Formulae involving ∇ Vector Identities with Proofs: Nabla Formulae ...

(10) Prove ∇x(AxB) = (B.∇)A - B(∇.A) - (A.∇)B + A(∇.B)
∇
=
i
( AxB)
= ∇x
A A A = ∇x[
( A B − A B ) i − ( A B − A B ) j + ( A B − A B ) k]
B
j
B
k
x 1 2 3
2 3 3 2 1 3 3 1 1 2 2 1
i
∂
∂x
A B − A B
2
3
3
2
1
3
1
2
B
1
3
j
∂
∂y
A B − A B
3
k
∂
∂z
A B − A B
1
2
2
1
= ⎛ ∂( A1
B2
− A2
B1
) ∂(
A3B1
− A1B
3 ) ⎞ ⎛ ∂(
A1
B2
− A2
B1
) ∂(
A2
B3
− A3B
2 ) ⎞ ⎛ ∂(
A3B1
− A1B
3 ) ∂(
A2B3
− A3B2
) ⎞
⎜
−
⎟i
−
−
j + ⎜
−
⎟k
⎝
= LHS
∂y
∂z
⎠
⎜
⎝
(B.∇)A - B(∇.A) = ⎛ ∂ ∂ ∂ ⎞
⎛ ∂A1
∂A2
∂A3
⎞
⎜ B1 + B2
+ B3
⎟(
A1i
+ A2
j + A3k
) −⎜
+ + ⎟(
B1i
+ B2
j + B3k
)
⎝ ∂x
∂y
∂z
⎠
⎝ ∂x
∂y
∂z
⎠
∂x
= ⎛ ∂A1 ∂A1
∂A2
∂A3
⎞ ⎛ ∂A2
∂A2
∂A1
∂A3
⎞ ⎛ ∂A3
∂A3
∂A1
∂A2
⎞
⎜ B B B B ⎟i
B B B B j + ⎜ B + B − B − B ⎟k
⎝
2
+
∂y
3
−
∂z
1
−
∂y
1
+ ⎜
∂z
⎠ ⎝
1
+
∂x
3
−
∂z
2
−
∂x
Similarly, by interchanging the variable of A and B, we have
(A.∇)B - A(∇.B) = ⎛ ∂ ∂ ∂ ⎞
⎛ ∂B1
∂B2
∂B3
⎞
⎜ A1 + A2
+ A3
⎟(
B1i
+ B2
j + B3k
) −⎜
+ + ⎟(
A1i
+ A2
j + A3k
)
⎝ ∂x
∂y
∂z
⎠
⎝ ∂x
∂y
∂z
⎠
= ⎛ ∂B1 ∂B1
∂B2
∂B3
⎞ ⎛ ∂B2
∂B2
∂B1
∂B3
⎞ ⎛ ∂B3
∂B3
∂B1
∂B2
⎞
⎜ A A A A ⎟i
A A A A j + ⎜ A + A − A − A ⎟k
⎝
2
+
∂y
3
−
∂z
1
−
∂y
1
+ ⎜
∂z
⎠ ⎝
(B.∇)A - B(∇.A) - (A.∇)B + A(∇.B)
1
+
∂x
3
−
∂z
2
−
∂x
5
2
2
∂z
⎟
∂z
⎠
⎟
∂z
⎠
= ⎡⎛
∂A B A B A B A B ⎤
1 ∂ 2 ⎞ ⎛ ∂ 1 ∂ 3 ⎞ ⎛ ∂ 2 ∂ 1 ⎞ ⎛ ∂ 3 ∂ 1 ⎞
⎜ A ⎟ B A ⎜ B A ⎟ B A i
⎢ B2 + 1 + ⎜ 3 + 1 ⎟ − 1 + 2 − ⎜ 1 + 3 ⎟⎥
⎣⎝
∂y
∂y
⎠ ⎝ ∂z
∂z
⎠ ⎝ ∂y
∂y
⎠ ⎝ ∂z
∂z
⎠⎦
⎡⎛
∂A2
∂B2
⎞ ⎛ ∂A2
∂B3
⎞ ⎛ ∂A1
∂B2
⎞ ⎛ ∂A3
∂B2
⎞⎤
+ ⎢⎜
B1 + A1
⎟ + ⎜ B3
+ A2
⎟ −⎜
B2
+ A1
⎟ −⎜
B2
+ A3
⎟ j
x x z z x x z z
⎥
⎣⎝
∂ ∂ ⎠ ⎝ ∂ ∂ ⎠ ⎝ ∂ ∂ ⎠ ⎝ ∂ ∂ ⎠⎦
⎡⎛
∂A
B A B A B A B ⎤
3 ∂ 1 ⎞ ⎛ ∂ 3 ∂ 2 ⎞ ⎛ ∂ 1 ∂ 3 ⎞ ⎛ ∂ 2 ∂ 3 ⎞
+ ⎢⎜
B1 + A3
⎟ + ⎜ B2
+ A3
⎟ −⎜
B3
+ A1
⎟ −⎜
B3
+ A2
⎟⎥k
⎣⎝
∂x
∂x
⎠ ⎝ ∂y
∂y
⎠ ⎝ ∂x
∂x
⎠ ⎝ ∂y
∂y
⎠⎦
= ⎛ ∂( A1
B2
− A2
B1
) ∂(
A1
B3
− A3B1
) ⎞ ⎛ ∂(
A1B
2 − A2B1
) ∂(
A3B
2 − A2
B3
) ⎞ ⎛ ∂(
A3B1
− A1B
3 ) ∂(
A3B2
− A2
B3
) ⎞
⎜
+
⎟i
−
+
j + ⎜
+
⎟k
⎝
∂y
∂z
⎠
⎜
⎝
∂x
= ⎛ ∂( A1
B2
− A2
B1
) ∂(
A3B1
− A1B
3 ) ⎞ ⎛ ∂(
A1
B2
− A2
B1
) ∂(
A2
B3
− A3B
2 ) ⎞ ⎛ ∂(
A3B1
− A1B
3 ) ∂(
A2B3
− A3B2
) ⎞
⎜
−
⎟i
−
−
j + ⎜
−
⎟k
⎝
= RHS
∂y
RHS = LHS
∂z
⎠
⎜
⎝
∂x
∂z
∂z
⎟
⎠
⎟
⎠
⎝
⎝
⎝
⎝
⎟
⎠
1
1
⎝
∂x
∂x
∂x
∂x
2
2
∂x
∂y
∂y
3
3
∂x
∂x
∂y
∂y
3
∂y
3
∂y
⎠
∂y
⎠
⎠
⎠
⎠