Vector Properties

Vector PropertiesSwapnil Sunil JainJuly 18, 2006

Vector Properties

Some Unconventional Syntax:

$A^x{\widehat{e}}_x+A^y{\widehat{e}}_y+A^z{\widehat{e}}_z$$=$$A_x{\widehat{e}}_x+A_y{\widehat{e}}_y+A_{\varphi }{\widehat{e}}_z$

$dx\wedge dy$$=$$dxdy$

$dx\wedge dy\wedge dz$$=$$dxdydz$

${\displaystyle {\int }_{t_1}^{t_2}}𝑑tf(t)g(t)$$=$${\displaystyle {\int }_{t_1}^{t_2}}f(t)g(t)𝑑t$

${\partial }_x$$=$${\displaystyle \frac{\partial }{\partial x}}$

$d\mathrm{\Sigma }$$=$material surface element

$f(\overrightarrow{r})$$=$$f(x,y,z)$

${𝔼}^3$$=$${ℝ}^3$

Definitions:

$\overrightarrow{A}\circ \overrightarrow{B}$$\equiv$${\displaystyle \sum _{i=1}^3}{A}_i{B}_i$

$\overrightarrow{A}\times \overrightarrow{B}$$\equiv$

$[\overrightarrow{A},\overrightarrow{B},\overrightarrow{C}]$$\equiv$

$\mathrm{\Delta }$$\equiv$${\nabla }^2\mathit{\hspace{1em}}\text{(Laplace operator or Laplacian)}$

${\mathrm{\square }}^2$$\equiv$${\nabla }^2-{\displaystyle \frac{1}{c^2}}\partial _t{}^2\mathit{\hspace{1em}}\text{(D’Alembert or wave operator or D’Alembertian)}$

${\nabla }^2\overrightarrow{A}\equiv \nabla (\nabla \circ \overrightarrow{A})-\nabla \times (\nabla \times \overrightarrow{A})$$=$$({\nabla }^2{A}_x,{\nabla }^2{A}_y,{\nabla }^2{A}_z)={\displaystyle \frac{\partial \overrightarrow{A}}{\partial x}}+{\displaystyle \frac{\partial \overrightarrow{A}}{\partial y}}+{\displaystyle \frac{\partial \overrightarrow{A}}{\partial z}}\mathit{\hspace{1em}}\text{(Vector Laplacian)}$

$A\circ \nabla$$\equiv$$(a_1{\displaystyle \frac{\partial }{\partial x_1}}+a_2{\displaystyle \frac{\partial }{\partial x_2}}+\mathrm{\dots }+a_n{\displaystyle \frac{\partial }{\partial x_n}})$

$\nabla \circ \overrightarrow{F}$$=$$0\Rightarrow \overrightarrow{F}\text{ is incompressible or solenoidal or divergence-free}$

$\nabla \times \overrightarrow{F}$$=$$0\Rightarrow \overrightarrow{F}\text{ is irrotational or conservative or curl-free}$

$laplacian(\overrightarrow{F})$$\equiv$$div(grad(\overrightarrow{F}))$

$D_{\overrightarrow{u}}f$$\equiv$$\nabla f\circ \widehat{u}\mathit{\hspace{1em}\hspace{1em}}\text{(Directional Derivative)}$

$ro{t}_{\overrightarrow{u}}\overrightarrow{F}$$\equiv$$curl(\overrightarrow{F})\cdot \widehat{u}\mathit{\hspace{1em}\hspace{1em}}\text{(Rotational Derivative)}$

$helicity(\overrightarrow{F})$$\equiv$$curl(\overrightarrow{F})\circ \overrightarrow{F}$

$\nabla \circ \overrightarrow{F}$$=$$\underset{\delta V\to 0}{lim}{\displaystyle \frac{1}{\delta V}}{\displaystyle {∯}_{\delta S}}\overrightarrow{F}\circ \widehat{n}𝑑S$

