notes on vector

Vector

position vector

$\vec{r}_{12} = \vec{r}\;to\;1\;from\;2$

Force vector

$\vec{F}_{12} = \vec{F}\;on\;1\;by\;2$

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Scalar triple product

$\vec{A}\cdot(\vec{B}\times\vec{C})$

• volume of parallel-piped defined by $\vec{A},\vec{B},\vec{C}$
• $\vec{A}\cdot(\vec{B}\times\vec{C})=(\vec{A}\times\vec{B})\cdot\vec{C}=-\vec{B}\cdot(\vec{A}\times\vec{C})=$…

Vector triple product

BAC-CAB rule

$\vec{A}\times(\vec{B}\times\vec{C})=\vec{B}(\vec{A}\cdot\vec{C})-\vec{C}(\vec{A}\cdot\vec{B})$

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Differential Vector Operators

irrotational

$\nabla\times\vec{A}=0 \;\; everywhere$

solenoidal

$\nabla\cdot\vec{A}=0 \;\; everywhere$

any gradient has a vanishing curl and is solenoidal

$\nabla \times \nabla \varphi =0,\; all\; \varphi$

any curl has a vanishing divergence and is irrotational

$\nabla\cdot(\nabla\times\vec{V})=0, \; all\;V$

vector Laplacian $\nabla^2\vec{V}$ or $\nabla\cdot\nabla\vec{V}$

$\nabla\times(\nabla\times\vec{V})=\nabla(\nabla\cdot\vec{V})-\nabla\cdot\nabla\vec{V}=\nabla(\nabla\cdot\vec{V})-\nabla^2\vec{V}$

Laplacian for Spherical coordinate

$\nabla^2 \equiv \frac{1}{r^2}\frac{\partial}{\partial r}(r^2\frac{\partial}{\partial r})+\frac{1}{r^2\sin^2{\phi}}\frac{\partial^2}{\partial\theta^2}+\frac{1}{r^2\sin{\phi}}\frac{\partial}{\partial\phi}(\sin{\phi}\frac{\partial}{\partial\phi})$

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Miscellaneous

$f$:scalar function, $\vec{V}$:vector function

$\nabla\cdot(f\vec{V})=(\nabla f)\cdot\vec{V}+f\nabla\vec{V}$

$\nabla\times(f\vec{V})=f(\nabla\times\vec{V})+(\nabla f)\times\vec{V}$

$\nabla\times(\vec{A}\times\vec{B})=\vec{A}(\nabla\cdot\vec{B})-\vec{B}(\nabla\cdot\vec{A})-(\vec{A}\cdot\nabla)\vec{B}+(\vec{B}\cdot\nabla)\vec{A}$

$\nabla \cdot (\vec{A}\times\vec{B}) = \vec{B}\cdot(\nabla\times\vec{A})-\vec{A}\cdot(\nabla\times\vec{B})$

$\vec{A}\times(\nabla\times\vec{A})=\frac{1}{2}\nabla(A^2)-(\vec{A}\cdot\nabla)\vec{A}$

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Del in cylindrical and spherical coordinates - wiki

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Vector Integration

Integration by parts

scalar function $f(\vec{r})$, vector function $\vec{A}(\vec{r})$
$f(\vec{r})$ or $\vec{A}(\vec{r})$ vanish sufficiently strongly at infinity, i.e. integrated terms vanish at infinity

$\int{\vec{A}(\vec{r})\cdot\nabla f(\vec{r})d^3r}=-\int{f(\vec{r})\nabla\cdot\vec{A}(\vec{r})d^3r}$

$\int{f(\vec{r})\nabla\cdot\vec{A}(\vec{r})d^3r}=-\int{\vec{A}(\vec{r})\cdot\nabla f(\vec{r})}$

$\int{\vec{C}(\vec{r})\cdot(\nabla\times\vec{A}(\vec{r}))d^3r}=\int{\vec{A}(\vec{r})\cdot(\nabla\times\vec{C}(\vec{r}))d^3r}$