1. ## Vector Question

if $\displaystyle F = x^3yi +(y^2x -zx)j +(z^3 x^2 +y^2)k$ verify the identity

$\displaystyle \nabla \wedge (\nabla\wedge F) = \nabla(\nabla\cdot F) - \nabla^2 F$

my attempt

i found $\displaystyle \nabla\wedge(\nabla \wedge F)$ to be $\displaystyle (2y+6xz^2)i -(1+3x^2)j +(-2z^3-2)k$

but when i calculated $\displaystyle \nabla(\nabla\cdot F)$ i got $\displaystyle (6x+2y+6z^2 x, 2x, 6zx^2)$ and

$\displaystyle \nabla^2 F$ = $\displaystyle (6+6z^2,0,6x^2)$

can someone tell me where i went wrong, am i using the wrong fomula?

2. Originally Posted by iLikeMaths
if $\displaystyle F = x^3yi +(y^2x -zx)j +(z^3 x^2 +y^2)k$ verify the identity

$\displaystyle \nabla \wedge (\nabla\wedge F) = \nabla(\nabla\cdot F) - \nabla^2 F$

my attempt

i found $\displaystyle \nabla\wedge(\nabla \wedge F)$ to be $\displaystyle (2y+6xz^2)i -(1+3x^2)j +(-2z^3-2)k$

but when i calculated $\displaystyle \nabla(\nabla\cdot F)$ i got $\displaystyle (6x+2y+6z^2 x, 2x, 6zx^2)$ and

$\displaystyle \nabla^2 F$ = $\displaystyle (6+6z^2,0,6x^2)$

can someone tell me where i went wrong, am i using the wrong fomula?
The formula is right. Some of your terms are wrong. Here's what I got.

$\displaystyle \nabla\times(\nabla \times F) = (2y+6xz^2)i +3x^2 j +(-2z^3-2)k$

$\displaystyle \nabla(\nabla\cdot F) = (6xy+2y+6xz^2)i + (2x+3x^2)j + (6 x^2z)K$

$\displaystyle \nabla^2 F = 6xyi + 2x j +(2 + 6x^2z + 2z^3)k$