$(\nabla \times \overrightarrow{F})\circ \overrightarrow{n}$$=$$\underset{\delta S\to 0}{lim}{\displaystyle \frac{1}{\delta S}}{\displaystyle {\oint }_{\delta C}}\overrightarrow{F}\circ \widehat{t}𝑑s$

$\nabla \times \overrightarrow{F}$$=$$\underset{\delta V\to 0}{lim}{\displaystyle \frac{1}{\delta V}}{\displaystyle {∯}_{\delta S}}\widehat{n}\times \overrightarrow{F}𝑑S$

$\nabla f$$=$$\underset{\delta V\to 0}{lim}{\displaystyle \frac{1}{\delta V}}{\displaystyle {∯}_{\delta S}}\widehat{n}f𝑑S$

$\nabla$$\equiv$${\displaystyle \sum _i}{\displaystyle \frac{{\widehat{e}}_i}{h_i}}{\displaystyle \frac{\partial }{\partial v_i}}$

$\nabla \circ$$\equiv$${\displaystyle \sum _i}{\displaystyle \frac{{\widehat{e}}_i}{h_i}}\circ {\displaystyle \frac{\partial }{\partial v_i}}$

$\nabla \times$$\equiv$${\displaystyle \sum _i}{\displaystyle \frac{{\widehat{e}}_i}{h_i}}\times {\displaystyle \frac{\partial }{\partial v_i}}$

Gradient, Divergence and Curl in Curvilinear Coordinates

$\nabla f={\displaystyle \frac{1}{h_1}}{\displaystyle \frac{\partial f}{\partial u_1}}{\widehat{e}}_1+{\displaystyle \frac{1}{h_2}}{\displaystyle \frac{\partial f}{\partial u_2}}{\widehat{e}}_2+{\displaystyle \frac{1}{h_3}}{\displaystyle \frac{\partial f}{\partial u_3}}{\widehat{e}}_3$

$\nabla \circ \overrightarrow{F}={\displaystyle \frac{1}{h_1{h}_2{h}_3}}\left[{\displaystyle \frac{\partial }{\partial u_1}}(h_2{h}_3{F}_1)+{\displaystyle \frac{\partial }{\partial u_2}}(h_1{h}_3{F}_2)+{\displaystyle \frac{\partial }{\partial u_3}}(h_1{h}_2{F}_3)\right]$

${\nabla }^{2}f={\displaystyle \frac{1}{h_1{h}_2{h}_3}}\left[{\displaystyle \frac{\partial }{\partial u_1}}\left({\displaystyle \frac{h_2{h}_3}{h_1}}{\displaystyle \frac{\partial f}{\partial u_1}}\right)+{\displaystyle \frac{\partial }{\partial u_2}}\left({\displaystyle \frac{h_1{h}_3}{h_2}}{\displaystyle \frac{\partial f}{\partial u_2}}\right)+{\displaystyle \frac{\partial }{\partial u_3}}\left({\displaystyle \frac{h_1{h}_2}{h_3}}{\displaystyle \frac{\partial f}{\partial u_3}}\right)\right]$

$\text{For Cartesian coordinates: }{h}_x=h_y=h_z=1$

$\text{For Cylindrical coordinates: }{h}_r=1,h_{\varphi }=r,h_z=1$

$\text{For Spherical coordinates: }{h}_r=1,h_{\theta }=r,h_{\varphi }=r\sin (\theta )$

$\nabla f$$=$${\displaystyle \sum _i}{\displaystyle \frac{1}{h_i}}{\displaystyle \frac{\partial f}{\partial q_i}}{\widehat{e}}_i$

$\nabla \times \overrightarrow{F}$$=$${\displaystyle \frac{1}{\mathrm{\Omega }}}{\displaystyle \sum _i}{\widehat{e}}_i{\displaystyle \sum _{jk}}{ϵ}_{ijk}{h}_i{\displaystyle \frac{\partial (h_k{F}_k)}{\partial q_j}}$

