ApendicesEMMC

337

Apéndice A

Identidades
Tensor identidad
I=1=g
ˆ g
ˆ = g
ˆ g
ˆ =    g
ˆ g
ˆ =  g
ˆ g
ˆ =   g
ˆ g
ˆ

(A.1)

Identidades de productos

ˆ = g
ˆ · g
ˆ =  = 
g
ˆ · g

(A.2)

g
ˆ · g
ˆ = g
ˆ · g
ˆ =    =  

(A.4)

g
ˆ × g
ˆ =  g
ˆ =  g
ˆ

(A.5)

g
ˆ × g
ˆ =   g
ˆ =    g
ˆ

(A.7)

ˆ =    g
ˆ =   g
ˆ
g
ˆ × g(A.8)

³
´
g
ˆ · g
ˆ × g
ˆ = 

(A.9)











g
ˆ ·g
ˆ =g
ˆ ·g
ˆ = =





g
ˆ × g
ˆ =  g
ˆ =  g
ˆ



(A.3)

(A.6)

g
ˆ · (ˆ
g × g
ˆ ) = 

(A.10)

  =       −      

(A.11)

a·b=b·a

(A.12)

a b · v = a (b · v)

(A.13)

v · a b = (v · a) b

(A.14)

u · v = Tr (u v) = (u v) : 1 = 1 : (u v)

(A.15)

338
a × b = −b × a
¡
¢
A × b = − b × A(A.16)
(A.17)

a b × v = a (b × v)

(A.18)

v × a b = (v × a) b

(A.19)

a · (b × A) = (a × b) · A = − (b × a) · A

(A.20)

a · (A × b) = (a · A) × b = −b × (a · A)

(A.21)

a b : v w = (a · w) (b · v)

(A.22)

A : B = B : A = A : B

(A.23)

A : u v = v · A · u = u v : A = v u : A

(A.24)

u v w · x = u v (w · x)

(A.25)

u v w : x y = u v w · ·x w = u (w · x) (v · y)
¯
¯
¯
¯
¯ g
¯ g
ˆ1 g
ˆ1 g
ˆ2 gˆ3 ¯¯
ˆ2 g
ˆ3 ¯¯
¯
¯
1 ¯
¯ √ ¯
¯
v × w = √ ¯ 1 2 3 ¯ =  ¯ 1 2 3 ¯
¯
¯
¯ 1
¯
¯ 1 2 3 ¯
¯  2 3 ¯
¯
¯
¯    ¯
1
2
3 ¯
¯
1 ¯
¯
u · (v × w) = v · (w × u) = w · (u × v) = √ ¯ 1 2 3 ¯
¯
¯
¯ 1 2 3 ¯

(A.26)
(A.27)

(A.28)

a × (b × c) = (a · c) b − (a · b) c

(A.29)

(a × b) × c = (a · c) b − (b · c) a

(A.30)

(a × b) · (c × d) = (a · c) (b · d) − (a · d) (b · c)

(A.31)

2

2

2(a × b) · (a × b) = kak kbk − (a · b)

(A.32)

(a × b) (c × d) = d a (b · c) + c b (d · a) − d b (c · a) − c a (d · b)

+ ((a · c) (b · d) − (a · d) (b · c)) 1 (A.33)

(a × b) × (c × d) = [c · (d × a)] b − [c · (d × b)] a

= [a · (b × d)] c − [a · (b × c)] d

[a · (b × c)] d = [d · (b × c)] a + [a · (d × c)] b + [a · (b × d)] c
(a · d) (b × c) + (b · d) (c × a) + (c · d) (a × b)

(A.34)
(A.35)(A.36)
(A.37)

339
a × (b × c) + b × (c × a) + c × (a × b) = 0

(A.38)

Derivadas de un vector de posición p
p
=g
ˆ


(A.39)

p
=g
ˆ · g
ˆ =    = Tr 1

p
ˆ g
ˆ = 1
∇p = g
ˆ  = g

p
∇×p=g
ˆ ×  = g
ˆ × g
ˆ =   g
ˆ = 0

∇·p=g
ˆ ·

(A.40)
(A.41)
(A.42)

Propiedades distributivas del operador ∇
∇ (A + B) = ∇A + ∇B

(A.43)

∇ · (A + B) = ∇ · A + ∇ · B

(A.44)

∇ × (A +B) = ∇ × A + ∇ × B

(A.45)

Identidades entre operadores diferenciales
∇ · ( v) =  ∇ · v + v · ∇

(A.46)

∇ · ( ∇) =  ∇2  + ∇ · ∇

(A.47)

∇ · ( ∇v) =  ∇2 v + ∇ · ∇v

(A.48)

en  3

(A.49)

¡
¢
∇ · ∇v = ∇ (∇ · v)

∇ · ( 1) = ∇

(A.50)

∇ · ( T) =  ∇ · T + ∇ · T

(A.51)

∇ · (u v) = (∇ · u) v + u · ∇v

(A.52)

∇ · (u ⊗ T) = (∇ · u) T + u · ∇T

(A.53)

∇ · (A · u) = (∇ · A) · u +A : ∇u

(A.54)

∇ · (u · ∇v) = ∇u : ∇v + u · (∇ (∇ · v))

(A.55)

∇ · (T × p) = −p × (∇ · T) + ε : T,

con T = T

∇ · (T × p) = (∇ · T) × p + T : ε
¯
¯ g
g
ˆ2
g
ˆ3
¯ ˆ1
¯
1



ˆ = √ ¯¯
∇ × v =  ; g
1
2
 ¯ 

3
¯ 1
2
3

¯
¯
¯
¯
¯
¯
¯
¯

(A.56)
(A.57)

(A.58)

340
∇ · (u × v) = v · (∇ × u) − u · (∇ × v)

(A.59)

∇ · (A × u) = (∇ · A) × u + A · (∇ × u)

(A.60)

∇ · (u × A) = (∇× u) · A − u · (∇ × A)

(A.61)

u × (∇ × v) = ∇v · u − u · ∇v

(A.62)

∇ × (u × v) = u ∇ · v − u · ∇v − v ∇ · u + v · ∇u

(A.63)

∇ × ( v) =  ∇ × v + ∇ × v

(A.64)

∇ × (u v) = (∇ × u) v − u × ∇v

(A.65)

∇ ( ) =  ∇ +  ∇

(A.66)

∇ ( v) =  (∇v) + (∇) v

(A.67)

∇ (u · v) = ∇u · v + ∇v · u

= u × (∇ × v) + v × (∇ × u) + u · ∇v + v · ∇u
1
∇ (v · v) − v × (∇ × v) = v · ∇v
2
∇ · (∇ × u) =0

(A.68)
(A.69)
(A.70)

∇ × ∇ = 0

(A.71)

∇ × (∇ × v) = ∇ (∇ · v) − ∇2 v

(A.72)

∇ · ∇ = ∇2 

(A.73)

∇ · ∇u = ∇2 u
2

∇ · ∇A = ∇ A

¡
¢
∇2 (∇ · v) = ∇ · ∇2 v

(A.74)
(A.75)
(A.76)

 ∇2  −  ∇2  = ∇ · ( ∇ −  ∇)

(A.77)

∇2 ( ) =  ∇2  + 2∇ · ∇ +  ∇2 
µ ¶

 ∇ −  ∇

1
=
= ∇ − 2 ∇
∇
2




µ ¶
u
 ∇ · u − u · ∇
∇·
=

2
µ ¶
u
 ∇ × u + u × ∇
∇×
=

2

(A.78)...