Spisak vektorskih identiteta

Šablon:Siroče Šablon:Infinitezimalni račun Ovaj članak je spisak nekoliko korisnih matematičkih identiteta koji se koriste u vektorskoj algebri.

Trostruki proizvodi

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

koje se mogu dokazati dajući proizvoljne komponente $A,$ $B,$ and $C,$

$\overrightarrow{A}=(a_x,a_y,a_z)$

$\overrightarrow{B}=(b_x,b_y,b_z)$

$\overrightarrow{C}=(c_x,c_y,c_z)$

te računajući vrijednosti svakog iskaza, kao što je $\overrightarrow{B}\times \overrightarrow{C}$, će pokazati da su obe strane jednačine jednake.

Ostali proizvodi

• $(\overrightarrow{A}\times \overrightarrow{B})\cdot (\overrightarrow{A}\times \overrightarrow{B})=A^2{B}^2-(\overrightarrow{A}\cdot \overrightarrow{B}{)}^2=\overrightarrow{B}\cdot \overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{A})$ (napomena: Ovo je drugi oblik Lagrangeove formule).

Pravila proizvoda

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

Greenov prvi identitet

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

Fundametnalni teoremi

Teorem divergencije

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

Stokesov teorem

• $\underset{S}{\int }(\mathrm{\nabla }\times \overrightarrow{A})\cdot d\overrightarrow{a}={\oint }_C\overrightarrow{A}\cdot d\overrightarrow{l}$

Također pogledajte

• Spisak matematičkih identiteta
• Identiteti vektorskog kalkulusa