$\nabla \cdot \overrightarrow{F}$$=$${\displaystyle \sum _i}{\displaystyle \frac{1}{\mathrm{\Omega }}}{\displaystyle \frac{\partial }{\partial q_i}}\left({\displaystyle \frac{\mathrm{\Omega }{F}_i}{h_i}}\right)$

${\nabla }^{2}f$$=$${\displaystyle \frac{1}{\mathrm{\Omega }}}{\displaystyle \sum _i}{\displaystyle \frac{\partial }{\partial q_i}}\left({\displaystyle \frac{\mathrm{\Omega }}{h_i^2}}{\displaystyle \frac{\partial f}{\partial q_i}}\right)$

$\text{where }\mathrm{\Omega }\equiv \mathrm{\Pi }{h}_i$

Inequalities:

$\parallel \overrightarrow{u}+\overrightarrow{v}\parallel \le \parallel \overrightarrow{u}\parallel +\parallel \overrightarrow{v}\parallel \mathit{\hspace{1em}\hspace{1em}}\text{Triangle Inequality}$

$|\overrightarrow{u}\circ \overrightarrow{v}|\le \parallel \overrightarrow{u}\parallel \parallel \overrightarrow{v}\parallel \mathit{\hspace{1em}\hspace{1em}}\text{Cauchy-Schwarz Inequality}$

Product identities:

$\overrightarrow{A}\circ (\overrightarrow{B}\times \overrightarrow{C})$$=$$\overrightarrow{B}\circ (\overrightarrow{C}\times \overrightarrow{A})=\overrightarrow{C}\circ (\overrightarrow{A}\times \overrightarrow{B})\mathit{\hspace{1em}\hspace{1em}}\text{Scalar Triple Product}$

$\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C})$$=$$\overrightarrow{B}(\overrightarrow{A}\circ \overrightarrow{C})-\overrightarrow{C}(\overrightarrow{A}\circ \overrightarrow{B})\mathit{\hspace{1em}\hspace{1em}}\text{Vector Triple Product}$

$\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C})+\overrightarrow{B}\times (\overrightarrow{C}\times \overrightarrow{A})+\overrightarrow{C}\times (\overrightarrow{A}\times \overrightarrow{B})$$=$$0\mathit{\hspace{1em}\hspace{1em}}\text{Jacobi Identity}$

$(\overrightarrow{A}\times \overrightarrow{B})\circ (\overrightarrow{C}\times \overrightarrow{D})$$=$$\overrightarrow{A}\circ [\overrightarrow{B}\times (\overrightarrow{C}\times \overrightarrow{D})]=(\overrightarrow{A}\circ \overrightarrow{C})(\overrightarrow{B}\circ \overrightarrow{D})-(\overrightarrow{B}\circ \overrightarrow{C})(\overrightarrow{A}\circ \overrightarrow{D})\mathit{\hspace{1em}}\text{Scalar Quadruple Product}$

$(\overrightarrow{A}\times \overrightarrow{B})\times (\overrightarrow{C}\times \overrightarrow{D})$$=$$(\overrightarrow{A}\times \overrightarrow{B}\circ \overrightarrow{D})\overrightarrow{C}-(\overrightarrow{A}\times \overrightarrow{B}\circ \overrightarrow{C})\overrightarrow{D}=[\overrightarrow{C},\overrightarrow{D},\overrightarrow{A}]\overrightarrow{B}-[\overrightarrow{C},\overrightarrow{D},\overrightarrow{B}]\overrightarrow{A}\mathit{\hspace{1em}}\text{Vector Quadruple Product}$

$\overrightarrow{A}\times [\overrightarrow{B}\times (\overrightarrow{C}\times \overrightarrow{D})]$$=$$(\overrightarrow{A}\times \overrightarrow{C})(\overrightarrow{B}\circ \overrightarrow{D})-(\overrightarrow{B}\circ \overrightarrow{C})(\overrightarrow{A}\times \overrightarrow{D})$

