# Vector Properties

Vector Properties Swapnil Sunil Jain July 18, 2006

Vector Properties

Some Unconventional Syntax:

 $\displaystyle A^{x}\hat{e}_{x}+A^{y}\hat{e}_{y}+A^{z}\hat{e}_{z}$ $\displaystyle=$ $\displaystyle A_{x}\hat{e}_{x}+A_{y}\hat{e}_{y}+A_{\phi}\hat{e}_{z}$ $\displaystyle dx\wedge dy$ $\displaystyle=$ $\displaystyle dxdy$ $\displaystyle dx\wedge dy\wedge dz$ $\displaystyle=$ $\displaystyle dxdydz$ $\displaystyle\int_{t_{1}}^{t_{2}}dtf(t)g(t)$ $\displaystyle=$ $\displaystyle\int_{t_{1}}^{t_{2}}f(t)g(t)dt$ $\displaystyle{\partial}_{x}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial x}$ $\displaystyle d\Sigma$ $\displaystyle=$ material surface element $\displaystyle f(\vec{r})$ $\displaystyle=$ $\displaystyle f(x,y,z)$ $\displaystyle\mathbb{E}^{3}$ $\displaystyle=$ $\displaystyle\mathbb{R}^{3}$

 $\displaystyle\vec{A}\circ\vec{B}$ $\displaystyle\equiv$ $\displaystyle\sum_{i=1}^{3}A_{i}B_{i}$
 $\displaystyle\vec{A}\times\vec{B}$ $\displaystyle\equiv$ $\displaystyle\left|\begin{array}[]{ccc}\mathbb{\hat{e}_{x}}&\mathbb{\hat{e}_{y% }}&\mathbb{\hat{e}_{z}}\\ A_{x}&A_{y}&A_{z}\\ B_{x}&B_{y}&B_{z}\end{array}\right|$
 $\displaystyle[\vec{A},\vec{B},\vec{C}]$ $\displaystyle\equiv$ $\displaystyle\vec{A}\circ(\vec{B}\times\vec{C})=(\vec{A}\times\vec{B})\circ% \vec{C}=\Bigg{|}\begin{array}[]{ccc}A_{x}&A_{y}&A_{z}\\ B_{x}&B_{y}&B_{z}\\ C_{x}&C_{y}&C_{z}\end{array}\Bigg{|}$
 $\displaystyle\Delta$ $\displaystyle\equiv$ $\displaystyle{\nabla}^{2}\quad\mbox{(Laplace operator or Laplacian)}$ $\displaystyle{\Box}^{2}$ $\displaystyle\equiv$ $\displaystyle{\nabla}^{2}-\frac{1}{c^{2}}{{\partial}_{t}}^{2}\quad\mbox{(D'% Alembert or wave operator or D'Alembertian)}$ $\displaystyle{\nabla}^{2}\vec{A}\equiv\nabla(\nabla\circ\vec{A})-\nabla\times(% \nabla\times\vec{A})$ $\displaystyle=$ $\displaystyle({\nabla}^{2}A_{x},{\nabla}^{2}A_{y},{\nabla}^{2}A_{z})=\frac{% \partial\vec{A}}{\partial x}+\frac{\partial\vec{A}}{\partial y}+\frac{\partial% \vec{A}}{\partial z}\quad\mbox{(Vector Laplacian)}$ $\displaystyle A\circ\nabla$ $\displaystyle\equiv$ $\displaystyle(a_{1}\frac{\partial}{\partial x_{1}}+a_{2}\frac{\partial}{% \partial x_{2}}+\ldots+a_{n}\frac{\partial}{\partial x_{n}})$
