Formulae involving ∇ Vector Identities with Proofs: Nabla Formulae ...
Formulae involving ∇ Vector Identities with Proofs: Nabla Formulae ...
Formulae involving ∇ Vector Identities with Proofs: Nabla Formulae ...
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(B.<strong>∇</strong>)A + (A.<strong>∇</strong>)B + Bx(<strong>∇</strong>xA) + Ax(<strong>∇</strong>xB)<br />
= ⎛ ∂( A1<br />
B1<br />
) ∂(<br />
A2B<br />
2 ) ∂(<br />
A3B3<br />
) ⎞ ⎛ ∂(<br />
A2B<br />
2 ) ∂(<br />
A1B1<br />
) ∂(<br />
A3B3<br />
) ⎞ ⎛ ∂(<br />
A3B<br />
3 ) ∂(<br />
A1B<br />
1)<br />
∂(<br />
A2B<br />
2 ) ⎞<br />
⎜ + + ⎟i<br />
+ ⎜ + + ⎟ j + ⎜ + + ⎟k<br />
⎝ ∂x<br />
∂x<br />
∂x<br />
⎠ ⎝ ∂y<br />
∂y<br />
∂y<br />
⎠ ⎝ ∂z<br />
∂z<br />
∂z<br />
⎠<br />
=<br />
( A1<br />
B1<br />
+ A2B<br />
2 + A3B<br />
3 ) ∂(<br />
A1B<br />
1 + A2B<br />
2 + A3B<br />
3)<br />
∂(<br />
A1B<br />
1 + A2B2<br />
+ A3B<br />
)<br />
i +<br />
j +<br />
k<br />
∂ 3<br />
⎛<br />
⎝ ∂x<br />
∂x<br />
∂y<br />
= ∂ ∂ ∂<br />
⎜ i + j + k ⎟(<br />
A B + A B + A B ) = RHS<br />
∂y<br />
LHS = RHS<br />
∂z<br />
⎞<br />
⎠<br />
1<br />
1<br />
2<br />
2<br />
∴ <strong>∇</strong>(A.B) = (B.<strong>∇</strong>)A + (A.<strong>∇</strong>)B + Bx(<strong>∇</strong>xA) + Ax(<strong>∇</strong>xB)<br />
(12) Prove <strong>∇</strong>.(<strong>∇</strong>φ) = <strong>∇</strong> 2 φ<br />
<strong>∇</strong>.(<strong>∇</strong>φ) = ⎛ ∂ ∂ ∂ ⎞ ⎛ ∂φ<br />
∂φ<br />
∂φ<br />
⎞<br />
⎜i<br />
+ j + k ⎟.<br />
⎜i<br />
+ j + k ⎟<br />
⎝ ∂x<br />
∂y<br />
∂z<br />
⎠ ⎝ ∂x<br />
∂y<br />
∂z<br />
⎠<br />
2 2 2<br />
=<br />
∂ φ ∂ φ ∂ φ 2<br />
+ + = <strong>∇</strong> φ<br />
2 2 2<br />
∂x<br />
∂y<br />
∂z<br />
∴ <strong>∇</strong>.(<strong>∇</strong>φ) = <strong>∇</strong> 2 φ<br />
(13) Prove <strong>∇</strong>x(<strong>∇</strong>φ) = 0<br />
<strong>∇</strong>x(<strong>∇</strong>φ) =<br />
3<br />
⎛ ∂ ∂ ∂ ⎞ ⎛ ∂φ<br />
∂φ<br />
⎜i<br />
+ j + k ⎟x⎜<br />
i + j + k<br />
⎝ ∂x<br />
∂y<br />
∂z<br />
⎠ ⎝ ∂x<br />
∂y<br />
=<br />
i<br />
∂<br />
∂x<br />
∂φ<br />
∂x<br />
j<br />
∂<br />
∂y<br />
∂φ<br />
∂y<br />
3<br />
k<br />
∂<br />
∂z<br />
∂φ<br />
∂z<br />
7<br />
∂φ<br />
⎞<br />
⎟<br />
∂z<br />
⎠<br />
∂z<br />
= ( φ − φ ) − ( φ −φ<br />
) j + ( φ −φ<br />
)k<br />
zy<br />
yz i zx xz yx xy<br />
Since φ has continuous second order partial derivatives, we have<br />
∴ <strong>∇</strong>x(<strong>∇</strong>φ) = 0<br />
φxy = φyx φyz = φzy φzx = φxz
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