Vektorianalyysin kaavat

Kokeneet kirjoittajat eivät ole vielä tarkistaneet sivun nykyistä versiota, ja se voi poiketa merkittävästi 27.9.2021 tarkistetusta versiosta . vahvistus vaatii 1 muokkauksen .

Merkintä

nabla operaattori : $\nabla$
skalaarikentän gradientti : $\nabla \psi =\operaattorinimi {grad}\ \psi$
vektorikentän divergentti : $\nabla \cdot {\mathbf {A}}=\operaattorinimi {div}\ {\mathbf {A}}$
vektorikentän curl : $\nabla \times \mathbf {A} =\operaattorinimi {rot} \ \mathbf {A}$
laplalainen : $\Delta =\nabla ^{2}=\nabla \cdot \nabla$

Lineaarisuus

Mille tahansa numerolle : $\alpha$

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

Toisen asteen operaattorit

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

Kenttien tuotteiden eriyttäminen

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

Katso myös