Gradient Identities:

$\nabla (f+g)$$=$$\nabla f+\nabla g$

$\nabla (fg)$$=$$f\nabla g+g\nabla f$

$\nabla (\overrightarrow{A}\circ \overrightarrow{B})$$=$$\overrightarrow{A}\times (\nabla \times \overrightarrow{B})+\overrightarrow{B}\times (\nabla \times \overrightarrow{A})+(\overrightarrow{A}\circ \nabla )\overrightarrow{B}+(\overrightarrow{B}\circ \nabla )\overrightarrow{A}$

$\nabla (\overrightarrow{A}\times \overrightarrow{B})$$=$$(\nabla \overrightarrow{A})\times \overrightarrow{B}-(\nabla \overrightarrow{B})\times \overrightarrow{A}\mathit{\hspace{1em}\hspace{1em}}\text{gradient of vector??}$

$\nabla (f\overrightarrow{A})$$=$$(\nabla f)\overrightarrow{A}+f(\nabla \overrightarrow{A})\mathit{\hspace{1em}\hspace{1em}}\text{gradient of vector??}$

Divergence Identities:

$\nabla \circ (\overrightarrow{A}+\overrightarrow{B})$$=$$\nabla \circ \overrightarrow{A}+\nabla \circ \overrightarrow{B}$

$\nabla \circ (f\overrightarrow{A})$$=$$f(\nabla \circ \overrightarrow{A})+\overrightarrow{A}\circ (\nabla f)$

$\nabla \circ (\overrightarrow{A}\times \overrightarrow{B})$$=$$\overrightarrow{B}\circ (\nabla \times \overrightarrow{A})-\overrightarrow{A}\circ (\nabla \times \overrightarrow{B})$

$\nabla \circ (\overrightarrow{A}\overrightarrow{B})$$=$$(\nabla \circ \overrightarrow{A})\overrightarrow{B}+\overrightarrow{A}\circ (\nabla \overrightarrow{B})=(\nabla \circ \overrightarrow{A})\overrightarrow{B}+(\overrightarrow{A}\circ \nabla )\overrightarrow{B}$

Curl Identities:

$\nabla \times (\overrightarrow{A}+\overrightarrow{B})$$=$$\nabla \times \overrightarrow{A}+\nabla \times \overrightarrow{B}$

$\nabla \times (f\overrightarrow{A})$$=$$f(\nabla \times \overrightarrow{A})-\overrightarrow{A}\times (\nabla f)$

$\nabla \times (\overrightarrow{A}\times \overrightarrow{B})$$=$$(\nabla \circ \overrightarrow{B})\overrightarrow{A}-(\overrightarrow{A}\circ \nabla )\overrightarrow{B}+(\overrightarrow{B}\circ \nabla )\overrightarrow{A}-(\nabla \circ \overrightarrow{A})\overrightarrow{B}$

$\nabla \times (\overrightarrow{A}\overrightarrow{B})$$=$$(\nabla \circ \overrightarrow{A})\overrightarrow{B}-\overrightarrow{A}\times (\nabla \overrightarrow{B})$

Laplacian Identities:

${\nabla }^2(fg)$$=$$g{\nabla }^{2}f+2\nabla f\circ \nabla g+f{\nabla }^{2}g$

${\nabla }^2(f\overrightarrow{A})$$=$$=f{\nabla }^2\overrightarrow{A}+\overrightarrow{A}{\nabla }^{2}f+2(\nabla f\circ \nabla )\overrightarrow{A}$

Mixed Identities:

$\nabla \circ (\nabla \times \overrightarrow{A})$$=$$0$

$\nabla \times \nabla f$$=$$0$

$\nabla \times (f\nabla g)$$=$$\nabla f\times \nabla g$

$\nabla \circ f\nabla g$$=$$f{\nabla }^{2}g+\nabla f\circ \nabla g$

$\nabla \times (\nabla \times \overrightarrow{A})$$=$$\nabla (\nabla \circ \overrightarrow{A})-{\nabla }^2\overrightarrow{A}\mathit{\hspace{1em}}\text{only true for rectangular coordinates}$

Differential Identities:

${\displaystyle \frac{d}{dt}}(f\overrightarrow{A})$$=$$f{\displaystyle \frac{d\overrightarrow{A}}{dt}}+{\displaystyle \frac{df}{dt}}\overrightarrow{A}$

${\displaystyle \frac{d}{dt}}(\overrightarrow{A}\circ \overrightarrow{B})$$=$$\overrightarrow{A}\circ {\displaystyle \frac{d\overrightarrow{B}}{dt}}+\overrightarrow{B}\circ {\displaystyle \frac{d\overrightarrow{A}}{dt}}$

${\displaystyle \frac{d}{dt}}(\overrightarrow{A}\times \overrightarrow{B})$$=$$\overrightarrow{A}\times {\displaystyle \frac{d\overrightarrow{B}}{dt}}+\overrightarrow{B}\times {\displaystyle \frac{d\overrightarrow{A}}{dt}}$

Integral Identities:

Gauss’/Divergence Thm:

Standard Form:

${\displaystyle {∯}_{𝒮}}\overrightarrow{F}\circ 𝑑\overrightarrow{A}$$=$${\displaystyle {\iiint }_{𝒱}}(\nabla \cdot \overrightarrow{F})𝑑V\mathit{\hspace{1em}}(𝒮\mathrm{encloses}𝒱)$

Variants:

${\displaystyle {∯}_{𝒮}}\overrightarrow{F}\circ \overrightarrow{n}𝑑S$$=$${\displaystyle {\iiint }_{𝒱}}(\nabla \cdot \overrightarrow{F})𝑑V\mathit{\hspace{1em}}(𝒮\mathrm{encloses}𝒱)$

Stokes’ Thm:

Standard Form:

${\displaystyle {\oint }_{𝒞}}\overrightarrow{F}\circ 𝑑\overrightarrow{l}$$=$${\displaystyle {\iint }_{𝒮}}(\nabla \times \overrightarrow{F})\circ \overrightarrow{n}𝑑S\mathit{\hspace{1em}}(𝒮\mathrm{bounded}\mathrm{by}𝒞)$

Variants:

${\displaystyle {\oint }_{𝒞}}\overrightarrow{F}\circ 𝑑\overrightarrow{l}$$=$${\displaystyle {\iint }_{𝒮}}(\nabla \times \overrightarrow{F})\circ 𝑑\overrightarrow{A}\mathit{\hspace{1em}}(𝒮\mathrm{bounded}\mathrm{by}𝒞)$

${\displaystyle {\oint }_{𝒞}}\overrightarrow{F}\circ \overrightarrow{t}𝑑s$$=$${\displaystyle {\iint }_{𝒮}}(\nabla \times \overrightarrow{F})\circ \overrightarrow{n}𝑑S\mathit{\hspace{1em}}(𝒮\mathrm{bounded}\mathrm{by}𝒞)$

${\displaystyle {\oint }_{𝒞}}\overrightarrow{F}\circ \overrightarrow{t}𝑑s$$=$${\displaystyle {\iint }_{𝒮}}(\nabla \times \overrightarrow{F})\circ 𝑑\overrightarrow{A}\mathit{\hspace{1em}}(𝒮\mathrm{bounded}\mathrm{by}𝒞)$

Gradient theorem:

Standard Form:

${\displaystyle {\int }_{𝒞}}(\nabla f)\circ 𝑑\overrightarrow{l}$$=$$f(r_2)-f(r_1)\mathit{\hspace{1em}}(𝒞\mathrm{goes}\mathrm{from}{\mathrm{r}}_1\mathrm{to}{\mathrm{r}}_2)$