 $\displaystyle\nabla\circ\vec{F}$ $\displaystyle=$ $\displaystyle 0\Rightarrow\vec{F}\mbox{ is incompressible or solenoidal or % divergence-free}$ $\displaystyle\nabla\times\vec{F}$ $\displaystyle=$ $\displaystyle 0\Rightarrow\vec{F}\mbox{ is irrotational or conservative or % curl-free}$ $\displaystyle laplacian(\vec{F})$ $\displaystyle\equiv$ $\displaystyle div(grad(\vec{F}))$ $\displaystyle D_{\vec{u}}f$ $\displaystyle\equiv$ $\displaystyle\nabla f\circ\hat{u}\qquad\mbox{(Directional Derivative)}$ $\displaystyle rot_{\vec{u}}\vec{F}$ $\displaystyle\equiv$ $\displaystyle curl(\vec{F})\cdot\hat{u}\qquad\mbox{(Rotational Derivative)}$ $\displaystyle helicity(\vec{F})$ $\displaystyle\equiv$ $\displaystyle curl(\vec{F})\circ\vec{F}$
 $\displaystyle\nabla\circ\vec{F}$ $\displaystyle=$ $\displaystyle\lim_{\delta V\to 0}\frac{1}{\delta V}\oiint_{\delta S}\vec{F}% \circ\hat{n}\;dS$ $\displaystyle(\nabla\times\vec{F})\circ\vec{n}$ $\displaystyle=$ $\displaystyle\lim_{\delta S\to 0}\frac{1}{\delta S}\oint_{\delta C}\vec{F}% \circ\hat{t}\;ds$ $\displaystyle\nabla\times\vec{F}$ $\displaystyle=$ $\displaystyle\lim_{\delta V\to 0}\frac{1}{\delta V}\oiint_{\delta S}\hat{n}% \times\vec{F}\;dS$ $\displaystyle\nabla f$ $\displaystyle=$ $\displaystyle\lim_{\delta V\to 0}\frac{1}{\delta V}\oiint_{\delta S}\hat{n}f\;dS$ $\displaystyle\nabla$ $\displaystyle\equiv$ $\displaystyle\sum_{i}\frac{\hat{e}_{i}}{h_{i}}\frac{\partial}{\partial v_{i}}$ $\displaystyle\nabla\circ$ $\displaystyle\equiv$ $\displaystyle\sum_{i}\frac{\hat{e}_{i}}{h_{i}}\circ\frac{\partial}{\partial v_% {i}}$ $\displaystyle\nabla\times$ $\displaystyle\equiv$ $\displaystyle\sum_{i}\frac{\hat{e}_{i}}{h_{i}}\times\frac{\partial}{\partial v% _{i}}$

