2} \Psi}{\partial \mathrm{z}^{2}}
\end{aligned}\]
Spherical coordinates (r,θ,φ):
\[\begin{aligned} \nabla \Psi &=\hat{r} \frac{\partial \Psi}{\partial \mathrm{r}}+\hat{\theta} \frac{1}{\mathrm{r}} \frac{\partial \Psi}{\partial \theta}+\hat{\phi} \frac{1}{\mathrm{r} \sin \theta} \frac{\partial \Psi}{\partial \phi} \\[4pt] \nabla \bullet \overrightarrow{\mathrm{A}} &=\frac{1}{\mathrm{r}^{2}} \frac{\partial\left(\mathrm{r}^{2} \mathrm{A}_{\mathrm{r}}\right)}{\partial \mathrm{r}}+\frac{1}{\mathrm{r} \sin \theta} \frac{\partial\left(\sin \theta \mathrm{A}_{\theta}\right)}{\partial \theta}+\frac{1}{\mathrm{r} \sin \theta} \frac{\partial \mathrm{A}_{\phi}}{\partial \phi} \\[4pt] \nabla \times \overrightarrow{\mathrm{A}} &=\hat{r} \frac{1}{\mathrm{r} \sin \theta}\left(\frac{\partial\left(\mathrm{r} \sin \theta \mathrm{A}_{\phi}\right)}{\partial \theta}-\frac{\partial \mathrm{A}_{\theta}}{\partial \phi}\right)+\hat{\theta}\left(\frac{1}{\mathrm{r} \sin \theta} \frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \phi}-\frac{1}{\mathrm{r}} \frac{\partial\left(\mathrm{r} \mathrm{A}_{\phi}\right)}{\partial \mathrm{r}}\right)+\hat{\phi} \frac{1}{\mathrm{r}}\left(\frac{\partial\left(\mathrm{r} \mathrm{A}_{\theta}\right)}{\partial \mathrm{r}}-\frac{\partial \mathrm{A}_{\mathrm{r}}}{\partial \theta}\right) \\[4pt] &=\frac{1}{\mathrm{r}^{2} \sin \theta} \operatorname{det}\left|\begin{array}{ccc} \hat{r} & r \hat{\theta} & \mathrm{r} \sin \theta \hat{\phi} \\[4pt] \partial / \partial \mathrm{r} & \partial / \partial \theta & \partial / \partial \phi \\[4pt] \mathrm{A}_{\mathrm{r}} & \mathrm{r} \mathrm{A}_{\theta} & \mathrm{r} \sin \theta \mathrm{A}_{\phi} \end{array}\right| \\[4pt] \nabla^{2} \Psi &=\mathrm{\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \Psi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \Psi}{\partial \theta}\right)+\frac{1}{r^{2} \sin ^{2} \theta} \frac{\partial^{2} \Psi}{\partial \phi^{2}}} \end{aligned}\]
Gauss’ Divergence Theorem:
\[\int_{V} \nabla \bullet \overrightarrow{G} d v=\oint_{A} \overrightarrow{G} \bullet \hat{n} d a \nonumber\]
Stokes’ Theorem:
\[ \mathrm{\int_{A}(\nabla \times \overrightarrow{G}) \bullet \hat{n} \ d a=\oint_{C} \overrightarrow{G} \bullet d} \overrightarrow{\ell} \nonumber\]
Fourier Transforms for pulse signals h(t):
\[\begin{aligned} \mathrm{\underline{H}(f)} &= \mathrm{\int_{-\infty}^{\infty} h(t) e^{-j 2 \pi t} d t} \\[4pt] \mathrm{h(t)} &= \mathrm{\int_{-\infty}^{\infty} \underline{H}(f) e^{+j 2 \pi f t} d f} \end{aligned}\]