Interested Article - Формулы векторного анализа

Обозначения

оператор набла : $\nabla$ $\nabla$
градиент скалярного поля : $\nabla \psi =\operatorname {grad} \ \psi$ $\nabla \psi =\operatorname {grad}\ \psi$
дивергенция векторного поля : $\nabla \cdot \mathbf {A} =\operatorname {div} \ \mathbf {A}$ $\nabla \cdot {\mathbf {A}}=\operatorname {div}\ {\mathbf {A}}$
ротор векторного поля : $\nabla \times \mathbf {A} =\operatorname {rot} \ \mathbf {A}$ $\nabla \times \mathbf {A} =\operatorname {rot} \ \mathbf {A}$
лапласиан : $\Delta =\nabla ^{2}=\nabla \cdot \nabla$ $\Delta =\nabla ^{2}=\nabla \cdot \nabla$

Линейность

Для любого числа $\alpha$ $\alpha$ :

$\ \nabla (\alpha \phi +\psi)=\alpha \nabla \phi +\nabla \psi$ $\ \nabla (\alpha \phi +\psi)=\alpha \nabla \phi +\nabla \psi$	$\ \mathbf {grad} (\alpha \phi +\psi)=\alpha \ \mathbf {grad} \ \phi +\mathbf {grad} \ \psi$ $\ {\mathbf {grad}}(\alpha \phi +\psi)=\alpha \ {\mathbf {grad}}\ \phi +{\mathbf {grad}}\ \psi$
$\ \nabla \cdot (\alpha \mathbf {A} +\mathbf {B})=\alpha \nabla \cdot \mathbf {A} +\nabla \cdot \mathbf {B}$ $\ \nabla \cdot (\alpha {\mathbf {A}}+{\mathbf {B}})=\alpha \nabla \cdot {\mathbf {A}}+\nabla \cdot {\mathbf {B}}$	$\ \mathbf {div} \ (\alpha \mathbf {A} +\mathbf {B})=\alpha \ \mathbf {div} \ \mathbf {A} +\mathbf {div} \ \mathbf {B}$ $\ {\mathbf {div}}\ (\alpha {\mathbf {A}}+{\mathbf {B}})=\alpha \ {\mathbf {div}}\ {\mathbf {A}}+{\mathbf {div}}\ {\mathbf {B}}$
$\ \nabla \times (\alpha \mathbf {A} +\mathbf {B})=\alpha \nabla \times \mathbf {A} +\nabla \times \mathbf {B}$ $\ \nabla \times (\alpha {\mathbf {A}}+{\mathbf {B}})=\alpha \nabla \times {\mathbf {A}}+\nabla \times {\mathbf {B}}$	$\ \mathbf {rot} (\alpha \mathbf {A} +\mathbf {B})=\alpha \ \mathbf {rot} \ \mathbf {A} +\mathbf {rot} \ \mathbf {B}$ $\ {\mathbf {rot}}(\alpha {\mathbf {A}}+{\mathbf {B}})=\alpha \ {\mathbf {rot}}\ {\mathbf {A}}+{\mathbf {rot}}\ {\mathbf {B}}$

Операторы второго порядка

$\ \nabla \times (\nabla \psi)=0$ $\ \nabla \times (\nabla \psi)=0$	$\ \mathbf {rot} (\mathbf {grad} \ \psi)=0$ $\ {\mathbf {rot}}({\mathbf {grad}}\ \psi)=0$
$\ \nabla \cdot (\nabla \times \mathbf {A})=0$ $\ \nabla \cdot (\nabla \times {\mathbf {A}})=0$	$\ \mathbf {div} \ (\mathbf {rot} \ \mathbf {A})=0$ $\ {\mathbf {div}}\ ({\mathbf {rot}}\ {\mathbf {A}})=0$
$\ \Delta \ \psi =\nabla \cdot (\nabla \psi)=\nabla ^{2}\psi$ $\ \Delta \ \psi =\nabla \cdot (\nabla \psi)=\nabla ^{2}\psi$	$\ \Delta \ \psi =\mathbf {div} \ (\mathbf {grad} \ \psi)$ $\ \Delta \ \psi ={\mathbf {div}}\ ({\mathbf {grad}}\ \psi)$
$\ \nabla \times \nabla \times \mathbf {A} =\nabla (\nabla \cdot \mathbf {A})-\nabla ^{2}\mathbf {A}$ $\ \nabla \times \nabla \times {\mathbf {A}}=\nabla (\nabla \cdot {\mathbf {A}})-\nabla ^{{2}}{\mathbf {A}}$	$\ \mathbf {rot} \ (\mathbf {rot} \ \mathbf {A})=\mathbf {grad} \ (\mathbf {div} \ \mathbf {A})-\Delta \mathbf {A}$ $\ {\mathbf {rot}}\ ({\mathbf {rot}}\ {\mathbf {A}})={\mathbf {grad}}\ ({\mathbf {div}}\ {\mathbf {A}})-\Delta {\mathbf {A}}$

Дифференцирование произведений полей

$\nabla \cdot (\psi \mathbf {A})=\mathbf {A} \cdot \nabla \psi +\psi \nabla \cdot \mathbf {A}$ $\nabla \cdot (\psi {\mathbf {A}})={\mathbf {A}}\cdot \nabla \psi +\psi \nabla \cdot {\mathbf {A}}$	$\mathbf {div} (\psi \mathbf {A})=\mathbf {A} \cdot \mathbf {grad} \psi +\psi \ \mathbf {div} \mathbf {A}$ ${\mathbf {div}}(\psi {\mathbf {A}})={\mathbf {A}}\cdot {\mathbf {grad}}\psi +\psi \ {\mathbf {div}}{\mathbf {A}}$
$\nabla \times (\psi \mathbf {A})=\nabla \psi \times \mathbf {A} +\psi \nabla \times \mathbf {A}$ $\nabla \times (\psi {\mathbf {A}})=\nabla \psi \times {\mathbf {A}}+\psi \nabla \times {\mathbf {A}}$	$\mathbf {rot} (\psi \mathbf {A})=\mathbf {grad} \psi \times \mathbf {A} +\psi \ \mathbf {rot} \mathbf {A}$ ${\mathbf {rot}}(\psi {\mathbf {A}})={\mathbf {grad}}\psi \times {\mathbf {A}}+\psi \ {\mathbf {rot}}{\mathbf {A}}$
$\nabla (\mathbf {A} \cdot \mathbf {B})=(\mathbf {A} \cdot \nabla)\mathbf {B} +(\mathbf {B} \cdot \nabla)\mathbf {A} +$ $\nabla ({\mathbf {A}}\cdot {\mathbf {B}})=({\mathbf {A}}\cdot \nabla){\mathbf {B}}+({\mathbf {B}}\cdot \nabla){\mathbf {A}}+$ $+\mathbf {A} \times (\nabla \times \mathbf {B})+\mathbf {B} \times (\nabla \times \mathbf {A})$ $+{\mathbf {A}}\times (\nabla \times {\mathbf {B}})+{\mathbf {B}}\times (\nabla \times {\mathbf {A}})$	$\ \mathbf {grad} (\mathbf {A} \cdot \mathbf {B})=(\mathbf {A} \cdot \nabla)\mathbf {B} +(\mathbf {B} \cdot \nabla)\mathbf {A} +$ $\ {\mathbf {grad}}({\mathbf {A}}\cdot {\mathbf {B}})=({\mathbf {A}}\cdot \nabla){\mathbf {B}}+({\mathbf {B}}\cdot \nabla){\mathbf {A}}+$ $+\mathbf {A} \times \mathbf {rot} \mathbf {B} +\mathbf {B} \times \mathbf {rot} \mathbf {A}$ $+{\mathbf {A}}\times {\mathbf {rot}}{\mathbf {B}}+{\mathbf {B}}\times {\mathbf {rot}}{\mathbf {A}}$
${\frac {1}{2}}\nabla A^{2}=\mathbf {A} \times (\nabla \times \mathbf {A})+(\mathbf {A} \cdot \nabla)\mathbf {A}$ ${\frac {1}{2}}\nabla A^{2}={\mathbf {A}}\times (\nabla \times {\mathbf {A}})+({\mathbf {A}}\cdot \nabla){\mathbf {A}}$	${\frac {1}{2}}\ \mathbf {grad} A^{2}=\mathbf {A} \times (\mathbf {rot} \mathbf {A})+(\mathbf {A} \cdot \nabla)\mathbf {A}$ ${\frac {1}{2}}\ {\mathbf {grad}}A^{2}={\mathbf {A}}\times ({\mathbf {rot}}{\mathbf {A}})+({\mathbf {A}}\cdot \nabla){\mathbf {A}}$
$\nabla \cdot (\mathbf {A} \times \mathbf {B})=\mathbf {B} \cdot \nabla \times \mathbf {A} -\mathbf {A} \cdot \nabla \times \mathbf {B}$ $\nabla \cdot ({\mathbf {A}}\times {\mathbf {B}})={\mathbf {B}}\cdot \nabla \times {\mathbf {A}}-{\mathbf {A}}\cdot \nabla \times {\mathbf {B}}$	$\mathbf {div} \ (\mathbf {A} \times \mathbf {B})=\mathbf {B} \cdot \mathbf {rot} \ \mathbf {A} -\mathbf {A} \cdot \mathbf {rot} \ \mathbf {B}$ ${\mathbf {div}}\ ({\mathbf {A}}\times {\mathbf {B}})={\mathbf {B}}\cdot {\mathbf {rot}}\ {\mathbf {A}}-{\mathbf {A}}\cdot {\mathbf {rot}}\ {\mathbf {B}}$
$\ \nabla \times (\mathbf {A} \times \mathbf {B})=\mathbf {A} (\nabla \cdot \mathbf {B})-\mathbf {B} (\nabla \cdot \mathbf {A})+$ $\ \nabla \times ({\mathbf {A}}\times {\mathbf {B}})={\mathbf {A}}(\nabla \cdot {\mathbf {B}})-{\mathbf {B}}(\nabla \cdot {\mathbf {A}})+$ $\;+(\mathbf {B} \cdot \nabla)\mathbf {A} -(\mathbf {A} \cdot \nabla)\mathbf {B}$ $\;+({\mathbf {B}}\cdot \nabla){\mathbf {A}}-({\mathbf {A}}\cdot \nabla){\mathbf {B}}$	$\ \mathbf {rot} (\mathbf {A} \times \mathbf {B})=\mathbf {A} \ (\mathbf {div} \ \mathbf {B})-\mathbf {B} \ (\mathbf {div} \ \mathbf {A})+$ $\ {\mathbf {rot}}({\mathbf {A}}\times {\mathbf {B}})={\mathbf {A}}\ ({\mathbf {div}}\ {\mathbf {B}})-{\mathbf {B}}\ ({\mathbf {div}}\ {\mathbf {A}})+$ $\;+(\mathbf {B} \cdot \nabla)\mathbf {A} -(\mathbf {A} \cdot \nabla)\mathbf {B}$ $\;+({\mathbf {B}}\cdot \nabla){\mathbf {A}}-({\mathbf {A}}\cdot \nabla){\mathbf {B}}$