Gradient, Divergence and Curl in Curvilinear Coordinates

 $\displaystyle\nabla f=\frac{1}{h_{1}}\frac{\partial f}{\partial u_{1}}\hat{e}_% {1}+\frac{1}{h_{2}}\frac{\partial f}{\partial u_{2}}\hat{e}_{2}+\frac{1}{h_{3}% }\frac{\partial f}{\partial u_{3}}\hat{e}_{3}$ $\displaystyle\nabla\circ\vec{F}=\frac{1}{h_{1}h_{2}h_{3}}\bigg{[}\frac{% \partial}{\partial u_{1}}(h_{2}h_{3}F_{1})+\frac{\partial}{\partial u_{2}}(h_{% 1}h_{3}F_{2})+\frac{\partial}{\partial u_{3}}(h_{1}h_{2}F_{3})\bigg{]}$ $\displaystyle{\nabla}^{2}f=\frac{1}{h_{1}h_{2}h_{3}}\bigg{[}\frac{\partial}{% \partial u_{1}}\bigg{(}\frac{h_{2}h_{3}}{h_{1}}\frac{\partial f}{\partial u_{1% }}\bigg{)}+\frac{\partial}{\partial u_{2}}\bigg{(}\frac{h_{1}h_{3}}{h_{2}}% \frac{\partial f}{\partial u_{2}}\bigg{)}+\frac{\partial}{\partial u_{3}}\bigg% {(}\frac{h_{1}h_{2}}{h_{3}}\frac{\partial f}{\partial u_{3}}\bigg{)}\bigg{]}$ $\displaystyle\nabla\times\vec{F}=\frac{1}{h_{1}h_{2}h_{3}}\left|\begin{array}[% ]{ccc}h_{1}\hat{e}_{1}&h_{2}\hat{e}_{2}&h_{3}\hat{e}_{3}\\ \frac{\partial}{\partial u_{1}}&\frac{\partial}{\partial u_{2}}&\frac{\partial% }{\partial u_{3}}\\ h_{1}F_{1}&h_{2}F_{2}&h_{3}F_{3}\end{array}\right|$ $\displaystyle\text{ For Cartesian coordinates: }h_{x}=h_{y}=h_{z}=1$ $\displaystyle\text{ For Cylindrical coordinates: }h_{r}=1,h_{\phi}=r,h_{z}=1$ $\displaystyle\text{ For Spherical coordinates: }h_{r}=1,h_{\theta}=r,h_{\phi}=% r\sin(\theta)$
 $\displaystyle\nabla f$ $\displaystyle=$ $\displaystyle\sum_{i}\frac{1}{h_{i}}\frac{\partial f}{\partial q_{i}}\hat{e}_{i}$ $\displaystyle\nabla\times\vec{F}$ $\displaystyle=$ $\displaystyle\frac{1}{\Omega}\sum_{i}\hat{e}_{i}\sum_{jk}\epsilon_{ijk}h_{i}% \frac{\partial(h_{k}F_{k})}{\partial q_{j}}$ $\displaystyle\nabla\cdot\vec{F}$ $\displaystyle=$ $\displaystyle\sum_{i}\frac{1}{\Omega}\frac{\partial}{\partial q_{i}}\Big{(}% \frac{\Omega F_{i}}{h_{i}}\Big{)}$ $\displaystyle\nabla^{2}f$ $\displaystyle=$ $\displaystyle\frac{1}{\Omega}\sum_{i}\frac{\partial}{\partial q_{i}}\Big{(}% \frac{\Omega}{h_{i}^{2}}\frac{\partial f}{\partial q_{i}}\Big{)}$ $\displaystyle\text{ where }\Omega\equiv\Pi h_{i}$

 $\displaystyle\parallel\vec{u}+\vec{v}\parallel\;\leq\;\parallel\vec{u}% \parallel+\parallel\vec{v}\parallel\qquad\mbox{Triangle Inequality}$ $\displaystyle|\vec{u}\circ\vec{v}|\;\leq\;\parallel\vec{u}\parallel\parallel% \vec{v}\parallel\qquad\mbox{Cauchy-Schwarz Inequality}$

Product identities:

 $\displaystyle\vec{A}\circ(\vec{B}\times\vec{C})$ $\displaystyle=$ $\displaystyle\vec{B}\circ(\vec{C}\times\vec{A})=\vec{C}\circ(\vec{A}\times\vec% {B})\qquad\mbox{Scalar Triple Product}$ $\displaystyle\vec{A}\times(\vec{B}\times\vec{C})$ $\displaystyle=$ $\displaystyle\vec{B}(\vec{A}\circ\vec{C})-\vec{C}(\vec{A}\circ\vec{B})\qquad% \mbox{Vector Triple Product}$ $\displaystyle\vec{A}\times(\vec{B}\times\vec{C})+\vec{B}\times(\vec{C}\times% \vec{A})+\vec{C}\times(\vec{A}\times\vec{B})$ $\displaystyle=$ $\displaystyle 0\qquad\mbox{Jacobi Identity}$
 $\displaystyle(\vec{A}\times\vec{B})\circ(\vec{C}\times\vec{D})$ $\displaystyle=$ $\displaystyle\vec{A}\circ[\vec{B}\times(\vec{C}\times\vec{D})]=(\vec{A}\circ% \vec{C})(\vec{B}\circ\vec{D})-(\vec{B}\circ\vec{C})(\vec{A}\circ\vec{D})\quad% \mbox{Scalar Quadruple Product}$ $\displaystyle(\vec{A}\times\vec{B})\times(\vec{C}\times\vec{D})$ $\displaystyle=$ $\displaystyle(\vec{A}\times\vec{B}\circ\vec{D})\vec{C}-(\vec{A}\times\vec{B}% \circ\vec{C})\vec{D}=[\vec{C},\vec{D},\vec{A}]\vec{B}-[\vec{C},\vec{D},\vec{B}% ]\vec{A}\quad\mbox{Vector Quadruple Product}$ $\displaystyle\vec{A}\times[\vec{B}\times(\vec{C}\times\vec{D})]$ $\displaystyle=$ $\displaystyle(\vec{A}\times\vec{C})(\vec{B}\circ\vec{D})-(\vec{B}\circ\vec{C})% (\vec{A}\times\vec{D})$

