- #1

MatinSAR

- 597

- 182

- 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).##

##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).##

Last edited: