Operacioj en vektora kalkulo

Jenaj operacioj estas gravaj en vektora kalkulo:

Kirlo de gradiento de skalara kampo ${\displaystyle \ \psi }$ estas ĉiam nulo:

${\displaystyle \nabla \times (\nabla \psi )=0}$

Diverĝenco de kirlo de vektora kampo ${\displaystyle \ \mathbf {A} }$ estas ĉiam nulo:

${\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}$

${\displaystyle \nabla \times \nabla \times \mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }$

${\displaystyle \nabla \cdot (\mathbf {A} +\mathbf {B} )=\nabla \cdot \mathbf {A} +\nabla \cdot \mathbf {B} }$

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

${\displaystyle \nabla (\mathbf {A} \cdot \mathbf {B} )=(\mathbf {A} \cdot \nabla )\mathbf {B} +(\mathbf {B} \cdot \nabla )\mathbf {A} +\mathbf {A} \times (\nabla \times \mathbf {B} )+\mathbf {B} \times (\nabla \times \mathbf {A} )}$

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

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

${\displaystyle \nabla \cdot (\psi \mathbf {A} )=\mathbf {A} \cdot \nabla \psi +\psi \nabla \cdot \mathbf {A} }$

${\displaystyle \nabla \times (\psi \mathbf {A} )=\nabla \psi \times \mathbf {A} +\psi \nabla \times \mathbf {A} }$

• Vektora kalkulo