 $\displaystyle\nabla(f+g)$ $\displaystyle=$ $\displaystyle\nabla f+\nabla g$ $\displaystyle\nabla(fg)$ $\displaystyle=$ $\displaystyle f\nabla g+g\nabla f$ $\displaystyle\nabla(\vec{A}\circ\vec{B})$ $\displaystyle=$ $\displaystyle\vec{A}\times(\nabla\times\vec{B})+\vec{B}\times(\nabla\times\vec% {A})+(\vec{A}\circ\nabla)\vec{B}+(\vec{B}\circ\nabla)\vec{A}$ $\displaystyle\nabla(\vec{A}\times\vec{B})$ $\displaystyle=$ $\displaystyle(\nabla\vec{A})\times\vec{B}-(\nabla\vec{B})\times\vec{A}\qquad% \mbox{gradient of vector??}$ $\displaystyle\nabla(f\vec{A})$ $\displaystyle=$ $\displaystyle(\nabla f)\vec{A}+f(\nabla\vec{A})\qquad\mbox{gradient of vector??}$

Divergence Identities:

 $\displaystyle\nabla\circ(\vec{A}+\vec{B})$ $\displaystyle=$ $\displaystyle\nabla\circ\vec{A}+\nabla\circ\vec{B}$ $\displaystyle\nabla\circ(f\vec{A})$ $\displaystyle=$ $\displaystyle f(\nabla\circ\vec{A})+\vec{A}\circ(\nabla f)$ $\displaystyle\nabla\circ(\vec{A}\times\vec{B})$ $\displaystyle=$ $\displaystyle\vec{B}\circ(\nabla\times\vec{A})-\vec{A}\circ(\nabla\times\vec{B})$ $\displaystyle\nabla\circ(\vec{A}\vec{B})$ $\displaystyle=$ $\displaystyle(\nabla\circ\vec{A})\vec{B}+\vec{A}\circ(\nabla\vec{B})=(\nabla% \circ\vec{A})\vec{B}+(\vec{A}\circ\nabla)\vec{B}$

Curl Identities:

 $\displaystyle\nabla\times(\vec{A}+\vec{B})$ $\displaystyle=$ $\displaystyle\nabla\times\vec{A}+\nabla\times\vec{B}$ $\displaystyle\nabla\times(f\vec{A})$ $\displaystyle=$ $\displaystyle f(\nabla\times\vec{A})-\vec{A}\times(\nabla f)$ $\displaystyle\nabla\times(\vec{A}\times\vec{B})$ $\displaystyle=$ $\displaystyle(\nabla\circ\vec{B})\vec{A}-(\vec{A}\circ\nabla)\vec{B}+(\vec{B}% \circ\nabla)\vec{A}-(\nabla\circ\vec{A})\vec{B}$ $\displaystyle\nabla\times(\vec{A}\vec{B})$ $\displaystyle=$ $\displaystyle(\nabla\circ\vec{A})\vec{B}-\vec{A}\times(\nabla\vec{B})$

Laplacian Identities:

 $\displaystyle{\nabla}^{2}(fg)$ $\displaystyle=$ $\displaystyle g{\nabla}^{2}f+2\nabla f\circ\nabla g+f{\nabla}^{2}g$ $\displaystyle{\nabla}^{2}(f\vec{A})$ $\displaystyle=$ $\displaystyle=f{\nabla}^{2}\vec{A}+\vec{A}{\nabla}^{2}f+2(\nabla f\circ\nabla)% \vec{A}$

Mixed Identities:

 $\displaystyle\nabla\circ(\nabla\times\vec{A})$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle\nabla\times\nabla f$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle\nabla\times(f\nabla g)$ $\displaystyle=$ $\displaystyle\nabla f\times\nabla g$ $\displaystyle\nabla\circ f\nabla g$ $\displaystyle=$ $\displaystyle f{\nabla}^{2}g+\nabla f\circ\nabla g$ $\displaystyle\nabla\times(\nabla\times\vec{A})$ $\displaystyle=$ $\displaystyle\nabla(\nabla\circ\vec{A})-{\nabla}^{2}\vec{A}\quad\mbox{only % true for rectangular coordinates}$

Differential Identities:

 $\displaystyle\frac{d}{dt}(f\vec{A})$ $\displaystyle=$ $\displaystyle f\frac{d\vec{A}}{dt}+\frac{df}{dt}\vec{A}$ $\displaystyle\frac{d}{dt}(\vec{A}\circ\vec{B})$ $\displaystyle=$ $\displaystyle\vec{A}\circ\frac{d\vec{B}}{dt}+\vec{B}\circ\frac{d\vec{A}}{dt}$ $\displaystyle\frac{d}{dt}(\vec{A}\times\vec{B})$ $\displaystyle=$ $\displaystyle\vec{A}\times\frac{d\vec{B}}{dt}+\vec{B}\times\frac{d\vec{A}}{dt}$

Integral Identities:

Gauss’/Divergence Thm:

Standard Form:

 $\displaystyle\oiint_{\mathcal{S}}\vec{F}\circ d\vec{A}$ $\displaystyle=$ $\displaystyle\iiint_{\mathcal{V}}(\nabla\cdot\vec{F})\;dV\quad\mathrm{(% \mathcal{S}\,encloses\mathcal{V})}$

Variants:

 $\displaystyle\oiint_{\mathcal{S}}\vec{F}\circ\vec{n}\;dS$ $\displaystyle=$ $\displaystyle\iiint_{\mathcal{V}}(\nabla\cdot\vec{F})\;dV\quad\mathrm{(% \mathcal{S}\,encloses\mathcal{V})}$

Stokes’ Thm:

Standard Form:

 $\displaystyle\oint_{\mathcal{C}}\vec{F}\circ d\vec{l}$ $\displaystyle=$ $\displaystyle\iint_{\mathcal{S}}(\nabla\times\vec{F})\circ\vec{n}\;dS\quad% \mathrm{(\mathcal{S}\,bounded\,by\,\mathcal{C})}$

Variants:

 $\displaystyle\oint_{\mathcal{C}}\vec{F}\circ d\vec{l}$ $\displaystyle=$ $\displaystyle\iint_{\mathcal{S}}(\nabla\times\vec{F})\circ d\vec{A}\quad% \mathrm{(\mathcal{S}\,bounded\,by\,\mathcal{C})}$ $\displaystyle\oint_{\mathcal{C}}\vec{F}\circ\vec{t}\;ds$ $\displaystyle=$ $\displaystyle\iint_{\mathcal{S}}(\nabla\times\vec{F})\circ\vec{n}\;dS\quad% \mathrm{(\mathcal{S}\,bounded\,by\,\mathcal{C})}$ $\displaystyle\oint_{\mathcal{C}}\vec{F}\circ\vec{t}\;ds$ $\displaystyle=$ $\displaystyle\iint_{\mathcal{S}}(\nabla\times\vec{F})\circ d\vec{A}\quad% \mathrm{(\mathcal{S}\,bounded\,by\,\mathcal{C})}$

Standard Form:

