Algunas fórmulas comunes en análisis vectorial

1. ( )f g f∇ + = ∇ +∇g

2. ( )cf c f∇ = ∇ , para una constante c

3. ( )fg f g g∇ = ∇ + ∇f

4. ( ) ( ) 2f g g f f g g∇ = ∇ − ∇ , en los puntos donde ( ) 0g x ≠

5. ( )div div div+ = +F G F G

6. ( )rot rot rot+ = +F G F G

7. ( ) ( ) ( ) rot rot∇ ⋅ = ⋅∇ + ⋅∇ + × + ×F G F G G F F G G F

8. ( )div divf f f= + ⋅F F F ∇

9. ( )div rot rot× = ⋅ − ⋅F G G F F G

10. div rot 0=F

11. ( )rot rotf f f= +∇F F ×F

12. ( ) ( ) ( )rot div div × = − + ⋅∇ − ⋅∇F G F G G F G F F G

13. 2rot rot grad div= −F F ∇ F

14. rot 0f∇ =

15. ( ) ( )2 2∇ ⋅ = ⋅∇ + ×F F F F F Frot

16. ( ) ( )2 2 2 2fg f g g f f g∇ = ∇ + ∇ + ∇ ⋅∇

17. ( )div 0f g∇ ×∇ =

18. ( ) 2 2f g g f f g g f∇⋅ ∇ − ∇ = ∇ − ∇

19. ( ) ( ) ( )⋅ × = ⋅ × = ⋅ ×H F G G H F F G H

20. ( )( ) ( )( ) ( )⋅ ×∇ × = ⋅ ⋅∇ ⋅ − ⋅ ⋅∇H F G H G F H F G

21. ( ) ( ) ( )×⋅ × = ⋅ − ⋅F G H F H G H F G

NOTA: f y g denotan campos escalares; F, G y H denotan campos vectoriales