Deriving the Vector Identity: $\nabla(\vec{A} \cdot \vec{B})$

[Xsnac]

Homework Statement

I'm trying to derive the vector identity:
$$\nabla(\vec{A} \cdot \vec{B})$$

Homework Equations

$$ \nabla(\vec{A} \cdot \vec{B})=(\vec{B} \cdot \nabla) \vec{A} + ( \vec{A} \cdot \nabla ) \vec{B} + \vec{B} \times (\nabla \times \vec{A})+ \vec{A} \times ( \nabla \times \vec{B})$$

The Attempt at a Solution

I tried to do it using analitical methods and I think I hit a dead end.
I tried everything, even the reverse start from the $$(\vec{B} \cdot \nabla) \vec{A} + ( \vec{A} \cdot \nabla ) \vec{B} + \vec{B} \times (\nabla \times \vec{A})+ \vec{A} \times ( \nabla \times \vec{B})$$ part but this is the best I could get

At this point I'm even willing to learn a totally new method .. I have an exam tomorrow and this is the only one I can't get right.​

 

[vela]

Homework Statement

I'm trying to derive the vector identity:
$$\nabla(\vec{A} \cdot \vec{B})$$

This isn't an identity. Identities generally have equal signs in them.

Homework Equations

$$ \nabla(\vec{A} \cdot \vec{B})=(\vec{B} \cdot \nabla) \vec{A} + ( \vec{A} \cdot \nabla ) \vec{B} + \vec{B} \times (\nabla \times \vec{A})+ \vec{A} \times ( \nabla \times \vec{B})$$

The Attempt at a Solution

I tried to do it using analytical methods and I think I hit a dead end.
I tried everything, even the reverse start from the $$(\vec{B} \cdot \nabla) \vec{A} + ( \vec{A} \cdot \nabla ) \vec{B} + \vec{B} \times (\nabla \times \vec{A})+ \vec{A} \times ( \nabla \times \vec{B})$$ part but this is the best I could get

At this point I'm even willing to learn a totally new method .. I have an exam tomorrow and this is the only one I can't get right.​

Speaking personally, I'm not really prone to putting in the effort to follow your chicken scratches. Why don't you try organizing your work in digestible chunks and posting it in LaTeX.

 

[Xsnac]

This isn't an identity. Identities generally have equal signs in them.Speaking personally, I'm not really prone to putting in the effort to follow your chicken scratches. Why don't you try organizing your work in digestible chunks and posting it in LaTeX.

I don't have 3 hours to format a text... I'm practicing for tomorrow's exam... I put a lot of effort to write the small pieces of latex code in this post aswel. (forgot all the syntax and got to relearn it today..)
And I'm looking for some other way since mine I think is a dead-end.

 

[vela]

Well, the approach you're taking is the one I would use. Omit the unit vector stuff. It just clutters up the derivation.