 $\displaystyle\int_{\mathcal{C}}(\nabla f)\circ d\vec{l}$ $\displaystyle=$ $\displaystyle f(r_{2})-f(r_{1})\quad\mathrm{(\mathcal{C}\,goes\,from\,r_{1}\,% to\,r_{2})}$

Integration by Parts for Vectors

 $\displaystyle\iiint_{\mathcal{V}}f(\nabla\circ\vec{A})dV$ $\displaystyle=$ $\displaystyle\iint_{\mathcal{S}}f\vec{A}\circ d\vec{a}-\iiint_{\mathcal{V}}% \vec{A}\circ(\nabla f)dV$

Integral Form of Maxwell’s Equations:

 $\displaystyle\oiint_{\mathcal{S}}\vec{D}\circ\hat{n}\;d^{2}A$ $\displaystyle=$ $\displaystyle Q_{free}^{enc}$ $\displaystyle\oiint_{\mathcal{S}}\vec{B}\circ\hat{n}\;d^{2}A$ $\displaystyle=$ $\displaystyle 0$ $\displaystyle\oint_{\mathcal{C}}\vec{E}\circ d\vec{l}$ $\displaystyle=$ $\displaystyle-\frac{d{\Phi_{B}}}{dt}$ $\displaystyle\oint_{\mathcal{C}}\vec{H}\circ d\vec{l}$ $\displaystyle=$ $\displaystyle I_{free}^{enc}+\frac{d{\Phi_{D}}}{dt}$ $\displaystyle\Phi_{B}$ $\displaystyle=$ $\displaystyle\iint_{\mathcal{S}}\vec{B}\circ\hat{n}\;d^{2}A$ $\displaystyle\Phi_{D}$ $\displaystyle=$ $\displaystyle\iint_{\mathcal{S}}\vec{D}\circ\hat{n}\;d^{2}A$

Differential Form of Maxwell’s Equations:

 $\displaystyle\nabla\circ\vec{D}=\rho_{free}$ $\displaystyle\nabla\circ\vec{B}=0$ $\displaystyle\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$ $\displaystyle\nabla\times\vec{H}=\vec{J}_{free}+\frac{\partial\vec{D}}{% \partial t}$

Complex Differential Form of Maxwell’s Equations:

 $\displaystyle\nabla\circ\vec{M}=\frac{i\rho}{\epsilon_{o}}$ $\displaystyle\nabla\times\vec{M}=c\mu_{0}\vec{J}-\frac{i}{c}\frac{\partial\vec% {M}}{\partial t}$ $\displaystyle\text{ where }\vec{M}\equiv c\vec{B}+i\vec{E}$

EM Equations:

 $\displaystyle{\nabla}^{2}\vec{A}-\frac{1}{c^{2}}\frac{{\partial}^{2}\vec{A}}{% \partial t^{2}}$ $\displaystyle=$ $\displaystyle-{\mu}_{0}\vec{J}\qquad\mbox{Vector Poisson's Equation}$ $\displaystyle{\nabla}^{2}\vec{V}-\frac{1}{c^{2}}\frac{{\partial}^{2}V}{% \partial t^{2}}$ $\displaystyle=$ $\displaystyle\frac{-\rho}{\epsilon}\qquad\mbox{Scalar Poisson's Equation}$ $\displaystyle\vec{A}(\vec{r})$ $\displaystyle=$ $\displaystyle\frac{{\mu}_{0}}{4\pi}\int\frac{\vec{J}(\vec{r^{\prime}})}{|\vec{% r}-\vec{r^{\prime}}|}d^{3}\vec{r^{\prime}}$ $\displaystyle\phi(\vec{r})$ $\displaystyle=$ $\displaystyle\frac{1}{4\pi{\epsilon}_{0}}\int\frac{\rho(\vec{r^{\prime}})}{|% \vec{r}-\vec{r^{\prime}}|}d^{3}\vec{r^{\prime}}$
Title Vector Properties VectorProperties1 2013-03-11 19:25:33 2013-03-11 19:25:33 swapnizzle (13346) (0) 1 swapnizzle (0) Definition