Vektoraj identoj

Ŝablono:Polurinda Ĉi tiuj artikolaj listaj kelkaj helpemaj matematikaj identoj kiu estas utila en vektora algebro.

Triopaj produtoj

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

Produtaj reguloj

• $\overrightarrow{\mathrm{\nabla }}(fg)=f(\overrightarrow{\mathrm{\nabla }}g)+g(\overrightarrow{\mathrm{\nabla }}f)$
• $\overrightarrow{\mathrm{\nabla }}(\overrightarrow{A}\cdot \overrightarrow{B})=\overrightarrow{A}\times (\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{B})+\overrightarrow{B}\times (\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{A})+(\overrightarrow{A}\cdot \overrightarrow{\mathrm{\nabla }})\overrightarrow{B}+(\overrightarrow{B}\cdot \overrightarrow{\mathrm{\nabla }})\overrightarrow{A}$
• $\overrightarrow{\mathrm{\nabla }}\cdot (f\overrightarrow{A})=f(\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{A})+\overrightarrow{A}\cdot (\overrightarrow{\mathrm{\nabla }}f)$
• $\overrightarrow{\mathrm{\nabla }}\cdot (\overrightarrow{A}\times \overrightarrow{B})=\overrightarrow{B}\cdot (\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{A})-\overrightarrow{A}\cdot (\overrightarrow{\mathrm{\nabla }}\times \overrightarrow{B})$
• $\mathrm{\nabla }\times (\overrightarrow{A}\times \overrightarrow{B})=(\overrightarrow{B}\cdot \mathrm{\nabla })\overrightarrow{A}-(\overrightarrow{A}\cdot \mathrm{\nabla })\overrightarrow{B}+\overrightarrow{A}(\mathrm{\nabla }\cdot \overrightarrow{B})-\overrightarrow{B}(\mathrm{\nabla }\cdot \overrightarrow{A})$
• $\mathrm{\nabla }\times (f\overrightarrow{A})=f(\mathrm{\nabla }\times \overrightarrow{A})-\overrightarrow{A}\times (\mathrm{\nabla }f)$

Unua idento de Green

• $\overrightarrow{\mathrm{\nabla }}\cdot \left(f\overrightarrow{\mathrm{\nabla }}f\right)=f\overrightarrow{\mathrm{\nabla }}\cdot \left(\overrightarrow{\mathrm{\nabla }}f\right)+(\overrightarrow{\mathrm{\nabla }}f\cdot \overrightarrow{\mathrm{\nabla }}f)=f{\mathrm{\nabla }}^{2}f+{\left(\overrightarrow{\mathrm{\nabla }}f\right)}^2$ pro tio
  $f{\mathrm{\nabla }}^{2}f=\overrightarrow{\mathrm{\nabla }}\cdot \left(f\overrightarrow{\mathrm{\nabla }}f\right)-{\left(\overrightarrow{\mathrm{\nabla }}f\right)}^2$

Fundamentaj teoremoj

Diverĝenca teoremo:

• $\int (\overrightarrow{\mathrm{\nabla }}\cdot \overrightarrow{A})dv={\oint }_S\overrightarrow{A}\cdot d\overrightarrow{a}$

Teoremo de Stokes:

• ${\oint }_C\overrightarrow{A}\cdot d\overrightarrow{l}={\oint }_S(\mathrm{\nabla }\times \overrightarrow{A})\cdot \overrightarrow{n}da$