Different forms of Stokes' theorem

[MatinSAR]

Homework Statement

Find the different forms using $\vec V=\vec a \phi$ and $\vec V=\vec a \times \vec P$ for constant $\vec a$.

Relevant Equations

Stokes' theorem.

What am I trying to do for $\vec V=\vec a \phi$ :
$R.H.S= \oint \vec V \cdot d \vec \lambda=\oint \vec a \phi \cdot d \vec \lambda=\vec a \cdot \oint \phi d \vec \lambda$

$L.H.S= \iint_S \vec \nabla \times \vec V \cdot \vec d \sigma=\iint_S \vec \nabla \times (\vec a \phi) \cdot \vec d \sigma=\iint_S (\phi \vec \nabla \times \vec a + (\vec \nabla \phi) \times \vec a) \cdot \vec d \sigma= ?$
I think $\phi \vec \nabla \times \vec a + (\vec \nabla \phi) \times \vec a$ should be 0. why is this wrong?

$\phi \vec \nabla \times \vec a$ is 0 because $\vec a$ is a constant vector.
$(\vec \nabla \phi) \times \vec a$ is 0 because $\vec \nabla$ acts on both $\phi$ and $\vec a$ so it should be zero.

Edit:
Now I think $(\vec \nabla \phi) \times \vec a$ is not 0 because $\vec \nabla$ acts only on $\phi$ so we can rewrite it as $- \vec a \times (\vec \nabla \phi).$

 

[haruspex]

Now I think $(\vec \nabla \phi) \times \vec a$ is not 0 because $\vec \nabla$ acts only on $\phi$ so we can rewrite it as $\vec a \times (\vec \nabla \phi).$

Right.

 

[MatinSAR]

Right.

Thanks for the reply @haruspex .
$L.H.S= \iint_S \vec \nabla \times \vec V \cdot d \vec \sigma=\iint_S \vec \nabla \times (\vec a \phi) \cdot d \vec \sigma=\iint_S (\phi \vec \nabla \times \vec a + (\vec \nabla \phi) \times \vec a) \cdot d \vec \sigma=$
$- \iint_S \vec a \times (\vec \nabla \phi) \cdot d \vec \sigma=- \iint_S \vec a \cdot (\vec \nabla \phi) \times d \vec \sigma=\vec a \cdot \iint_S d \vec \sigma \times (\vec \nabla \phi)$
L.H.S = R.H.S
$\vec a \cdot \iint_S d \vec \sigma \times (\vec \nabla \phi) = \vec a \cdot \oint \phi d \vec \lambda$
$\iint_S d \vec \sigma \times (\vec \nabla \phi) = \oint \phi d \vec \lambda$

I hope I won't have problem with other part. ($\vec V=\vec a \times \vec P$)

 

[MatinSAR]

I've managed to prove 2nd part using what I've learnt here : https://www.physicsforums.com/threads/vector-operators-grad-div-and-curl.1057533/

But I'm not sure if my proof is mathematically true or it is nonsense. Picture of my work:

I would be grateful if someone could point out the problem with my proof.