Integration by Parts for Vectors

${\displaystyle {\iiint }_{𝒱}}f(\nabla \circ \overrightarrow{A})𝑑V$$=$${\displaystyle {\iint }_{𝒮}}f\overrightarrow{A}\circ 𝑑\overrightarrow{a}-{\displaystyle {\iiint }_{𝒱}}\overrightarrow{A}\circ (\nabla f)𝑑V$

Integral Form of Maxwell’s Equations:

${\displaystyle {∯}_{𝒮}}\overrightarrow{D}\circ \widehat{n}{d}^{2}A$$=$$Q_{free}^{enc}$

${\displaystyle {∯}_{𝒮}}\overrightarrow{B}\circ \widehat{n}{d}^{2}A$$=$$0$

${\displaystyle {\oint }_{𝒞}}\overrightarrow{E}\circ 𝑑\overrightarrow{l}$$=$$-{\displaystyle \frac{d{\mathrm{\Phi }}_B}{dt}}$

${\displaystyle {\oint }_{𝒞}}\overrightarrow{H}\circ 𝑑\overrightarrow{l}$$=$$I_{free}^{enc}+{\displaystyle \frac{d{\mathrm{\Phi }}_D}{dt}}$

${\mathrm{\Phi }}_B$$=$${\displaystyle {\iint }_{𝒮}}\overrightarrow{B}\circ \widehat{n}{d}^{2}A$

${\mathrm{\Phi }}_D$$=$${\displaystyle {\iint }_{𝒮}}\overrightarrow{D}\circ \widehat{n}{d}^{2}A$

Differential Form of Maxwell’s Equations:

$\nabla \circ \overrightarrow{D}={\rho }_{free}$

$\nabla \circ \overrightarrow{B}=0$

$\nabla \times \overrightarrow{E}=-{\displaystyle \frac{\partial \overrightarrow{B}}{\partial t}}$

$\nabla \times \overrightarrow{H}={\overrightarrow{J}}_{free}+{\displaystyle \frac{\partial \overrightarrow{D}}{\partial t}}$

Complex Differential Form of Maxwell’s Equations:

$\nabla \circ \overrightarrow{M}={\displaystyle \frac{i\rho }{{ϵ}_o}}$

$\nabla \times \overrightarrow{M}=c{\mu }_0\overrightarrow{J}-{\displaystyle \frac{i}{c}}{\displaystyle \frac{\partial \overrightarrow{M}}{\partial t}}$

$\text{where }\overrightarrow{M}\equiv c\overrightarrow{B}+i\overrightarrow{E}$

EM Equations:

${\nabla }^2\overrightarrow{A}-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^2\overrightarrow{A}}{\partial t^2}}$$=$$-{\mu }_0\overrightarrow{J}\mathit{\hspace{1em}\hspace{1em}}\text{Vector Poisson’s Equation}$

${\nabla }^2\overrightarrow{V}-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\partial }^{2}V}{\partial t^2}}$$=$${\displaystyle \frac{-\rho }{ϵ}}\mathit{\hspace{1em}\hspace{1em}}\text{Scalar Poisson’s Equation}$

$\overrightarrow{A}(\overrightarrow{r})$$=$${\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \int \frac{\overrightarrow{J}(\overrightarrow{r^{\prime }})}{|\overrightarrow{r}-\overrightarrow{r^{\prime }}|}{d}^3\overrightarrow{r^{\prime }}}$

$\varphi (\overrightarrow{r})$$=$${\displaystyle \frac{1}{4\pi {ϵ}_0}}{\displaystyle \int \frac{\rho (\overrightarrow{r^{\prime }})}{|\overrightarrow{r}-\overrightarrow{r^{\prime }}|}{d}^3\overrightarrow{r^{\prime }}}$